This site requires javascript, which is disabled or not supported by your browser.
Most pages on this site will not work without javascript.
However, the simple paper list should be OK.

Publications

Publications (co)authored by S. Shelah

Click the Sh:-number to get to the papers' detail page (which may include pdf's of the paper).
Can't find an Sh:-number (in particular, an "F-number")? You can try your luck here.

Books (9 items)
Published research papers (1138 items)
Accepted research papers (10 items)
Articles in preparation (87 items)
Non-research articles (8 items)
Non-peer-reviewed research articles (5 items)
Corrections (18 items)
Retracted publications (3 items)
Unpublished notes (53 items)
Texts in other authors' publications (4 items)
Parts of books (56 items)
Preliminary versions, reprints (4 items)

Books

The books by S. Shelah

Sh:a
Shelah, S. (1978). Classification theory and the number of nonisomorphic models, Vol. 92, North-Holland Publishing Co., Amsterdam-New York, p. xvi+544. MR: 513226
See [Sh:c]
type: book
abstract: (none)
keywords:
Sh:b
Shelah, S. (1982). Proper forcing, Vol. 940, Springer-Verlag, Berlin-New York, p. xxix+496. MR: 675955
type: book
abstract: (none)
keywords:
Sh:c
Shelah, S. (1990). Classification theory and the number of nonisomorphic models, 2nd edn, Vol. 92, North-Holland Publishing Co., Amsterdam, p. xxxiv+705. MR: 1083551
Revised edition of [Sh:a]
type: book
abstract: (none)
keywords:
Sh:d
Shelah, S. (1986). Around classification theory of models, Vol. 1182, Springer-Verlag, Berlin, p. viii+279. DOI: 10.1007/BFb0098503 MR: 850051
Contains [Sh:171], [Sh:197], [Sh:212], [Sh:228], [Sh:229], [Sh:232], [Sh:233], [Sh:234], [Sh:237a], [Sh:237b], [Sh:237c], [Sh:237d], [Sh:237e], [Sh:247], [Sh:E8]
type: book
abstract: (none)
keywords:
Sh:e
Shelah, S. Non-structure theory, Oxford University Press. To appear.
Contains [Sh:309], [Sh:331], [Sh:363], [Sh:384], [Sh:482], [Sh:511], [Sh:E58], [Sh:E59], [Sh:E60], [Sh:E61], [Sh:E62], [Sh:E63]
type: book
abstract: (none)
keywords:
Sh:f
Shelah, S. (1998). Proper and improper forcing, 2nd edn, Springer-Verlag, Berlin, p. xlviii+1020. DOI: 10.1007/978-3-662-12831-2 MR: 1623206
See [Sh:253], [Sh:263]
type: book
abstract: (none)
keywords:
Sh:g
Shelah, S. (1994). Cardinal arithmetic, Vol. 29, The Clarendon Press, Oxford University Press, New York, p. xxxii+481. MR: 1318912
Contains [Sh:282a], [Sh:333], [Sh:345a], [Sh:345b], [Sh:355], [Sh:365], [Sh:371], [Sh:380], [Sh:386], [Sh:400]. See [Sh:E12]
type: book
abstract: (none)
keywords:
Sh:h
Shelah, S. (2009). Classification theory for abstract elementary classes, Vol. 18, College Publications, London, p. vi+813. MR: 2643267
Contains [Sh:88r], [Sh:300x], [Sh:600], [Sh:705], [Sh:734], [Sh:E53]. See [Sh:E54]
type: book
abstract: (none)
keywords:
Sh:i
Shelah, S. (2009). Classification theory for abstract elementary classes. Vol. 2, Vol. 20, College Publications, London, p. iii+694. MR: 2649290
Contains [Sh:300a], [Sh:300b], [Sh:300c], [Sh:300d], [Sh:300e], [Sh:300f], [Sh:300g], [Sh:300z], [Sh:838], [Sh:E46]
type: book
abstract: (none)
keywords:

Published research papers

Research articles (co)authored by S. Shelah published in peer reviewed journals.

Sh:1
Shelah, S. (1969). Stable theories. Israel J. Math., 7, 187–202. DOI: 10.1007/BF02787611 MR: 0253889
type: article
abstract: We define when a first order complete theory is stable/super stable, characterize the stability spectrum for countable theory, and investigate categoricity spectrum for elementary and pseudo elementary classes
keywords: M: cla, (mod), (sta)
Sh:2
Shelah, S. (1969). Note on a min-max problem of Leo Moser. J. Combinatorial Theory, 6, 298–300. MR: 241312
type: article
abstract: (none)
keywords: O: fin, (fc)
Sh:3
Shelah, S. Finite diagrams stable in power. Ann. Math. Logic, 2(1), 69–118. DOI: 10.1016/0003-4843(70)90007-0 MR: 0285374
type: article
abstract: (none)
keywords: M: cla, M: nec, (mod), (diam)
Sh:4
Shelah, S. (1970). On theories T categorical in |T|. J. Symbolic Logic, 35, 73–82. DOI: 10.2307/2271158 MR: 0282818
type: article
abstract: (none)
keywords: M: cla, (mod), (cat)
Sh:5
Shelah, S. (1970). On languages with non-homogeneous strings of quantifiers. Israel J. Math., 8, 75–79. DOI: 10.1007/BF02771553 MR: 0262064
type: article
abstract: In infinitary logics, we can consider linearly ordered though not well ordered string of quantifiers. The satisfaction relation is defined via Skolem functions. The main theorem said that such strings can be replaced by well ordered ones though the cardinality is somewhat increased. However this is fully proved for the case of anti-well ordered ones. The proofs of the full theorem in included in the author’s M,Sc. thesis or see [Sh:E40]
keywords: M: smt, (mod), (inf-log)
Sh:6
Shelah, S. (1970). A note on Hanf numbers. Pacific J. Math., 34, 541–545. http://projecteuclid.org/euclid.pjm/1102976446 MR: 0268033
type: article
abstract: (none)
keywords: M: odm, (mod)
Sh:7
Shelah, S. (1970). On the cardinality of ultraproduct of finite sets. J. Symbolic Logic, 35, 83–84. DOI: 10.2307/2271159 MR: 0325388
type: article
abstract: (none)
keywords: S: ods, (set), (mod), (up)
Sh:8
Shelah, S. (1971). Two cardinal compactness. Israel J. Math., 9, 193–198. DOI: 10.1007/BF02771584 MR: 0302437
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:9
Shelah, S. Remark to “local definability theory” of Reyes. Ann. Math. Logic, 2(4), 441–447. DOI: 10.1016/0003-4843(71)90004-0 MR: 0282822
type: article
abstract: (none)
keywords: M: odm, (mod)
Sh:10
Shelah, S. (1971). Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory. Ann. Math. Logic, 3(3), 271–362. DOI: 10.1016/0003-4843(71)90015-5 MR: 0317926
type: article
abstract: (none)
keywords: M: cla, (mod)
Sh:11
Shelah, S. (1971). On the number of non-almost isomorphic models of T in a power. Pacific J. Math., 36, 811–818. http://projecteuclid.org/euclid.pjm/1102970932 MR: 0285375
type: article
abstract: (none)
keywords: M: smt, (mod), (inf-log)
Sh:12
Shelah, S. (1971). The number of non-isomorphic models of an unstable first-order theory. Israel J. Math., 9, 473–487. DOI: 10.1007/BF02771463 MR: 0278926
type: article
abstract: (none)
keywords: M: non, (mod), (nni)
Sh:13
Shelah, S. (1971). Every two elementarily equivalent models have isomorphic ultrapowers. Israel J. Math., 10, 224–233. DOI: 10.1007/BF02771574 MR: 0297554
type: article
abstract: (none)
keywords: M: odm, (mod), (up)
Sh:14
Shelah, S. (1972). Saturation of ultrapowers and Keisler’s order. Ann. Math. Logic, 4, 75–114. DOI: 10.1016/0003-4843(72)90012-5 MR: 0294113
type: article
abstract: (none)
keywords: M: cla, (mod), (up)
Sh:15
Shelah, S. (1972). Uniqueness and characterization of prime models over sets for totally transcendental first-order theories. J. Symbolic Logic, 37, 107–113. DOI: 10.2307/2272553 MR: 0316239
type: article
abstract: (none)
keywords: M: cla, (mod)
Sh:16
Shelah, S. (1972). A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific J. Math., 41, 247–261. http://projecteuclid.org/euclid.pjm/1102968432 MR: 0307903
type: article
abstract: (none)
keywords: M: nec, S: ico, (mod)
Sh:17
Shelah, S. (1972). For what filters is every reduced product saturated? Israel J. Math., 12, 23–31. DOI: 10.1007/BF02764810 MR: 0304157
type: article
abstract: Fr a cardinal \lambda we characterize the class of D a filter on a set I such that for every model, with countable vocabulary for transparency, the model the reduced power M^I/D is \lambda-saturated.
keywords: M: odm, (mod)
Sh:18
Shelah, S. (1972). On models with power-like orderings. J. Symbolic Logic, 37, 247–267. DOI: 10.2307/2272971 MR: 0446955
type: article
abstract: (none)
keywords: M: odm, (mod), (inf-log)
Sh:19
Erdős, P., & Shelah, S. (1972). Separability properties of almost-disjoint families of sets. Israel J. Math., 12, 207–214. DOI: 10.1007/BF02764666 MR: 0319770
type: article
abstract: (none)
keywords: S: ico, (cont)
Sh:20
Schmerl, J. H., & Shelah, S. (1972). On power-like models for hyperinaccessible cardinals. J. Symbolic Logic, 37, 531–537. DOI: 10.2307/2272739 MR: 0317925
type: article
abstract: (none)
keywords: S: ico, (inf-log)
Sh:21
Erdős, P., & Shelah, S. (1972). On problems of Moser and Hanson. In Graph theory and applications (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1972; dedicated to the memory of J. W. T. Youngs), Vol. 303, Springer, Berlin, pp. 75–79. MR: 0337646
type: article
abstract: (none)
keywords: O: fin, (fc)
Sh:22
Shelah, S. (1972). A note on model complete models and generic models. Proc. Amer. Math. Soc., 34, 509–514. DOI: 10.2307/2038398 MR: 294114
type: article
abstract: (none)
keywords: M: odm, (mod)
Sh:23
Galvin, F., & Shelah, S. (1973). Some counterexamples in the partition calculus. J. Combinatorial Theory Ser. A, 15, 167–174. DOI: 10.1016/s0097-3165(73)80004-4 MR: 0329900
type: article
abstract: (none)
keywords: S: ico, (pc), (cont)
Sh:24
Shelah, S. (1973). First order theory of permutation groups. Israel J. Math., 14, 149–162; errata, ibid.15 (1973), 437–441. DOI: 10.1007/BF02762670 MR: 0416909
See [Sh:25]
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:26
Shelah, S. (1973). Notes on combinatorial set theory. Israel J. Math., 14, 262–277. DOI: 10.1007/BF02764885 MR: 0327522
type: article
abstract: (none)
keywords: S: ico, (pc), (graph), (univ)
Sh:27
Moran, G., & Shelah, S. (1973). Size direction games over the real line. III. Israel J. Math., 14, 442–449. DOI: 10.1007/BF02764720 MR: 0321551
type: article
abstract: (none)
keywords: S: ico, (cont)
Sh:28
Shelah, S. (1973). There are just four second-order quantifiers. Israel J. Math., 15, 282–300. DOI: 10.1007/BF02787572 MR: 0335237
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:29
Shelah, S. (1974). A substitute for Hall’s theorem for families with infinite sets. J. Combinatorial Theory Ser. A, 16, 199–208. DOI: 10.1016/0097-3165(74)90044-2 MR: 0332497
type: article
abstract: (none)
keywords: S: ico, (fc)
Sh:30
McKenzie, R. N., & Shelah, S. (1974). The cardinals of simple models for universal theories. In Proceedings of the Tarski Symposium, Vol. XXV, Amer. Math. Soc., Providence, R.I., pp. 53–74. MR: 0360261
type: article
abstract: (none)
keywords: O: alg, (mod), (ua)
Sh:31
Shelah, S. (1974). Categoricity of uncountable theories. In Proceedings of the Tarski Symposium, Vol. XXV, Amer. Math. Soc., Providence, R.I., pp. 187–203. MR: 0373874
type: article
abstract: (none)
keywords: M: cla, (mod), (cat)
Sh:32
Erdős, P., Hajnal, A., & Shelah, S. (1974). On some general properties of chromatic numbers. In Topics in topology (Proc. Colloq., Keszthely, 1972), Vol. 8, North-Holland, Amsterdam, pp. 243–255. MR: 0357194
type: article
abstract: (none)
keywords: S: ico
Sh:33
Shelah, S. (1974). The Hanf number of omitting complete types. Pacific J. Math., 50, 163–168. http://projecteuclid.org/euclid.pjm/1102913702 MR: 0363877
type: article
abstract: (none)
keywords: M: odm, (mod)
Sh:34
Shelah, S. (1973). Weak definability in infinitary languages. J. Symbolic Logic, 38, 399–404. DOI: 10.2307/2273033 MR: 0369027
type: article
abstract: (none)
keywords: M: smt, (mod), (nni), (inf-log)
Sh:35
Milner, E. C., & Shelah, S. (1974). Sufficiency conditions for the existence of transversals. Canadian J. Math., 26, 948–961. DOI: 10.4153/CJM-1974-089-8 MR: 373907
type: article
abstract: (none)
keywords: S: ico
Sh:36
Shelah, S. (1977). Remarks on cardinal invariants in topology. General Topology and Appl., 7(3), 251–259. MR: 0482614
type: article
abstract: (none)
keywords: O: top, (inv), (gt)
Sh:37
Shelah, S. (1975). A two-cardinal theorem. Proc. Amer. Math. Soc., 48, 207–213. DOI: 10.2307/2040719 MR: 357105
type: article
abstract: (none)
keywords: M: odm, S: ico, (mod), (fc)
Sh:38
Shelah, S. (1975). Graphs with prescribed asymmetry and minimal number of edges. In Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. III, Vol. 10, North-Holland, Amsterdam, pp. 1241–1256. MR: 0371727
type: article
abstract: (none)
keywords: O: fin, (auto), (fc)
Sh:39
Shelah, S. (1973). Differentially closed fields. Israel J. Math., 16, 314–328. DOI: 10.1007/BF02756711 MR: 0344116
type: article
abstract: (none)
keywords: M: cla, (mod), (sta)
Sh:40
Shelah, S. (1975). Notes on partition calculus. In Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. III, Vol. 10, North-Holland, Amsterdam, pp. 1257–1276. MR: 0406798
type: article
abstract: (none)
keywords: S: ico, (pc)
Sh:41
Milner, E. C., & Shelah, S. (1975). Some theorems on transversals. In Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol III, Vol. 10, North Holland, Amsterdam, pp. 1115–1126. MR: 0376358
type: article
abstract: (none)
keywords: S: ico
Sh:42
Shelah, S. (1975). The monadic theory of order. Ann. Of Math. (2), 102(3), 379–419. arXiv: 2305.00968 DOI: 10.2307/1971037 MR: 0491120
type: article
abstract: (none)
keywords: M: smt, (mod), (mon)
Sh:43
Shelah, S. (1975). Generalized quantifiers and compact logic. Trans. Amer. Math. Soc., 204, 342–364. DOI: 10.2307/1997362 MR: 376334
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:44
Shelah, S. (1974). Infinite abelian groups, Whitehead problem and some constructions. Israel J. Math., 18, 243–256. DOI: 10.1007/BF02757281 MR: 0357114
type: article
abstract: (none)
keywords: M: non, (ab), (wh)
Sh:45
Shelah, S. (1975). Existence of rigid-like families of abelian p-groups. In Model theory and algebra (A memorial tribute to Abraham Robinson), Vol. 498, Springer, Berlin, pp. 384–402. MR: 0412299
type: article
abstract: (none)
keywords: M: non, (ab)
Sh:46
Shelah, S. (1975). Colouring without triangles and partition relation. Israel J. Math., 20, 1–12. DOI: 10.1007/BF02756751 MR: 0427073
type: article
abstract: (none)
keywords: S: ico, (pc)
Sh:47
Makowsky, J. A., Shelah, S., & Stavi, J. (1976). \Delta-logics and generalized quantifiers. Ann. Math. Logic, 10(2), 155–192. DOI: 10.1016/0003-4843(76)90021-8 MR: 0457146
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:48
Shelah, S. (1975). Categoricity in \aleph_1 of sentences in L_{\omega_1,\omega}(Q). Israel J. Math., 20(2), 127–148. DOI: 10.1007/BF02757882 MR: 0379177
type: article
abstract: (none)
keywords: M: cla, M: nec, (mod), (inf-log), (cat)
Sh:49
Shelah, S. (1976). A two-cardinal theorem and a combinatorial theorem. Proc. Amer. Math. Soc., 62(1), 134–136 (1977). DOI: 10.2307/2041962 MR: 434800
type: article
abstract: (none)
keywords: M: odm, S: ico, (mod), (pc)
Sh:50
Shelah, S. (1976). Decomposing uncountable squares to countably many chains. J. Combinatorial Theory Ser. A, 21(1), 110–114. DOI: 10.1016/0097-3165(76)90053-4 MR: 0409196
type: article
abstract: (none)
keywords: S: ico, (cont), (trees), (linear order), (aron)
Sh:51
Shelah, S. (1975). Why there are many nonisomorphic models for unsuperstable theories. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., pp. 259–263. MR: 0422015
type: article
abstract: (none)
keywords: M: non, (mod), (ba)
Sh:52
Shelah, S. (1975). A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals. Israel J. Math., 21(4), 319–349. DOI: 10.1007/BF02757993 MR: 0389579
type: article
abstract: (none)
keywords: S: ico, (ab)
Sh:53
Litman, A., & Shelah, S. (1977). Models with few isomorphic expansions. Israel J. Math., 28(4), 331–338. DOI: 10.1007/BF02760639 MR: 0469741
type: article
abstract: (none)
keywords: M: odm, (mod)
Sh:54
Shelah, S. (1975). The lazy model-theoretician’s guide to stability. Logique et Analyse (N.S.), 18(71-72), 241–308. MR: 0539969
See [Sh:54a]
type: article
abstract: (none)
keywords: M: cla, M: nec, (mod)
Sh:55
Macintyre, A. J., & Shelah, S. (1976). Uncountable universal locally finite groups. J. Algebra, 43(1), 168–175. DOI: 10.1016/0021-8693(76)90150-2 MR: 0439625
type: article
abstract: (none)
keywords: M: non, O: alg, (mod), (nni), (stal)
Sh:56
Shelah, S. (1976). Refuting Ehrenfeucht conjecture on rigid models. Israel J. Math., 25(3-4), 273–286. DOI: 10.1007/BF02757005 MR: 0485326
type: article
abstract: (none)
keywords: M: odm, (mod)
Sh:57
Amit, R., & Shelah, S. (1976). The complete finitely axiomatized theories of order are dense. Israel J. Math., 23(3-4), 200–208. DOI: 10.1007/BF02761800 MR: 0485315
type: article
abstract: (none)
keywords: M: odm, (mod), (linear order)
Sh:58
Shelah, S. (1977). Decidability of a portion of the predicate calculus. Israel J. Math., 28(1-2), 32–44. DOI: 10.1007/BF02759780 MR: 0505410
type: article
abstract: We show decidability of the existence of a model (a finite model) for sentences with the string of quantifiers Vx(3y ...y,), for a language with equality, one one-place function, predicates and constants.

A simpified version appear in a book by Borger, Gradel Gurevich, The cclassical decision problem
keywords: M: odm, O: fin, (mod), (fmt)
Sh:59
Hiller, H. L., & Shelah, S. (1977). Singular cohomology in L. Israel J. Math., 26(3–4), 313–319. DOI: 10.1007/BF03007650 MR: 0444469
type: article
abstract: (none)
keywords: O: alg, (ab), (stal), (wh)
Sh:60
Hodges, W., Lachlan, A. H., & Shelah, S. (1977). Possible orderings of an indiscernible sequence. Bull. London Math. Soc., 9(2), 212–215. DOI: 10.1112/blms/9.2.212 MR: 0476525
type: article
abstract: (none)
keywords: S: ico, O: fin, (mod), (auto), (fc)
Sh:61
Shelah, S. (1976). Interpreting set theory in the endomorphism semi-group of afree algebra or in a category. Ann. Sci. Univ. Clermont, (60 Math. No. 13), 1–29. MR: 0505511
type: article
abstract: (none)
keywords: M: odm, O: alg, (mod), (ua)
Sh:62
Makowsky, J. A., & Shelah, S. (1979). The theorems of Beth and Craig in abstract model theory. I. The abstract setting. Trans. Amer. Math. Soc., 256, 215–239. DOI: 10.2307/1998109 MR: 546916
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:63
Shelah, S., & Stern, J. (1978). The Hanf number of the first order theory of Banach spaces. Trans. Amer. Math. Soc., 244, 147–171. DOI: 10.2307/1997892 MR: 506613
type: article
abstract: (none)
keywords: M: odm, O: top, (mod)
Sh:64
Shelah, S. (1977). Whitehead groups may be not free, even assuming CH. I. Israel J. Math., 28(3), 193–204. DOI: 10.1007/BF02759809 MR: 0469757
type: article
abstract: (none)
keywords: S: for, (set), (ab)
Sh:65
Devlin, K. J., & Shelah, S. (1978). A weak version of \diamondsuit which follows from 2^{\aleph_0}<2^{\aleph_1}. Israel J. Math., 29(2-3), 239–247. DOI: 10.1007/BF02762012 MR: 0469756
type: article
abstract: (none)
keywords: S: ico, (set), (cont), (wd)
Sh:66
Shelah, S. (1978). End extensions and numbers of countable models. J. Symbolic Logic, 43(3), 550–562. DOI: 10.2307/2273531 MR: 503792
type: article
abstract: The answer to the question from page 562 (the end). is negative; have known a solution but not sure if have Not record it.

For any countable model M with countable vocabulary with predicates only. Not including < and E. First we choose a function F from Q the rationals onto M such that the pre-image of any element is dense Second we define a model N Universe. The rationals <. Is interpreted. As the rational order E is interpreted as the equivalence relation xEy iff F(x)=F(y) For any predicate P of the vocabulary of M is interpreted as it’s pre-image by F No Th(N) is a countable fo theory with the same number of countable models up to isomorphism as Th(M) So we are done giving a negative answer to the question
keywords: M: odm, (mod), (nni)
Sh:67
Shelah, S. (1978). On the number of minimal models. J. Symbolic Logic, 43(3), 475–480. DOI: 10.2307/2273522 MR: 0491148
type: article
abstract: (none)
keywords: M: odm, (mod)
Sh:68
Shelah, S. (1978). Jonsson algebras in successor cardinals. Israel J. Math., 30(1-2), 57–64. DOI: 10.1007/BF02760829 MR: 0505434
type: article
abstract: (none)
keywords: S: ico, (set)
Sh:69
Shelah, S. (1980). On a problem of Kurosh, Jónsson groups, and applications. In Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Vol. 95, North-Holland, Amsterdam-New York, pp. 373–394. MR: 579953
type: article
abstract: (none)
keywords: M: non, (mod), (grp)
Sh:70
Gurevich, Y., & Shelah, S. (1979). Modest theory of short chains. II. J. Symbolic Logic, 44(4), 491–502. DOI: 10.2307/2273288 MR: 550378
type: article
abstract: (none)
keywords: M: smt, (mod), (mon)
Sh:71
Shelah, S. (1980). A note on cardinal exponentiation. J. Symbolic Logic, 45(1), 56–66. DOI: 10.2307/2273354 MR: 560225
type: article
abstract: (none)
keywords: S: ico, S: pcf, (set)
Sh:72
Shelah, S. (1978). Models with second-order properties. I. Boolean algebras with no definable automorphisms. Ann. Math. Logic, 14(1), 57–72. DOI: 10.1016/0003-4843(78)90008-6 MR: 501097
type: article
abstract: (none)
keywords: M: non, (mod), (ba)
Sh:73
Shelah, S. (1978). Models with second-order properties. II. Trees with no undefined branches. Ann. Math. Logic, 14(1), 73–87. DOI: 10.1016/0003-4843(78)90009-8 MR: 501098
type: article
abstract: (none)
keywords: M: non, (mod), (trees)
Sh:74
Shelah, S. (1978). Appendix to: “Models with second-order properties. II. Trees with no undefined branches” (Ann. Math. Logic 14 (1978), no. 1, 73–87). Ann. Math. Logic, 14, 223–226. DOI: 10.1016/0003-4843(78)90017-7 MR: 506531
See [Sh:E28]
type: article
abstract: We know, [Sh:8] that two cardinal transfer and compactness theorem rely on so called identities. While the set of identities of pairs like (\beth_\omega,(\kappa) are clear from the proof using indiscernibles. See more in [Sh:E17].

this is was not clear for the gap one case, in particular (\aleph_1, \aleph_ 0).. Those is solved here. That is we characterize the set of identities for this case.
keywords: M: smt, S: ico, (mod), (pc)
Sh:75
Shelah, S. (1978). A Banach space with few operators. Israel J. Math., 30(1-2), 181–191. DOI: 10.1007/BF02760838 MR: 508262
type: article
abstract: (none)
keywords: M: non, (set)
Sh:76
Shelah, S. (1980). Independence of strong partition relation for small cardinals, and the free-subset problem. J. Symbolic Logic, 45(3), 505–509. DOI: 10.2307/2273418 MR: 583369
type: article
abstract: (none)
keywords: S: for, (set), (pc)
Sh:77
Shelah, S. (1977). Existentially-closed groups in \aleph_1 with special properties. Bull. Soc. Math. Grèce (N.S.), 18(1), 17–27. MR: 528419
type: article
abstract: (none)
keywords: O: alg, (mod), (grp)
Sh:78
Shelah, S. (1979). Hanf number of omitting type for simple first-order theories. J. Symbolic Logic, 44(3), 319–324. DOI: 10.2307/2273125 MR: 540663
type: article
abstract: (none)
keywords: M: cla, (mod)
Sh:79
Shelah, S. (1979). On uniqueness of prime models. J. Symbolic Logic, 44(2), 215–220. DOI: 10.2307/2273729 MR: 534571
type: article
abstract: (none)
keywords: M: cla, (mod)
Sh:80
Shelah, S. (1978). A weak generalization of MA to higher cardinals. Israel J. Math., 30(4), 297–306. DOI: 10.1007/BF02761994 MR: 0505492
type: article
abstract: (none)
keywords: S: for, (set)
Sh:81
Abraham, U., Devlin, K. J., & Shelah, S. (1978). The consistency with CH of some consequences of Martin’s axiom plus 2^{\aleph_0}>\aleph_1. Israel J. Math., 31(1), 19–33. DOI: 10.1007/BF02761378 MR: 0505488
type: article
abstract: (none)
keywords: S: for, (set)
Sh:82
Shelah, S. (1981). Models with second order properties. III. Omitting types for L(Q). Arch. Math. Logik Grundlag., 21(1-2), 1–11. DOI: 10.1007/BF02011630 MR: 625527
type: article
abstract: (none)
keywords: M: non, (mod)
Sh:83
Giorgetta, D., & Shelah, S. (1984). Existentially closed structures in the power of the continuum. Ann. Pure Appl. Logic, 26(2), 123–148. DOI: 10.1016/0168-0072(84)90013-7 MR: 739576
type: article
abstract: (none)
keywords: M: non, (mod), (grp)
Sh:84
Rubin, M., & Shelah, S. (1980). On the elementary equivalence of automorphism groups of Boolean algebras; downward Skolem-Löwenheim theorems and compactness of related quantifiers. J. Symbolic Logic, 45(2), 265–283. DOI: 10.2307/2273187 MR: 569397
type: article
abstract: (none)
keywords: M: non, (mod), (ba), (auto)
Sh:85
Devlin, K. J., & Shelah, S. (1979). A note on the normal Moore space conjecture. Canadian J. Math., 31(2), 241–251. DOI: 10.4153/CJM-1979-025-8 MR: 528801
type: article
abstract: (none)
keywords: O: top, (gt), (normal), (wd)
Sh:86
Devlin, K. J., & Shelah, S. (1979). Souslin properties and tree topologies. Proc. London Math. Soc. (3), 39(2), 237–252. DOI: 10.1112/plms/s3-39.2.237 MR: 548979
type: article
abstract: (none)
keywords: O: top, (set), (gt), (trees), (wd), (aron)
Sh:87a
Shelah, S. (1983). Classification theory for nonelementary classes. I. The number of uncountable models of \psi \in L_{\omega_1,\omega }. Part A. Israel J. Math., 46(3), 212–240. DOI: 10.1007/BF02761954 MR: 733351
type: article
abstract: (none)
keywords: M: nec, M: non, (cat), (wd)
Sh:87b
Shelah, S. (1983). Classification theory for nonelementary classes. I. The number of uncountable models of \psi \in L_{\omega_1,\omega }. Part B. Israel J. Math., 46(4), 241–273. DOI: 10.1007/BF02762887 MR: 730343
type: article
abstract: (none)
keywords: M: nec, M: non, (cat), (wd)
Sh:88
Shelah, S. (1987). Classification of nonelementary classes. II. Abstract elementary classes. In Classification theory (Chicago, IL, 1985), Vol. 1292, Springer, Berlin, pp. 419–497. DOI: 10.1007/BFb0082243 MR: 1033034
Contains [Sh:88a]
type: article
abstract: (none)
keywords: M: nec, M: smt, (mod), (cat), (wd), (aec)
Sh:89
Shelah, S. (1979). Boolean algebras with few endomorphisms. Proc. Amer. Math. Soc., 74(1), 135–142. DOI: 10.2307/2042119 MR: 521887
type: article
abstract: (none)
keywords: M: non, S: ico, (ba), (cont)
Sh:90
Shelah, S. (1977). Remarks on \lambda-collectionwise Hausdorff spaces. Topology Proc., 2(2), 583–592 (1978). MR: 540629
type: article
abstract: (none)
keywords: S: for, (gt)
Sh:91
Hiller, H. L., Huber, M. K., & Shelah, S. (1978). The structure of \mathrm{Ext}(A, \mathbf Z) and V=L. Math. Z., 162(1), 39–50. DOI: 10.1007/BF01437821 MR: 0492007
type: article
abstract: (none)
keywords: O: alg, (ab), (stal), (wh)
Sh:92
Shelah, S. (1980). Remarks on Boolean algebras. Algebra Universalis, 11(1), 77–89. DOI: 10.1007/BF02483083 MR: 593014
type: article
abstract: (none)
keywords: S: ico, (ba)
Sh:93
Shelah, S. (1980). Simple unstable theories. Ann. Math. Logic, 19(3), 177–203. DOI: 10.1016/0003-4843(80)90009-1 MR: 595012
type: article
abstract: (none)
keywords: M: cla
Sh:94
Shelah, S. (1979). Weakly compact cardinals: a combinatorial proof. J. Symbolic Logic, 44(4), 559–562. DOI: 10.2307/2273294 MR: 550384
type: article
abstract: (none)
keywords: S: ico
Sh:95
Shelah, S. (1981). Canonization theorems and applications. J. Symbolic Logic, 46(2), 345–353. DOI: 10.2307/2273626 MR: 613287
type: article
abstract: (none)
keywords: S: ico, (pc)
Sh:96
Shelah, S., & Ziegler, M. (1979). Algebraically closed groups of large cardinality. J. Symbolic Logic, 44(4), 522–532. DOI: 10.2307/2273291 MR: 550381
type: article
abstract: (none)
keywords: M: non, (mod), (grp)
Sh:97
Rudin, M. E., & Shelah, S. (1978). Unordered types of ultrafilters. Topology Proc., 3(1), 199–204 (1979). MR: 540490
type: article
abstract: (none)
keywords: S: ico, (gt)
Sh:98
Shelah, S. (1980). Whitehead groups may not be free, even assuming CH. II. Israel J. Math., 35(4), 257–285. DOI: 10.1007/BF02760652 MR: 594332
type: article
abstract: (none)
keywords: S: for, (set), (ab), (wh)
Sh:99
Harrington, L. A., & Shelah, S. (1985). Some exact equiconsistency results in set theory. Notre Dame J. Formal Logic, 26(2), 178–188. DOI: 10.1305/ndjfl/1093870823 MR: 783595
type: article
abstract: (none)
keywords: S: for, S: dst, (trees), (aron)
Sh:100
Shelah, S. (1980). Independence results. J. Symbolic Logic, 45(3), 563–573. DOI: 10.2307/2273423 MR: 583374
type: article
abstract: We introduce proper forcing and oracle cc. The specific independence problems addressed are: prove that the results of [Sh:c, Ch.VIII] are in a sense best poosible, that is the occurance of many cases is necessary because we cannot prove the natural results implying them all. In particular: (A): consistently , CH fail and for some complete f.o. T \subseteq T_1, YT is countable super-stable not {\aleph_0}-stable T_1 of cardinality {\aleph_1} but the pseudo elementary class PC(T,T_1) is categorical in {\aleph_1}

(B) second a related result for T the theory of dense linear order but the conclusion is only having a universal model in {\aleph_1} < 2^ {\aleph_0}
keywords: S: for, (set), (iter), (nni), (univ), (set-mod), (linear order)
Sh:101
Makowsky, J. A., & Shelah, S. (1981). The theorems of Beth and Craig in abstract model theory. II. Compact logics. Arch. Math. Logik Grundlag., 21(1-2), 13–35. DOI: 10.1007/BF02011631 MR: 625528
type: article
abstract: (none)
keywords: M: smt
Sh:102
Abraham, U., & Shelah, S. (1982). Forcing with stable posets. J. Symbolic Logic, 47(1), 37–42. DOI: 10.2307/2273379 MR: 644751
type: article
abstract: (none)
keywords: S: for, (set), (iter), (cont), (trees), (aron)
Sh:103
Fremlin, D. H., & Shelah, S. (1979). On partitions of the real line. Israel J. Math., 32(4), 299–304. DOI: 10.1007/BF02760459 MR: 571084
type: article
abstract: (none)
keywords: S: str, (set), (inv), (meag)
Sh:104
Laver, R. J., & Shelah, S. (1981). The \aleph_2-Souslin hypothesis. Trans. Amer. Math. Soc., 264(2), 411–417. DOI: 10.2307/1998547 MR: 603771
type: article
abstract: (none)
keywords: S: for, (set), (trees)
Sh:105
Shelah, S. (1979). On uncountable abelian groups. Israel J. Math., 32(4), 311–330. DOI: 10.1007/BF02760461 MR: 571086
type: article
abstract: (none)
keywords: O: alg, (ab), (stal), (wh)
Sh:106
Abraham, U., & Shelah, S. (1981). Martin’s axiom does not imply that every two \aleph_1-dense sets of reals are isomorphic. Israel J. Math., 38(1-2), 161–176. DOI: 10.1007/BF02761858 MR: 599485
type: article
abstract: (none)
keywords: S: for, (set), (iter), (linear order)
Sh:107
Shelah, S. (1983). Models with second order properties. IV. A general method and eliminating diamonds. Ann. Pure Appl. Logic, 25(2), 183–212. DOI: 10.1016/0168-0072(83)90013-1 MR: 725733
type: article
abstract: (none)
keywords: M: non, (mod)
Sh:108
Shelah, S. (1979). On successors of singular cardinals. In Logic Colloquium ’78 (Mons, 1978), Vol. 97, North-Holland, Amsterdam-New York, pp. 357–380. MR: 567680
type: article
abstract: See [Sh:88a]
keywords: M: non, (set), (ab), (af)
Sh:109
Hodges, W., & Shelah, S. (1981). Infinite games and reduced products. Ann. Math. Logic, 20(1), 77–108. DOI: 10.1016/0003-4843(81)90012-7 MR: 611395
type: article
abstract: (none)
keywords: M: smt, (mod), (inf-log)
Sh:110
Shelah, S. (1982). Better quasi-orders for uncountable cardinals. Israel J. Math., 42(3), 177–226. DOI: 10.1007/BF02802723 MR: 687127
type: article
abstract: (none)
keywords: S: ico, (pure(large))
Sh:111
Shelah, S. (1986). On power of singular cardinals. Notre Dame J. Formal Logic, 27(2), 263–299. DOI: 10.1305/ndjfl/1093636617 MR: 842153
type: article
abstract: (none)
keywords: S: ico, S: pcf
Sh:112
Shelah, S., & Stanley, L. J. (1982). S-forcing. I. A “black-box” theorem for morasses, with applications to super-Souslin trees. Israel J. Math., 43(3), 185–224. DOI: 10.1007/BF02761942 MR: 689979
type: article
abstract: (none)
keywords: S: ico, (set), (trees)
Sh:113
Shelah, S. (1990). The theorems of Beth and Craig in abstract model theory. III. \Delta-logics and infinitary logics. Israel J. Math., 69(2), 193–213. DOI: 10.1007/BF02937304 MR: 1045373
type: article
abstract: (none)
keywords: M: smt, (mod), (inf-log)
Sh:114
Abraham, U., & Shelah, S. (1985). Isomorphism types of Aronszajn trees. Israel J. Math., 50(1-2), 75–113. DOI: 10.1007/BF02761119 MR: 788070
type: article
abstract: (none)
keywords: S: for, (set), (aron)
Sh:115
Cherlin, G. L., & Shelah, S. (1980). Superstable fields and groups. Ann. Math. Logic, 18(3), 227–270. DOI: 10.1016/0003-4843(80)90006-6 MR: 585519
type: article
abstract: (none)
keywords: M: cla, O: alg, (mod), (grp)
Sh:116
Makowsky, J. A., & Shelah, S. (1983). Positive results in abstract model theory: a theory of compact logics. Ann. Pure Appl. Logic, 25(3), 263–299. DOI: 10.1016/0168-0072(83)90021-0 MR: 730857
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:117
Rubin, M., & Shelah, S. (1987). Combinatorial problems on trees: partitions, \Delta-systems and large free subtrees. Ann. Pure Appl. Logic, 33(1), 43–81. DOI: 10.1016/0168-0072(87)90075-3 MR: 870686
type: article
abstract: (none)
keywords: S: ico, (pc)
Sh:118
Rubin, M., & Shelah, S. (1983). On the expressibility hierarchy of Magidor-Malitz quantifiers. J. Symbolic Logic, 48(3), 542–557. DOI: 10.2307/2273445 MR: 716614
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:119
Shelah, S. (1981). Iterated forcing and changing cofinalities. Israel J. Math., 40(1), 1–32. DOI: 10.1007/BF02761815 MR: 636904
type: article
abstract: (none)
keywords: S: for, (set), (iter), (cont), (ref)
Sh:120
Shelah, S. (1981). Free limits of forcing and more on Aronszajn trees. Israel J. Math., 38(4), 315–334. DOI: 10.1007/BF02762777 MR: 617678
type: article
abstract: (none)
keywords: S: for, (set), (iter), (trees), (aron)
Sh:121
Magidor, M., Shelah, S., & Stavi, J. (1983). On the standard part of nonstandard models of set theory. J. Symbolic Logic, 48(1), 33–38. DOI: 10.2307/2273317 MR: 693245
type: article
abstract: (none)
keywords: S: ods, (set), (ps-dst)
Sh:122
Shelah, S. (1981). On Fleissner’s diamond. Notre Dame J. Formal Logic, 22(1), 29–35. http://projecteuclid.org/euclid.ndjfl/1093883337 MR: 603754
type: article
abstract: (none)
keywords: S: for, (set), (iter)
Sh:123
Gurevich, Y., & Shelah, S. (1982). Monadic theory of order and topology in ZFC. Ann. Math. Logic, 23(2-3), 179–198 (1983). DOI: 10.1016/0003-4843(82)90004-3 MR: 701125
type: article
abstract: (none)
keywords: S: ico, (mod), (mon), (cont)
Sh:124
Shelah, S. (1981). \aleph_\omega may have a strong partition relation. Israel J. Math., 38(4), 283–288. DOI: 10.1007/BF02762774 MR: 617675
type: article
abstract: (none)
keywords: S: for, (set), (pc)
Sh:125
Shelah, S. (1981). The consistency of \mathrm{Ext}(G,\,\mathbf Z)=\mathbf Q. Israel J. Math., 39(1-2), 74–82. DOI: 10.1007/BF02762854 MR: 617291
type: article
abstract: (none)
keywords: S: for, (set), (ab), (iter)
Sh:126
Shelah, S. (1981). On saturation for a predicate. Notre Dame J. Formal Logic, 22(3), 239–248. http://projecteuclid.org/euclid.ndjfl/1093883458 MR: 614121
type: article
abstract: (none)
keywords: M: cla, (mod)
Sh:127
Shelah, S. (1981). On uncountable Boolean algebras with no uncountable pairwise comparable or incomparable sets of elements. Notre Dame J. Formal Logic, 22(4), 301–308. http://projecteuclid.org/euclid.ndjfl/1093883511 MR: 622361
type: article
abstract: (none)
keywords: S: ico, (set), (cont), (inv(ba))
Sh:128
Shelah, S. (1985). Uncountable constructions for B.A., e.c. groups and Banach spaces. Israel J. Math., 51(4), 273–297. DOI: 10.1007/BF02764721 MR: 804487
type: article
abstract: (none)
keywords: M: non, (mod), (ba), (grp), (diam)
Sh:129
Shelah, S. (1981). On the number of nonisomorphic models of cardinality \lambda L_{\infty \lambda }-equivalent to a fixed model. Notre Dame J. Formal Logic, 22(1), 5–10. http://projecteuclid.org/euclid.ndjfl/1093883334 MR: 603751
type: article
abstract: (none)
keywords: M: non, (mod), (nni), (inf-log), (diam)
Sh:130
Pillay, A., & Shelah, S. (1985). Classification theory over a predicate. I. Notre Dame J. Formal Logic, 26(4), 361–376. DOI: 10.1305/ndjfl/1093870929 MR: 799506
type: article
abstract: (none)
keywords: M: cla, (mod)
Sh:131
Shelah, S. (1982). The spectrum problem. I. \aleph_\varepsilon-saturated models, the main gap. Israel J. Math., 43(4), 324–356. DOI: 10.1007/BF02761237 MR: 693353
type: article
abstract: (none)
keywords: M: cla, (mod), (nni), (sta)
Sh:132
Shelah, S. (1982). The spectrum problem. II. Totally transcendental and infinite depth. Israel J. Math., 43(4), 357–364. DOI: 10.1007/BF02761238 MR: 693354
type: article
abstract: (none)
keywords: M: cla, (mod), (nni)
Sh:133
Shelah, S. (1982). On the number of nonisomorphic models in L_{\infty,\kappa } when \kappa is weakly compact. Notre Dame J. Formal Logic, 23(1), 21–26. http://projecteuclid.org/euclid.ndjfl/1093883562 MR: 634740
type: article
abstract: (none)
keywords: M: smt, (nni), (diam)
Sh:134
Gabbay, D. M., Pnueli, A., Shelah, S., & Stavi, J. (1980). On the Temporal Analysis of Fairness. In Proceedings of the 7th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, Association Comp. Machinery, NY, pp. 163–173. DOI: 10.1145/567446.567462
type: article
abstract: (none)
keywords: M: odm, O: fin, (fmt)
Sh:135
Glass, A. M. W., Gurevich, Y., Holland, W. C., & Shelah, S. (1981). Rigid homogeneous chains. Math. Proc. Cambridge Philos. Soc., 89(1), 7–17. DOI: 10.1017/S0305004100057881 MR: 591966
type: article
abstract: (none)
keywords: S: ico, (set), (leb), (linear order)
Sh:136
Shelah, S. (1983). Constructions of many complicated uncountable structures and Boolean algebras. Israel J. Math., 45(2-3), 100–146. DOI: 10.1007/BF02774012 MR: 719115
type: article
abstract: (none)
keywords: M: non, (mod), (ba)
Sh:137
Shelah, S. (1983). The singular cardinals problem: independence results. In Surveys in set theory, Vol. 87, Cambridge Univ. Press, Cambridge, pp. 116–134. DOI: 10.1017/CBO9780511758867.004 MR: 823777
type: article
abstract: (none)
keywords: S: for, (set)
Sh:138
Sageev, G., & Shelah, S. (1985). On the structure of \mathrm{Ext}(A,\mathbf Z) in ZFC^+. J. Symbolic Logic, 50(2), 302–315. DOI: 10.2307/2274216 MR: 793108
type: article
abstract: (none)
keywords: O: alg, (ab), (stal), (wh)
Sh:139
Shelah, S. (1983). On the number of nonconjugate subgroups. Algebra Universalis, 16(2), 131–146. DOI: 10.1007/BF01191760 MR: 692252
type: article
abstract: (none)
keywords: M: non, O: alg, (mod), (nni), (grp)
Sh:140
Shelah, S. (1981). On endo-rigid, strongly \aleph_1-free abelian groups in \aleph_1. Israel J. Math., 40(3-4), 291–295 (1982). DOI: 10.1007/BF02761369 MR: 654584
type: article
abstract: (none)
keywords: O: alg, (ab), (stal)
Sh:141
Gurevich, Y., Magidor, M., & Shelah, S. (1983). The monadic theory of \omega_2. J. Symbolic Logic, 48(2), 387–398. DOI: 10.2307/2273556 MR: 704093
type: article
abstract: (none)
keywords: M: smt, (mod), (mon)
Sh:142
Baldwin, J. T., & Shelah, S. (1983). The structure of saturated free algebras. Algebra Universalis, 17(2), 191–199. DOI: 10.1007/BF01194528 MR: 726272
type: article
abstract: (none)
keywords: M: odm, O: alg, (mod), (ua)
Sh:143
Gurevich, Y., & Shelah, S. (1984). The monadic theory and the “next world”. Israel J. Math., 49(1-3), 55–68. DOI: 10.1007/BF02760646 MR: 788265
type: article
abstract: (none)
keywords: M: odm, S: for, (mod), (mon), (cont), (pure(for))
Sh:144
Magidor, M., Shelah, S., & Stavi, J. (1984). Countably decomposable admissible sets. Ann. Pure Appl. Logic, 26(3), 287–361. DOI: 10.1016/0168-0072(84)90006-X MR: 747687
type: article
abstract: (none)
keywords: S: ods, (set), (inf-log), (ps-dst)
Sh:145
Eklof, P. C., Mekler, A. H., & Shelah, S. (1984). Almost disjoint abelian groups. Israel J. Math., 49(1-3), 34–54. DOI: 10.1007/BF02760645 MR: 788264
type: article
abstract: (none)
keywords: O: alg, (ab), (stal)
Sh:146
Abraham, U., & Shelah, S. (1983). Forcing closed unbounded sets. J. Symbolic Logic, 48(3), 643–657. DOI: 10.2307/2273456 MR: 716625
type: article
abstract: (none)
keywords: S: for, (set), (cont)
Sh:147
Harrington, L. A., & Shelah, S. (1982). The undecidability of the recursively enumerable degrees. Bull. Amer. Math. Soc. (N.S.), 6(1), 79–80. DOI: 10.1090/S0273-0979-1982-14970-9 MR: 634436
type: article
abstract: (none)
keywords: M: odm, O: odo
Sh:148
Sageev, G., & Shelah, S. (1981). Weak compactness and the structure of \mathrm{Ext}(A,\,\mathbf Z). In Abelian group theory (Oberwolfach, 1981), Vol. 874, Springer, Berlin-New York, pp. 87–92. MR: 645920
type: article
abstract: (none)
keywords: O: alg, (ab), (stal), (wh)
Sh:149
Friedman, S.-D., & Shelah, S. (1983). Tall \alpha-recursive structures. Proc. Amer. Math. Soc., 88(4), 672–678. DOI: 10.2307/2045460 MR: 702297
type: article
abstract: (none)
keywords: M: odm, O: odo
Sh:150
Kaufmann, M., & Shelah, S. (1986). The Hanf number of stationary logic. Notre Dame J. Formal Logic, 27(1), 111–123. DOI: 10.1305/ndjfl/1093636530 MR: 819653
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:151
Gurevich, Y., & Shelah, S. (1983). Interpreting second-order logic in the monadic theory of order. J. Symbolic Logic, 48(3), 816–828. DOI: 10.2307/2273475 MR: 716644
type: article
abstract: (none)
keywords: M: smt
Sh:152
Harrington, L. A., & Shelah, S. (1982). Counting equivalence classes for co-\kappa-Souslin equivalence relations. In Logic Colloquium ’80 (Prague, 1980), Vol. 108, North-Holland, Amsterdam-New York, pp. 147–152. MR: 673790
type: article
abstract: (none)
keywords: S: dst
Sh:153
Abraham, U., Rubin, M., & Shelah, S. (1985). On the consistency of some partition theorems for continuous colorings, and the structure of \aleph_1-dense real order types. Ann. Pure Appl. Logic, 29(2), 123–206. DOI: 10.1016/0168-0072(84)90024-1 MR: 801036
type: article
abstract: (none)
keywords: S: for, (set), (iter), (cont), (linear order)
Sh:154
Shelah, S., & Stanley, L. J. (1982). Generalized Martin’s axiom and Souslin’s hypothesis for higher cardinals. Israel J. Math., 43(3), 225–236. DOI: 10.1007/BF02761943 MR: 689980
See [Sh:154a]
type: article
abstract: (none)
keywords: S: for, (set)
Sh:155
Shelah, S. (1986). The spectrum problem. III. Universal theories. Israel J. Math., 55(2), 229–256. DOI: 10.1007/BF02801997 MR: 868182
type: article
abstract: (none)
keywords: M: cla, M: non, (mod), (nni)
Sh:156
Baldwin, J. T., & Shelah, S. (1985). Second-order quantifiers and the complexity of theories. Notre Dame J. Formal Logic, 26(3), 229–303. DOI: 10.1305/ndjfl/1093870870 MR: 796638
type: article
abstract: (none)
keywords: M: smt, (mod), (mon)
Sh:157
Lachlan, A. H., & Shelah, S. (1984). Stable structures homogeneous for a finite binary language. Israel J. Math., 49(1-3), 155–180. DOI: 10.1007/BF02760648 MR: 788267
type: article
abstract: (none)
keywords: M: cla, (fmt)
Sh:158
Harrington, L. A., Makkai, M., & Shelah, S. (1984). A proof of Vaught’s conjecture for \omega-stable theories. Israel J. Math., 49(1-3), 259–280. DOI: 10.1007/BF02760651 MR: 788270
type: article
abstract: (none)
keywords: M: cla, M: non, (mod)
Sh:159
Shelah, S., & Woodin, W. H. (1984). Forcing the failure of CH by adding a real. J. Symbolic Logic, 49(4), 1185–1189. DOI: 10.2307/2274270 MR: 771786
type: article
abstract: (none)
keywords: S: for, (set), (cont)
Sh:160
Hodges, W., & Shelah, S. (1986). Naturality and definability. I. J. London Math. Soc. (2), 33(1), 1–12. DOI: 10.1112/jlms/s2-33.1.1 MR: 829382
type: article
abstract: (none)
keywords: S: ods, (set), (mod)
Sh:161
Shelah, S. (1985). Incompactness in regular cardinals. Notre Dame J. Formal Logic, 26(3), 195–228. DOI: 10.1305/ndjfl/1093870869 MR: 796637
type: article
abstract: (none)
keywords: S: ico, (ab), (af)
Sh:162
Hart, B. T., Laflamme, C., & Shelah, S. (1993). Models with second order properties. V. A general principle. Ann. Pure Appl. Logic, 64(2), 169–194. arXiv: math/9311211 DOI: 10.1016/0168-0072(93)90033-A MR: 1241253
type: article
abstract: We present a general framework for carrying out some constructions. The unifying factor is a combinatorial principle which we present in terms of a game in which the first player challenges the second player to carry out constructions which would be much easier in a generic extension of the universe, and the second player cheats with the aid of \Diamond. Section 1 contains an axiomatic framework suitable for the description of a number of related constructions, and the statement of the main theorem in terms of this framework. In §2 we illustrate the use of our combinatorial principle. The proof of the main result is then carried out in §§3-5.
keywords: S: ico, (mod), (diam)
Sh:163
Gurevich, Y., & Shelah, S. (1985). To the decision problem for branching time logic. In Foundations of logic and linguistics (Salzburg, 1983), Plenum, New York, pp. 181–198. MR: 797952
type: article
abstract: (none)
keywords: M: cla, M: non, (fmt), (trees)
Sh:164
Jarden, M., & Shelah, S. (1983). Pseudo-algebraically closed fields over rational function fields. Proc. Amer. Math. Soc., 87(2), 223–228. DOI: 10.2307/2043693 MR: 681825
type: article
abstract: (none)
keywords: O: alg, (leb)
Sh:165
Shelah, S., & Weiss, B. (1982). Measurable recurrence and quasi-invariant measures. Israel J. Math., 43(2), 154–160. DOI: 10.1007/BF02761726 MR: 689974
type: article
abstract: (none)
keywords: S: dst, O: top
Sh:166
Mekler, A. H., & Shelah, S. (1985). Stationary logic and its friends. I. Notre Dame J. Formal Logic, 26(2), 129–138. DOI: 10.1305/ndjfl/1093870821 MR: 783593
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:167
Shelah, S., & Stanley, L. J. (1986). S-forcing. IIa. Adding diamonds and more applications: coding sets, Arhangelskiı̆’s problem and {\mathcal L}[Q^{<\omega}_1,Q^1_2]. Israel J. Math., 56(1), 1–65. With an appendix by John P. Burgess DOI: 10.1007/BF02776239 MR: 879913
type: article
abstract: (none)
keywords: S: ico, (set), (gt), (diam)
Sh:168
Gurevich, Y., & Shelah, S. (1989). On the strength of the interpretation method. J. Symbolic Logic, 54(2), 305–323. DOI: 10.2307/2274850 MR: 997869
type: article
abstract: (none)
keywords: M: odm, (mod)
Sh:169
Eklof, P. C., Mekler, A. H., & Shelah, S. (1987). On strongly nonreflexive groups. Israel J. Math., 59(3), 283–298. DOI: 10.1007/BF02774142 MR: 920497
type: article
abstract: (none)
keywords: O: alg, (ab), (stal)
Sh:170
Shelah, S. (1984). On logical sentences in PA. In Logic colloquium ’82 (Florence, 1982), Vol. 112, North-Holland, Amsterdam, pp. 145–160. DOI: 10.1016/S0049-237X(08)71815-9 MR: 762109
type: article
abstract: (none)
keywords: M: odm
Sh:172
Shelah, S. (1984). A combinatorial principle and endomorphism rings. I. On p-groups. Israel J. Math., 49(1-3), 239–257. DOI: 10.1007/BF02760650 MR: 788269
type: article
abstract: (none)
keywords: S: ico, O: alg, (mod), (ab), (stal)
Sh:173
Aharoni, R., Nash-Williams, C. S. J. A., & Shelah, S. (1984). Marriage in infinite societies. In Progress in graph theory (Waterloo, Ont., 1982), Academic Press, Toronto, ON, pp. 71–79. MR: 776791
type: article
abstract: (none)
keywords: S: ico, (set)
Sh:174
Grossberg, R. P., & Shelah, S. (1983). On universal locally finite groups. Israel J. Math., 44(4), 289–302. DOI: 10.1007/BF02761988 MR: 710234
type: article
abstract: (none)
keywords: M: smt, (mod), (stal), (univ), (pure(large))
Sh:175
Shelah, S. (1984). On universal graphs without instances of CH. Ann. Pure Appl. Logic, 26(1), 75–87. DOI: 10.1016/0168-0072(84)90042-3 MR: 739914
type: article
abstract: (none)
keywords: S: for, (set), (iter), (graph), (univ)
Sh:175a
Shelah, S. (1990). Universal graphs without instances of CH: revisited. Israel J. Math., 70(1), 69–81. DOI: 10.1007/BF02807219 MR: 1057268
type: article
abstract: (none)
keywords: S: for, (set), (iter), (univ)
Sh:176
Shelah, S. (1984). Can you take Solovay’s inaccessible away? Israel J. Math., 48(1), 1–47. DOI: 10.1007/BF02760522 MR: 768264
type: article
abstract: (none)
keywords: S: dst, (set), (leb), (meag), (AC)
Sh:177
Shelah, S. (1984). More on proper forcing. J. Symbolic Logic, 49(4), 1034–1038. DOI: 10.2307/2274259 MR: 771775
type: article
abstract: (none)
keywords: S: for, (set), (iter)
Sh:178
Gurevich, Y., & Shelah, S. (1983). Random models and the Gödel case of the decision problem. J. Symbolic Logic, 48(4), 1120–1124 (1984). DOI: 10.2307/2273674 MR: 727799
type: article
abstract: (none)
keywords: M: odm, O: fin, (fmt)
Sh:179
Shelah, S., & Steinhorn, C. I. (1986). On the nonaxiomatizability of some logics by finitely many schemas. Notre Dame J. Formal Logic, 27(1), 1–11. DOI: 10.1305/ndjfl/1093636517 MR: 819640
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:180
Shelah, S., & Steinhorn, C. I. (1990). The nonaxiomatizability of L(Q^2_{\aleph_1}) by finitely many schemata. Notre Dame J. Formal Logic, 31(1), 1–13. DOI: 10.1305/ndjfl/1093635328 MR: 1043787
type: article
abstract: (none)
keywords: M: smt
Sh:181
Kaufmann, M., & Shelah, S. (1984). A nonconservativity result on global choice. Ann. Pure Appl. Logic, 27(3), 209–214. DOI: 10.1016/0168-0072(84)90026-5 MR: 765590
type: article
abstract: (none)
keywords: S: ods, (set), (AC)
Sh:182
Abraham, U., & Shelah, S. (1986). On the intersection of closed unbounded sets. J. Symbolic Logic, 51(1), 180–189. DOI: 10.2307/2273954 MR: 830084
type: article
abstract: (none)
keywords: S: for
Sh:183
Gurevich, Y., & Shelah, S. (1983). Rabin’s uniformization problem. J. Symbolic Logic, 48(4), 1105–1119 (1984). DOI: 10.2307/2273673 MR: 727798
type: article
abstract: (none)
keywords: M: odm, (mod), (mon)
Sh:184
Goldfarb, W. D., Gurevich, Y., & Shelah, S. (1984). A decidable subclass of the minimal Gödel class with identity. J. Symbolic Logic, 49(4), 1253–1261. DOI: 10.2307/2274275 MR: 771791
type: article
abstract: (none)
keywords: M: odm, O: fin, (mod)
Sh:185
Shelah, S. (1983). Lifting problem of the measure algebra. Israel J. Math., 45(1), 90–96. DOI: 10.1007/BF02760673 MR: 710248
type: article
abstract: (none)
keywords: S: for, (set), (leb)
Sh:186
Shelah, S. (1984). Diamonds, uniformization. J. Symbolic Logic, 49(4), 1022–1033. DOI: 10.2307/2274258 MR: 771774
type: article
abstract: (none)
keywords: S: ico, S: for, (set), (unif), (diam)
Sh:187
Mekler, A. H., & Shelah, S. (1986). Stationary logic and its friends. II. Notre Dame J. Formal Logic, 27(1), 39–50. DOI: 10.1305/ndjfl/1093636521 MR: 819644
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:188
Shelah, S. (1984). A pair of nonisomorphic \equiv_{\infty \lambda } models of power \lambda for \lambda singular with \lambda ^\omega =\lambda. Notre Dame J. Formal Logic, 25(2), 97–104. DOI: 10.1305/ndjfl/1093870570 MR: 733596
type: article
abstract: (none)
keywords: M: smt, (mod), (nni), (inf-log)
Sh:189
Shelah, S. (1985). On the possible number \mathrm{no}(M) = the number of nonisomorphic models L_{\infty,\lambda}-equivalent to M of power \lambda, for \lambda singular. Notre Dame J. Formal Logic, 26(1), 36–50. DOI: 10.1305/ndjfl/1093870759 MR: 766665
type: article
abstract: (none)
keywords: M: non, (mod), (nni), (inf-log)
Sh:190
Göbel, R., & Shelah, S. (1985). Semirigid classes of cotorsion-free abelian groups. J. Algebra, 93(1), 136–150. DOI: 10.1016/0021-8693(85)90178-4 MR: 780487
type: article
abstract: (none)
keywords: M: non, (ab)
Sh:191
Gitik, M., & Shelah, S. (1984). On the \mathbb I-condition. Israel J. Math., 48(2-3), 148–158. DOI: 10.1007/BF02761160 MR: 770697
type: article
abstract: (none)
keywords: S: for, (set), (iter), (cont)
Sh:192
Shelah, S. (1987). Uncountable groups have many nonconjugate subgroups. Ann. Pure Appl. Logic, 36(2), 153–206. DOI: 10.1016/0168-0072(87)90016-9 MR: 911580
type: article
abstract: (none)
keywords: S: ico, (nni), (grp), (wd)
Sh:193
Lehmann, D. J., & Shelah, S. (1982). Reasoning with time and chance. Inform. And Control, 53(3), 165–198. DOI: 10.1016/S0019-9958(82)91022-1 MR: 715529
type: article
abstract: (none)
keywords: M: smt, O: fin, (fmt)
Sh:194
Aharoni, R., Nash-Williams, C. S. J. A., & Shelah, S. (1983). A general criterion for the existence of transversals. Proc. London Math. Soc. (3), 47(1), 43–68. DOI: 10.1112/plms/s3-47.1.43 MR: 698927
type: article
abstract: (none)
keywords: S: ico, (set)
Sh:195
Droste, M., & Shelah, S. (1985). A construction of all normal subgroup lattices of 2-transitive automorphism groups of linearly ordered sets. Israel J. Math., 51(3), 223–261. DOI: 10.1007/BF02772666 MR: 804485
type: article
abstract: (none)
keywords: O: alg, (set), (stal), (auto), (grp), (linear order)
Sh:196
Aharoni, R., Nash-Williams, C. S. J. A., & Shelah, S. (1984). Another form of a criterion for the existence of transversals. J. London Math. Soc. (2), 29(2), 193–203. DOI: 10.1112/jlms/s2-29.2.193 MR: 744087
type: article
abstract: (none)
keywords: S: ico, (set)
Sh:198
Levinski, J.-P., Magidor, M., & Shelah, S. (1990). Chang’s conjecture for \aleph_\omega. Israel J. Math., 69(2), 161–172. DOI: 10.1007/BF02937302 MR: 1045371
type: article
abstract: (none)
keywords: S: for, (set)
Sh:199
Shelah, S. (1985). Remarks in abstract model theory. Ann. Pure Appl. Logic, 29(3), 255–288. DOI: 10.1016/0168-0072(85)90002-8 MR: 808815
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:200
Shelah, S. (1985). Classification of first order theories which have a structure theorem. Bull. Amer. Math. Soc. (N.S.), 12(2), 227–232. DOI: 10.1090/S0273-0979-1985-15354-6 MR: 776474
type: article
abstract: (none)
keywords: M: cla, (mod)
Sh:201
Kaufmann, M., & Shelah, S. (1985). On random models of finite power and monadic logic. Discrete Math., 54(3), 285–293. DOI: 10.1016/0012-365X(85)90112-8 MR: 790589
type: article
abstract: (none)
keywords: M: odm, O: fin, (fmt), (mon)
Sh:202
Shelah, S. (1984). On co-\kappa-Souslin relations. Israel J. Math., 47(2-3), 139–153. DOI: 10.1007/BF02760513 MR: 738165
type: article
abstract: (none)
keywords: S: dst, (linear order)
Sh:203
Ben-David, S., & Shelah, S. (1986). Souslin trees and successors of singular cardinals. Ann. Pure Appl. Logic, 30(3), 207–217. DOI: 10.1016/0168-0072(86)90020-5 MR: 836425
type: article
abstract: (none)
keywords: S: for, (trees), (aron)
Sh:204
Magidor, M., & Shelah, S. (1994). When does almost free imply free? (For groups, transversals, etc.). J. Amer. Math. Soc., 7(4), 769–830. DOI: 10.2307/2152733 MR: 1249391
type: article
abstract: (none)
keywords: S: ico, S: for, (set), (ab), (stal), (af)
Sh:205
Shelah, S. (1985). Monadic logic and Löwenheim numbers. Ann. Pure Appl. Logic, 28(2), 203–216. DOI: 10.1016/0168-0072(85)90026-0 MR: 779162
type: article
abstract: (none)
keywords: M: smt, (mod), (inf-log), (mon)
Sh:206
Shelah, S. (1988). Decomposing topological spaces into two rigid homeomorphic subspaces. Israel J. Math., 63(2), 183–211. DOI: 10.1007/BF02765038 MR: 968538
type: article
abstract: (none)
keywords: S: ico, (gt)
Sh:207
Shelah, S. (1984). On cardinal invariants of the continuum. In Axiomatic set theory (Boulder, Colo., 1983), Vol. 31, Amer. Math. Soc., Providence, RI, pp. 183–207. DOI: 10.1090/conm/031/763901 MR: 763901
type: article
abstract: (none)
keywords: S: for, S: str, (set), (iter), (inv)
Sh:208
Shelah, S. (1985). More on the weak diamond. Ann. Pure Appl. Logic, 28(3), 315–318. DOI: 10.1016/0168-0072(85)90019-3 MR: 790390
type: article
abstract: (none)
keywords: S: ico, S: for, (set), (wd)
Sh:209
Shelah, S., & Todorčević, S. (1986). A note on small Baire spaces. Canad. J. Math., 38(3), 659–665. DOI: 10.4153/CJM-1986-033-8 MR: 845670
type: article
abstract: (none)
keywords: S: for, (set), (gt)
Sh:210
Bonnet, R., & Shelah, S. (1985). Narrow Boolean algebras. Ann. Pure Appl. Logic, 28(1), 1–12. DOI: 10.1016/0168-0072(85)90028-4 MR: 776283
type: article
abstract: (none)
keywords: S: ico, (ba)
Sh:211
Shelah, S. (1992). The Hanf numbers of stationary logic. II. Comparison with other logics. Notre Dame J. Formal Logic, 33(1), 1–12. arXiv: math/9201243 DOI: 10.1305/ndjfl/1093636007 MR: 1149955
type: article
abstract: We show that the ordering of the Hanf number of L_{\omega,\omega}(wo) (well ordering), L^c_{\omega,\omega} (quantification on countable sets), L_{\omega, \omega}(aa) (stationary logic) and second order logic, have no more restraints provable in ZFC than previously known (those independence proofs assume CON(ZFC) only). We also get results on corresponding logics for L_{\lambda,\mu}.
keywords: M: smt, (mod), (inf-log)
Sh:213
Denenberg, L., Gurevich, Y., & Shelah, S. (1986). Definability by constant-depth polynomial-size circuits. Inform. And Control, 70(2-3), 216–240. DOI: 10.1016/S0019-9958(86)80006-7 MR: 859107
type: article
abstract: We investigate the expressive power of constant-depth polynomial-size circuit models. In particular, we construct a circuit model whose expressive power is precisely that of first-order logic.
keywords: O: fin, (fmt)
Sh:214
Mekler, A. H., & Shelah, S. (1986). \omega-elongations and Crawley’s problem. Pacific J. Math., 121(1), 121–132. http://projecteuclid.org/euclid.pjm/1102702803 MR: 815039
type: article
abstract: (none)
keywords: O: alg, (ab), (stal), (wh)
Sh:214a
Mekler, A. H., & Shelah, S. (1986). The solution to Crawley’s problem. Pacific J. Math., 121(1), 133–134. http://projecteuclid.org/euclid.pjm/1102702804 MR: 815040
type: article
abstract: (none)
keywords: O: alg, (ab), (stal), (wh)
Sh:215
Harrington, L. A., Marker, D. E., & Shelah, S. (1988). Borel orderings. Trans. Amer. Math. Soc., 310(1), 293–302. DOI: 10.2307/2001122 MR: 965754
type: article
abstract: (none)
keywords: S: dst, (linear order)
Sh:216
Holland, W. C., Mekler, A. H., & Shelah, S. (1985). Lawless order. Order, 1(4), 383–397. DOI: 10.1007/BF00582744 MR: 787550
type: article
abstract: (none)
keywords: M: non, S: ico, (grp), (linear order)
Sh:216a
Holland, W. C., Mekler, A. H., & Shelah, S. (1986). Total orders whose carried groups satisfy no laws. In Algebra and order (Luminy-Marseille, 1984), Vol. 14, Heldermann, Berlin, pp. 29–33. MR: 891446
type: article
abstract: (none)
keywords: M: non, S: ico, (grp), (linear order)
Sh:218
Shelah, S. (1985). On measure and category. Israel J. Math., 52(1-2), 110–114. DOI: 10.1007/BF02776084 MR: 815606
type: article
abstract: (none)
keywords: S: for, S: dst, (leb), (meag)
Sh:219
Göbel, R., & Shelah, S. (1985). Modules over arbitrary domains. Math. Z., 188(3), 325–337. DOI: 10.1007/BF01159179 MR: 771988
type: article
abstract: (none)
keywords: O: alg, (ab), (stal)
Sh:220
Shelah, S. (1987). Existence of many L_{\infty,\lambda}-equivalent, nonisomorphic models of T of power \lambda. Ann. Pure Appl. Logic, 34(3), 291–310. DOI: 10.1016/0168-0072(87)90005-4 MR: 899084
type: article
abstract: (none)
keywords: M: non, (mod), (nni), (inf-log)
Sh:221
Abraham, U., Shelah, S., & Solovay, R. M. (1987). Squares with diamonds and Souslin trees with special squares. Fund. Math., 127(2), 133–162. DOI: 10.4064/fm-127-2-133-162 MR: 882623
type: article
abstract: (none)
keywords: S: ico, S: for, (set)
Sh:222
Grossberg, R. P., & Shelah, S. (1986). On the number of nonisomorphic models of an infinitary theory which has the infinitary order property. I. J. Symbolic Logic, 51(2), 302–322. DOI: 10.2307/2274053 MR: 840407
type: article
abstract: (none)
keywords: M: non, (mod), (nni), (inf-log)
Sh:223
Droste, M., & Shelah, S. (1987). On the universality of systems of words in permutation groups. Pacific J. Math., 127(2), 321–328. http://projecteuclid.org/euclid.pjm/1102699565 MR: 881762
type: article
abstract: (none)
keywords: O: alg, (ua)
Sh:224
Göbel, R., & Shelah, S. (1986). Modules over arbitrary domains. II. Fund. Math., 126(3), 217–243. DOI: 10.4064/fm-126-3-217-243 MR: 882431
type: article
abstract: (none)
keywords: O: alg, (ab), (stal)
Sh:225
Shelah, S. (1987). On the number of strongly \aleph_\epsilon-saturated models of power \lambda. Ann. Pure Appl. Logic, 36(3), 279–287. DOI: 10.1016/0168-0072(87)90020-0 MR: 915901
See [Sh:225a]
type: article
abstract: (none)
keywords: M: cla, (mod), (sta)
Sh:225a
Shelah, S. (1988). Number of strongly \aleph_\epsilon saturated models—an addition. Ann. Pure Appl. Logic, 40(1), 89–91. DOI: 10.1016/0168-0072(88)90041-3 MR: 965589
improvement of [Sh:225]
type: article
abstract: (none)
keywords: M: cla, (mod), (sta)
Sh:226
Foreman, M. D., Magidor, M., & Shelah, S. (1986). 0^\sharp and some forcing principles. J. Symbolic Logic, 51(1), 39–46. DOI: 10.2307/2273940 MR: 830070
type: article
abstract: (none)
keywords: S: for, (set)
Sh:227
Shelah, S. (1984). A combinatorial theorem and endomorphism rings of abelian groups. II. In Abelian groups and modules (Udine, 1984), Vol. 287, Springer, Vienna, pp. 37–86. DOI: 10.1007/978-3-7091-2814-5_3 MR: 789808
type: article
abstract: (none)
keywords: M: non, (ab)
Sh:230
Gurevich, Y., & Shelah, S. (1985). The decision problem for branching time logic. J. Symbolic Logic, 50(3), 668–681. DOI: 10.2307/2274321 MR: 805676
type: article
abstract: (none)
keywords: M: smt, M: odm, O: fin
Sh:231
Juhász, I., & Shelah, S. (1986). How large can a hereditarily separable or hereditarily Lindelöf space be? Israel J. Math., 53(3), 355–364. DOI: 10.1007/BF02786567 MR: 852486
type: article
abstract: (none)
keywords: S: for, (gt)
Sh:235
Shelah, S., & Soifer, A. (1986). Two problems on \aleph_0-indecomposable abelian groups. J. Algebra, 99(2), 359–369. DOI: 10.1016/0021-8693(86)90033-5 MR: 837550
type: article
abstract: (none)
keywords: O: alg, (ab)
Sh:236
Ben-David, S., & Shelah, S. (1986). Nonspecial Aronszajn trees on \aleph_{\omega+1}. Israel J. Math., 53(1), 93–96. DOI: 10.1007/BF02772672 MR: 861900
type: article
abstract: (none)
keywords: S: for, (set), (aron)
Sh:238
Grossberg, R. P., & Shelah, S. (1986). A nonstructure theorem for an infinitary theory which has the unsuperstability property. Illinois J. Math., 30(2), 364–390. http://projecteuclid.org/euclid.ijm/1256044645 MR: 840135
type: article
abstract: (none)
keywords: M: cla, M: nec, M: non, (mod)
Sh:239
Shelah, S., & Soifer, A. (1986). Countable \aleph_0-indecomposable mixed abelian groups of finite torsion-free rank. J. Algebra, 100(2), 421–429. DOI: 10.1016/0021-8693(86)90085-2 MR: 840585
type: article
abstract: (none)
keywords: O: alg, (ab)
Sh:240
Foreman, M. D., Magidor, M., & Shelah, S. (1988). Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Ann. Of Math. (2), 127(1), 1–47. DOI: 10.2307/1971415 MR: 924672
See [Sh:240a]
type: article
abstract: (none)
keywords: S: for, (set), (iter)
Sh:241
Shelah, S., & Woodin, W. H. (1990). Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable. Israel J. Math., 70(3), 381–394. DOI: 10.1007/BF02801471 MR: 1074499
type: article
abstract: (none)
keywords: S: dst, (set), (leb)
Sh:242
Blass, A. R., & Shelah, S. (1987). There may be simple P_{\aleph_1}- and P_{\aleph_2}-points and the Rudin-Keisler ordering may be downward directed. Ann. Pure Appl. Logic, 33(3), 213–243. DOI: 10.1016/0168-0072(87)90082-0 MR: 879489
type: article
abstract: (none)
keywords: S: for, S: str, (set), (inv), (large)
Sh:243
Gurevich, Y., & Shelah, S. (1987). Expected computation time for Hamiltonian path problem. SIAM J. Comput., 16(3), 486–502. DOI: 10.1137/0216034 MR: 889404
type: article
abstract: (none)
keywords: O: fin, (fc)
Sh:244a
Gurevich, Y., & Shelah, S. (1986). Fixed-point extensions of first-order logic. Ann. Pure Appl. Logic, 32(3), 265–280. DOI: 10.1016/0168-0072(86)90055-2 MR: 865992
See [Sh:244]
type: article
abstract: (none)
keywords: M: odm, O: fin
Sh:245
Compton, K. J., Henson, C. W., & Shelah, S. (1987). Nonconvergence, undecidability, and intractability in asymptotic problems. Ann. Pure Appl. Logic, 36(3), 207–224. DOI: 10.1016/0168-0072(87)90017-0 MR: 915898
type: article
abstract: (none)
keywords: M: odm, O: fin, (fmt)
Sh:246
Shelah, S. (1991). On a problem in cylindric algebra. In Algebraic logic (Budapest, 1988), Vol. 54, North-Holland, Amsterdam, pp. 645–664. MR: 1153444
type: article
abstract: (none)
keywords: M: non, (mod)
Sh:249
Hajnal, A., Juhász, I., & Shelah, S. (1986). Splitting strongly almost disjoint families. Trans. Amer. Math. Soc., 295(1), 369–387. DOI: 10.2307/2000161 MR: 831204
type: article
abstract: (none)
keywords: S: ico, (set)
Sh:250
Shelah, S. (1988). Some notes on iterated forcing with 2^{\aleph_0}>\aleph_2. Notre Dame J. Formal Logic, 29(1), 1–17. DOI: 10.1305/ndjfl/1093637766 MR: 932690
type: article
abstract: (none)
keywords: S: for, (set), (iter)
Sh:251
Mekler, A. H., & Shelah, S. (1987). When \kappa-free implies strongly \kappa-free. In Abelian group theory (Oberwolfach, 1985), Gordon; Breach, New York, pp. 137–148. MR: 1011309
type: article
abstract: (none)
keywords: O: alg, (ab), (stal), (af)
Sh:252
Foreman, M. D., Magidor, M., & Shelah, S. (1988). Martin’s maximum, saturated ideals and nonregular ultrafilters. II. Ann. Of Math. (2), 127(3), 521–545. DOI: 10.2307/2007004 MR: 942519
type: article
abstract: (none)
keywords: S: ico, S: for, (set), (normal)
Sh:253
Shelah, S. (1987). Iterated forcing and normal ideals on \omega_1. Israel J. Math., 60(3), 345–380. DOI: 10.1007/BF02780398 MR: 937796
initial version of Ch. XIII of [Sh:f]
type: article
abstract: (none)
keywords: S: for, (set), (iter), (normal)
Sh:254
Baumgartner, J. E., & Shelah, S. (1987). Remarks on superatomic Boolean algebras. Ann. Pure Appl. Logic, 33(2), 109–129. DOI: 10.1016/0168-0072(87)90077-7 MR: 874021
type: article
abstract: (none)
keywords: S: ico, S: for, (set), (ba)
Sh:255
Eklof, P. C., & Shelah, S. (1987). On groups A such that A\oplus \mathbf Z^n\cong A. In Abelian group theory (Oberwolfach, 1985), Gordon; Breach, New York, pp. 149–163. MR: 1011310
type: article
abstract: (none)
keywords: O: alg, (ab), (stal)
Sh:256
Shelah, S. (1987). More on powers of singular cardinals. Israel J. Math., 59(3), 299–326. DOI: 10.1007/BF02774143 MR: 920498
type: article
abstract: (none)
keywords: S: pcf, (set), (normal)
Sh:257
Blass, A. R., & Shelah, S. (1989). Ultrafilters with small generating sets. Israel J. Math., 65(3), 259–271. DOI: 10.1007/BF02764864 MR: 1005010
type: article
abstract: (none)
keywords: S: for, (set), (inv)
Sh:258
Shelah, S., & Stanley, L. J. (1987). A theorem and some consistency results in partition calculus. Ann. Pure Appl. Logic, 36(2), 119–152. DOI: 10.1016/0168-0072(87)90015-7 MR: 911579
type: article
abstract: (none)
keywords: S: ico, S: for, (set), (pc)
Sh:260
Shelah, S., & Steprāns, J. (1987). Extraspecial p-groups. Ann. Pure Appl. Logic, 34(1), 87–97. DOI: 10.1016/0168-0072(87)90041-8 MR: 887554
type: article
abstract: (none)
keywords: S: ico, O: alg, (stal)
Sh:261
Shelah, S. (1988). A graph which embeds all small graphs on any large set of vertices. Ann. Pure Appl. Logic, 38(2), 171–183. DOI: 10.1016/0168-0072(88)90052-8 MR: 938374
type: article
abstract: (none)
keywords: S: ico, (set), (graph)
Sh:262
Shelah, S. (1989). The number of pairwise non-elementarily-embeddable models. J. Symbolic Logic, 54(4), 1431–1455. DOI: 10.2307/2274824 MR: 1026608
type: article
abstract: (none)
keywords: S: ico, S: for, (mod)
Sh:263
Shelah, S. (1987). Semiproper forcing axiom implies Martin maximum but not PFA^+. J. Symbolic Logic, 52(2), 360–367. DOI: 10.2307/2274385 MR: 890443
represented in Ch. XVII of [Sh:f]
type: article
abstract: (none)
keywords: S: ico, (set)
Sh:264
Shelah, S., & Steprāns, J. (1988). A Banach space on which there are few operators. Proc. Amer. Math. Soc., 104(1), 101–105. DOI: 10.2307/2047469 MR: 958051
type: article
abstract: (none)
keywords: M: non
Sh:265
Dugas, M. H., Fay, T. H., & Shelah, S. (1987). Singly cogenerated annihilator classes. J. Algebra, 109(1), 127–137. DOI: 10.1016/0021-8693(87)90168-2 MR: 898341
type: article
abstract: (none)
keywords: M: non, (ab)
Sh:266
Shelah, S. (2019). Compactness in singular cardinals revisited. Sarajevo J. Math., 15(28)(2), 201–208. arXiv: 1401.3175 DOI: 10.5644/sjm MR: 4069744
type: article
abstract: (none)
keywords: S: ico, O: alg
Sh:267
Fleissner, W. G., & Shelah, S. (1989). Collectionwise Hausdorff: incompactness at singulars. Topology Appl., 31(2), 101–107. DOI: 10.1016/0166-8641(89)90074-6 MR: 994403
type: article
abstract: (none)
keywords: S: ico, (gt)
Sh:268
Hajnal, A., Kanamori, A., & Shelah, S. (1987). Regressive partition relations for infinite cardinals. Trans. Amer. Math. Soc., 299(1), 145–154. DOI: 10.2307/2000486 MR: 869404
type: article
abstract: (none)
keywords: S: ico, (set), (pc)
Sh:269
Shelah, S. (1989). “Gap 1” two-cardinal principles and the omitting types theorem for \mathcal L (Q). Israel J. Math., 65(2), 133–152. DOI: 10.1007/BF02764857 MR: 998667
type: article
abstract: (none)
keywords: M: smt, M: odm
Sh:270
Shelah, S. (1989). Baire irresolvable spaces and lifting for a layered ideal. Topology Appl., 33(3), 217–221. DOI: 10.1016/0166-8641(89)90102-8 MR: 1026923
type: article
abstract: (none)
keywords: S: ico, (set), (gt)
Sh:271
Hodges, W., & Shelah, S. (1991). There are reasonably nice logics. J. Symbolic Logic, 56(1), 300–322. DOI: 10.2307/2274921 MR: 1131747
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:272
Shelah, S. (1987). On almost categorical theories. In Classification theory (Chicago, IL, 1985), Vol. 1292, Springer, Berlin, pp. 498–500. DOI: 10.1007/BFb0082244 MR: 1033035
type: article
abstract: (none)
keywords: M: cla, (mod), (sta), (cat)
Sh:273
Shelah, S. (1988). Can the fundamental (homotopy) group of a space be the rationals? Proc. Amer. Math. Soc., 103(2), 627–632. DOI: 10.2307/2047190 MR: 943095
type: article
abstract: (none)
keywords: S: dst, O: top
Sh:274
Mekler, A. H., & Shelah, S. (1989). Uniformization principles. J. Symbolic Logic, 54(2), 441–459. DOI: 10.2307/2274859 MR: 997878
type: article
abstract: (none)
keywords: S: ico, (set), (ab), (unif)
Sh:275
Mekler, A. H., & Shelah, S. (1989). L_{\infty\omega}-free algebras. Algebra Universalis, 26(3), 351–366. DOI: 10.1007/BF01211842 MR: 1044855
type: article
abstract: (none)
keywords: M: nec, (ua)
Sh:276
Shelah, S. (1988). Was Sierpiński right? I. Israel J. Math., 62(3), 355–380. DOI: 10.1007/BF02783304 MR: 955139
type: article
abstract: (none)
keywords: S: ico, S: for, (set), (iter), (pc)
Sh:277
Gurevich, Y., & Shelah, S. (1989). Nearly linear time. In Logic at Botik ’89 (Pereslavl-Zalesskiy, 1989), Vol. 363, Springer, Berlin, pp. 108–118. DOI: 10.1007/3-540-51237-3_10 MR: 1030571
type: article
abstract: (none)
keywords: M: odm, O: fin
Sh:278
Chatzidakis, Z. M., Cherlin, G. L., Shelah, S., Srour, G., & Wood, C. (1987). Orthogonality of types in separably closed fields. In Classification theory (Chicago, IL, 1985), Vol. 1292, Springer, Berlin, pp. 72–88. DOI: 10.1007/BFb0082232 MR: 1033023
type: article
abstract: (none)
keywords: M: cla, (mod), (sta)
Sh:279
Shelah, S., & Stanley, L. J. (1988). Weakly compact cardinals and nonspecial Aronszajn trees. Proc. Amer. Math. Soc., 104(3), 887–897. DOI: 10.2307/2046812 MR: 964870
type: article
abstract: (none)
keywords: S: ico, (set), (large), (trees), (aron)
Sh:280
Shelah, S. (1990). Strong negative partition above the continuum. J. Symbolic Logic, 55(1), 21–31. DOI: 10.2307/2274951 MR: 1043541
type: article
abstract: (none)
keywords: S: ico, (set)
Sh:281
Drezner, Z., & Shelah, S. (1987). On the complexity of the Elzinga-Hearn algorithm for the 1-center problem. Math. Oper. Res., 12(2), 255–261. DOI: 10.1287/moor.12.2.255 MR: 888974
type: article
abstract: (none)
keywords: O: fin, (fc)
Sh:282
Shelah, S. (1988). Successors of singulars, cofinalities of reduced products of cardinals and productivity of chain conditions. Israel J. Math., 62(2), 213–256. DOI: 10.1007/BF02787123 MR: 947823
type: article
abstract: (none)
keywords: S: pcf, (set)
Sh:283
Shelah, S. (1987). On reconstructing separable reduced p-groups with a given socle. Israel J. Math., 60(2), 146–166. DOI: 10.1007/BF02790788 MR: 931873
type: article
abstract: (none)
keywords: S: ico, O: alg, (ab)
Sh:284a
Shelah, S. (1988). Notes on monadic logic. Part A. Monadic theory of the real line. Israel J. Math., 63(3), 335–352. DOI: 10.1007/BF02778038 MR: 969946
type: article
abstract: (none)
keywords: M: smt
Sh:284b
Shelah, S. (1990). Notes on monadic logic. Part B: Complexity of linear orders in ZFC. Israel J. Math., 69(1), 94–116. DOI: 10.1007/BF02764732 MR: 1046176
type: article
abstract: (none)
keywords: M: smt
Sh:284c
Shelah, S. (1990). More on monadic logic. Part C. Monadically interpreting in stable unsuperstable \mathcal T and the monadic theory of {}^\omega\lambda. Israel J. Math., 70(3), 353–364. DOI: 10.1007/BF02801469 MR: 1074497
type: article
abstract: (none)
keywords: M: smt
Sh:284d
Shelah, S. (1989). More on monadic logic. Part D: A note on addition of theories. Israel J. Math., 68(3), 302–306. DOI: 10.1007/BF02764986 MR: 1039475
type: article
abstract: (none)
keywords: M: smt
Sh:285
Makkai, M., & Shelah, S. (1990). Categoricity of theories in L_{\kappa\omega}, with \kappa a compact cardinal. Ann. Pure Appl. Logic, 47(1), 41–97. DOI: 10.1016/0168-0072(90)90016-U MR: 1050561
type: article
abstract: (none)
keywords: M: cla, M: nec, (mod), (inf-log), (cat)
Sh:286
Judah, H. I., & Shelah, S. (1988). Q-sets do not necessarily have strong measure zero. Proc. Amer. Math. Soc., 102(3), 681–683. DOI: 10.2307/2047245 MR: 929002
type: article
abstract: (none)
keywords: S: str, (set), (leb)
Sh:287
Blass, A. R., & Shelah, S. (1989). Near coherence of filters. III. A simplified consistency proof. Notre Dame J. Formal Logic, 30(4), 530–538. DOI: 10.1305/ndjfl/1093635236 MR: 1036674
type: article
abstract: (none)
keywords: S: for, S: str, (set), (inv), (large)
Sh:288
Shelah, S. (1992). Strong partition relations below the power set: consistency; was Sierpiński right? II. In Sets, graphs and numbers (Budapest, 1991), Vol. 60, North-Holland, Amsterdam, pp. 637–668. arXiv: math/9201244 MR: 1218224
type: article
abstract: We continue here [Sh:276] (see the introduction there) but we do not rely on it. The motivation was a conjecture of Galvin stating that \big( 2^{\omega}\ge \omega_2 \big) + \big( \omega_2\to [\omega_1]^{n}_{h(n)} \big) is consistent for a suitable h : \omega \to \omega. In section 5 we disprove this and give similar negative results. In section 3 we prove the consistency of the conjecture replacing \omega_2 by 2^\omega, which is quite large, starting with an Erdős cardinal. In section 1 we present iteration lemmas which are needed when we replace \omega by a larger \lambda, and in section 4 we generalize a theorem of Halpern and Lauchli replacing \omega by a larger \lambda.

This is a slightly corrected version of an old work.
keywords: S: ico, S: for, (set), (iter), (pc)
Sh:289
Shelah, S. (1989). Consistency of positive partition theorems for graphs and models. In Set theory and its applications (Toronto, ON, 1987), Vol. 1401, Springer, Berlin, pp. 167–193. DOI: 10.1007/BFb0097339 MR: 1031773
type: article
abstract: (none)
keywords: S: for, (set), (pc)
Sh:290
Biró, B., & Shelah, S. (1988). Isomorphic but not lower base-isomorphic cylindric set algebras. J. Symbolic Logic, 53(3), 846–853. DOI: 10.2307/2274576 MR: 961003
type: article
abstract: (none)
keywords: M: odm, O: alg, (mod)
Sh:291
Mekler, A. H., Nelson, E. M., & Shelah, S. (1993). A variety with solvable, but not uniformly solvable, word problem. Proc. London Math. Soc. (3), 66(2), 225–256. arXiv: math/9301203 DOI: 10.1112/plms/s3-66.2.225 MR: 1199065
type: article
abstract: In the literature two notions of the word problem for a variety occur. A variety has a decidable word problem if every finitely presented algebra in the variety has a decidable word problem. It has a uniformly decidable word problem if there is an algorithm which given a finite presentation produces an algorithm for solving the word problem of the algebra so presented. A variety is given with finitely many axioms having a decidable, but not uniformly decidable, word problem. Other related examples are given as well.
keywords: O: alg, (ua)
Sh:292
Judah, H. I., & Shelah, S. (1988). Souslin forcing. J. Symbolic Logic, 53(4), 1188–1207. DOI: 10.2307/2274613 MR: 973109
type: article
abstract: (none)
keywords: S: for, S: str, (set)
Sh:293
Shelah, S., & Stanley, L. J. (1993). More consistency results in partition calculus. Israel J. Math., 81(1-2), 97–110. DOI: 10.1007/BF02761299 MR: 1231180
type: article
abstract: (none)
keywords: S: for, (set), (pc)
Sh:294
Shelah, S., & Stanley, L. J. (1992). Coding and reshaping when there are no sharps. In Set theory of the continuum (Berkeley, CA, 1989), Vol. 26, Springer, New York, pp. 407–416. arXiv: math/9201249 DOI: 10.1007/978-1-4613-9754-0_21 MR: 1233827
type: article
abstract: Assuming 0^\sharp does not exist, \kappa is an uncountable cardinal and for all cardinals \lambda with \kappa \leq \lambda <\kappa^{+\omega},\ 2^\lambda = \lambda^+, we present a ‘‘mini-coding between \kappa and \kappa^{+\omega}. This allows us to prove that any subset of \kappa^{+\omega} can be coded into a subset, W of \kappa^+ which, further, ‘‘reshapes the interval [\kappa,\ \kappa^+), i.e., for all \kappa < \delta < \kappa^+, \ \kappa = (card\ \delta)^{L[W \cap \delta]}. We sketch two applications of this result, assuming 0^\sharp does not exist. First, we point out that this shows that any set can be coded by a real, via a set forcing. The second application involves a notion of abstract condensation, due to Woodin. Our methods can be used to show that for any cardinal \mu, condensation for \mu holds in a generic extension by a set forcing.
keywords: S: for, (set)
Sh:296
Shelah, S., & Steprāns, J. (1989). Nontrivial homeomorphisms of \beta \mathbf N\setminus \mathbf N without the continuum hypothesis. Fund. Math., 132(2), 135–141. DOI: 10.4064/fm-132-2-135-141 MR: 1002627
type: article
abstract: (none)
keywords: S: for, S: str, (set), (gt)
Sh:297
Hodkinson, I. M., & Shelah, S. (1993). A construction of many uncountable rings using SFP domains and Aronszajn trees. Proc. London Math. Soc. (3), 67(3), 449–492. DOI: 10.1112/plms/s3-67.3.449 MR: 1238042
type: article
abstract: (none)
keywords: M: non, O: alg, (stal), (aron)
Sh:298
Eklof, P. C., & Shelah, S. (1987). A calculation of injective dimension over valuation domains. Rend. Sem. Mat. Univ. Padova, 78, 279–284. http://www.numdam.org/item?id=RSMUP_1987__78__279_0 MR: 934519
type: article
abstract: (none)
keywords: O: alg, (ab), (stal)
Sh:299
Shelah, S. (1987). Taxonomy of universal and other classes. In Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), Vol. 1, Amer. Math. Soc., Providence, RI, pp. 154–162. MR: 934221
type: article
abstract: (none)
keywords: M: cla, M: nec, (mod)
Sh:300
Shelah, S. (1987). Universal classes. In Classification theory (Chicago, IL, 1985), Vol. 1292, Springer, Berlin, pp. 264–418. DOI: 10.1007/BFb0082242 MR: 1033033
type: article
abstract: (none)
keywords: M: nec, (mod)
Sh:301
Hodges, W., & Shelah, S. (2019). Naturality and definability II. Cubo, 21(3), 9–27. arXiv: math/0102060 DOI: 10.4067/s0719-06462019000300009 MR: 4077584
type: article
abstract: In two papers we noted that in common practice many algebraic constructions are defined only ‘up to isomorphism’ rather than explicitly. We mentioned some questions raised by this fact, and we gave some partial answers. The present paper provides much fuller answers, though some questions remain open. Our main result says that there is a transitive model of Zermelo-Fraenkel set theory with choice (ZFC) in which every fully definable construction is ‘weakly natural’ (a weakening of the notion of a natural transformation). A corollary is that there are models of ZFC in which some well-known constructions, such as algebraic closure of fields, are not explicitly definable. We also show that there is no model of ZFC in which the explicitly definable constructions are precisely the natural ones.
keywords: M: odm, S: for, O: alg, (set), (mod), (set-mod)
Sh:302
Grossberg, R. P., & Shelah, S. (1989). On the structure of \mathrm{Ext}_p(G,\mathbf Z). J. Algebra, 121(1), 117–128. DOI: 10.1016/0021-8693(89)90088-4 MR: 992319
type: article
abstract: (none)
keywords: S: dst, O: alg, (ab), (wh), (ps-dst)
Sh:302a
Grossberg, R. P., & Shelah, S. (1998). On cardinalities in quotients of inverse limits of groups. Math. Japon., 47(2), 189–197. arXiv: math/9911225 MR: 1615081
type: article
abstract: Let \lambda be \aleph_0 or a strong limit of cofinality \aleph_0. Suppose that \langle G_m,\pi_{m,n}\;:\;m\leq n<\omega\rangle and \langle H_m,\pi^t_{m,n}\;:\;m\leq n<\omega\rangle are projective systems of groups of cardinality less than \lambda and suppose that for every n< \omega there is a homorphism \sigma:H_n\rightarrow G_n such that all the diagrams commute. If for every \mu< \lambda there exists \langle f_i\in G_{\omega} \;:\;i< \mu\rangle such that i\neq j\Longrightarrow f_if_j^{-1}\not\in\sigma_{\omega}(H_{\omega}) then there exists \langle f_i\in G_{\omega} \;:\;i < 2^{\lambda}\rangle such that i\neq j\Longrightarrow f_if_j^{-1}\not\in\sigma_{\omega}(H_{\omega}).
keywords: S: dst, O: alg, (ab), (wh), (ps-dst)
Sh:303
Komjáth, P., & Shelah, S. (1988). Forcing constructions for uncountably chromatic graphs. J. Symbolic Logic, 53(3), 696–707. DOI: 10.2307/2274566 MR: 960993
type: article
abstract: (none)
keywords: S: for, (set), (graph)
Sh:304
Shelah, S., & Spencer, J. H. (1988). Zero-one laws for sparse random graphs. J. Amer. Math. Soc., 1(1), 97–115. DOI: 10.2307/1990968 MR: 924703
type: article
abstract: (none)
keywords: M: odm, O: fin, (fmt), (graph)
Sh:305
Shelah, S., & Thomas, S. (1989). Subgroups of small index in infinite symmetric groups. II. J. Symbolic Logic, 54(1), 95–99. DOI: 10.2307/2275018 MR: 987325
type: article
abstract: (none)
keywords: S: ico
Sh:306
Mekler, A. H., & Shelah, S. (1990). Determining abelian p-groups from their n-socles. Comm. Algebra, 18(2), 287–307. DOI: 10.1080/00927879008823915 MR: 1047311
type: article
abstract: (none)
keywords: S: ico, (ab)
Sh:307
Buechler, S., & Shelah, S. (1989). On the existence of regular types. Ann. Pure Appl. Logic, 45(3), 277–308. DOI: 10.1016/0168-0072(89)90039-0 MR: 1032833
type: article
abstract: (none)
keywords: M: cla, (mod), (sta)
Sh:308
Judah, H. I., & Shelah, S. (1990). The Kunen-Miller chart (Lebesgue measure, the Baire property, Laver reals and preservation theorems for forcing). J. Symbolic Logic, 55(3), 909–927. DOI: 10.2307/2274464 MR: 1071305
type: article
abstract: (none)
keywords: S: for, S: str, (set), (leb), (inv), (meag)
Sh:310
Gitik, M., & Shelah, S. (1999). Cardinal preserving ideals. J. Symbolic Logic, 64(4), 1527–1551. arXiv: math/9605234 DOI: 10.2307/2586794 MR: 1780068
type: article
abstract: We give some general criteria, when \kappa-complete forcing preserves largeness properties – like \kappa-presaturation of normal ideals on \lambda (even when they concentrate on small cofinalities). Then we quite accurately obtain the consistency strength “NS_\lambda is \aleph_1-preserving", for \lambda >\aleph_2.
keywords: S: for, (set), (sta), (ref)
Sh:312
Shelah, S. (2017). Existentially closed locally finite groups (Sh312). In Beyond first order model theory, CRC Press, Boca Raton, FL, pp. 221–298. arXiv: 1102.5578 MR: 3729328
type: article
abstract: We investigate this class of groups originally called ulf (universal locally finite groups) of cardinality \lambda. We prove that for every locally finite group G there is a canonical existentially closed extention of the same cardinality, unique up to isomorphism and increasing with G. Also we get, e.g. existence of complete members (i.e. with no non-inner automorphisms) in many cardinals (provably in ZFC). We also get a parallel to stability theory in the sense of investigating definable types. Had appeared.
keywords: M: non, (stal), (grp)
Sh:313
Mekler, A. H., & Shelah, S. (1988). Diamond and \lambda-systems. Fund. Math., 131(1), 45–51. DOI: 10.4064/fm-131-1-45-51 MR: 970913
type: article
abstract: (none)
keywords: M: non, (diam)
Sh:314
Mekler, A. H., Rosłanowski, A., & Shelah, S. (1999). On the p-rank of Ext. Israel J. Math., 112, 327–356. arXiv: math/9806165 DOI: 10.1007/BF02773487 MR: 1714978
type: article
abstract: Assume V=L and \lambda is regular smaller than the first weakly compact cardinal. Under those circumstances and with arbitrary requirements on the structure of Ext(G,{\mathbb Z}) (under well known limitations), we construct an abelian group G of cardinality \lambda such that for no G'\subseteq G, |G'|< \lambda is G/G' free and Ext(G,{\mathbb Z}) realizes our requirements.
keywords: S: ico, (set), (ab), (wh)
Sh:315
Shelah, S., & Steprāns, J. (1988). PFA implies all automorphisms are trivial. Proc. Amer. Math. Soc., 104(4), 1220–1225. DOI: 10.2307/2047617 MR: 935111
type: article
abstract: (none)
keywords: S: for, (set)
Sh:316
Fuchs, L., & Shelah, S. (1989). Kaplansky’s problem on valuation rings. Proc. Amer. Math. Soc., 105(1), 25–30. DOI: 10.2307/2046728 MR: 929431
type: article
abstract: (none)
keywords: O: alg, (ab), (stal)
Sh:317
Becker, T., Fuchs, L., & Shelah, S. (1989). Whitehead modules over domains. Forum Math., 1(1), 53–68. DOI: 10.1515/form.1989.1.53 MR: 978975
type: article
abstract: (none)
keywords: O: alg, (ab), (stal), (wh)
Sh:318
Macpherson, D., Mekler, A. H., & Shelah, S. (1991). The number of infinite substructures. Math. Proc. Cambridge Philos. Soc., 109(1), 193–209. DOI: 10.1017/S0305004100069668 MR: 1075131
type: article
abstract: (none)
keywords: S: ico, (nni)
Sh:319
Judah, H. I., & Shelah, S. (1989). Martin’s axioms, measurability and equiconsistency results. J. Symbolic Logic, 54(1), 78–94. DOI: 10.2307/2275017 MR: 987324
type: article
abstract: (none)
keywords: S: for, S: str, (set), (leb)
Sh:320
Juhász, I., Shelah, S., & Soukup, L. (1988). More on countably compact, locally countable spaces. Israel J. Math., 62(3), 302–310. DOI: 10.1007/BF02783299 MR: 955134
type: article
abstract: (none)
keywords: O: top, (gt)
Sh:321
Judah, H. I., & Shelah, S. (1989). \Delta^1_2-sets of reals. Ann. Pure Appl. Logic, 42(3), 207–223. DOI: 10.1016/0168-0072(89)90016-X MR: 998607
type: article
abstract: (none)
keywords: S: str, S: dst, (set)
Sh:323
Hart, B. T., & Shelah, S. (1990). Categoricity over P for first order T or categoricity for \phi\in\mathcal L_{\omega_1\omega} can stop at \aleph_k while holding for \aleph_0,\cdots,\aleph_{k-1}. Israel J. Math., 70(2), 219–235. arXiv: math/9201240 DOI: 10.1007/BF02807869 MR: 1070267
type: article
abstract: Suppose L is a relational language and P\in L is a unary predicate. If M is an L-structure then P(M) is the L-structure formed as the substructure of M with domain \{a: M\models P(a)\}. Now suppose T is a complete first order theory in L with infinite models. Following Hodges, we say that T is relatively \lambda-categorical if whenever M, N\models T, P(M)=P(N), |P(M)|=\lambda then there is an isomorphism i:M\rightarrow N which is the identity on P(M). T is relatively categorical if it is relatively \lambda-categorical for every \lambda. The question arises whether the relative \lambda-categoricity of T for some \lambda>|T| implies that T is relatively categorical.

In this paper, we provide an example, for every k>0, of a theory T_k and an L_{\omega_1\omega} sentence \varphi_k so that T_k is relatively \aleph_n-categorical for n < k and \varphi_k is \aleph_n-categorical for n<k but T_k is not relatively \beth_k-categorical and \varphi_k is not \beth_k-categorical.
keywords: M: nec, (mod), (cat)
Sh:324
Magidor, M., & Shelah, S. (1996). The tree property at successors of singular cardinals. Arch. Math. Logic, 35(5-6), 385–404. arXiv: math/9501220 DOI: 10.1007/s001530050052 MR: 1420265
type: article
abstract: Assuming some large cardinals, a model of ZFC is obtained in which \aleph_{omega+1} carries no Aronszajn trees. It is also shown that if \lambda is a singular limit of strongly compact cardinals, then \lambda^+ carries no Aronszajn trees.
keywords: S: ico, S: for, (set), (large), (aron)
Sh:325
Dugas, M. H., & Shelah, S. (1989). E-transitive groups in L. In Abelian group theory (Perth, 1987), Vol. 87, Amer. Math. Soc., Providence, RI, pp. 191–199. DOI: 10.1090/conm/087/995276 MR: 995276
type: article
abstract: (none)
keywords: M: non, (ab)
Sh:326
Shelah, S. (1992). Vive la différence. I. Nonisomorphism of ultrapowers of countable models. In Set theory of the continuum (Berkeley, CA, 1989), Vol. 26, Springer, New York, pp. 357–405. arXiv: math/9201245 DOI: 10.1007/978-1-4613-9754-0_20 MR: 1233826
See [Sh:326a]
type: article
abstract: We show that it is not provable in ZFC that any two countable elementarily equivalent structures have isomorphic ultrapowers relative to some ultrafilter on \omega.
keywords: S: for, S: str, (iter), (set-mod), (up), (creatures)
Sh:327
Shelah, S. (1991). Strong negative partition relations below the continuum. Acta Math. Hungar., 58(1-2), 95–100. DOI: 10.1007/BF01903551 MR: 1152830
type: article
abstract: (none)
keywords: S: ico, (set)
Sh:328
Shelah, S., & Thomas, S. (1988). Implausible subgroups of infinite symmetric groups. Bull. London Math. Soc., 20(4), 313–318. DOI: 10.1112/blms/20.4.313 MR: 940283
type: article
abstract: (none)
keywords: S: ico
Sh:329
Shelah, S. (1988). Primitive recursive bounds for van der Waerden numbers. J. Amer. Math. Soc., 1(3), 683–697. DOI: 10.2307/1990952 MR: 929498
type: article
abstract: (none)
keywords: O: fin, (fc)
Sh:330
Baldwin, J. T., & Shelah, S. (1990). The primal framework. I. Ann. Pure Appl. Logic, 46(3), 235–264. arXiv: math/9201241 DOI: 10.1016/0168-0072(90)90005-M MR: 1049388
type: article
abstract: This the first of a series of articles dealing with abstract classification theory. The apparatus to assign systems of cardinal invariants to models of a first order theory (or determine its impossibility) is developed in [Sh:a]. It is natural to try to extend this theory to classes of models which are described in other ways. Work on the classification theory for nonelementary classes [Sh:88] and for universal classes [Sh:300] led to the conclusion that an axiomatic approach provided the best setting for developing a theory of wider application. In the first chapter we describe the axioms on which the remainder of the article depends and give some examples and context to justify this level of generality. The study of universal classes takes as a primitive the notion of closing a subset under functions to obtain a model. We replace that concept by the notion of a prime model. We begin the detailed discussion of this idea in Chapter II. One of the important contributions of classification theory is the recognition that large models can often be analyzed by means of a family of small models indexed by a tree of height at most \omega. More precisely, the analyzed model is prime over such a tree. Chapter III provides sufficient conditions for prime models over such trees to exist.
keywords: M: nec, (mod)
Sh:332
Gurevich, Y., & Shelah, S. (1988). Nondeterministic Linear Tasks May Require Substantially Nonlinear Deterministic Time in the Case of Sublinear Work Space. In Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, New York, NY, USA: ACM, pp. 281–289. DOI: 10.1145/62212.62239
type: article
abstract: (none)
keywords: O: fin
Sh:332a
Gurevich, Y., & Shelah, S. (1990). Nondeterministic linear-time tasks may require substantially nonlinear deterministic time in the case of sublinear work space. J. Assoc. Comput. Mach., 37(3), 674–687. DOI: 10.1145/79147.214070 MR: 1072274
type: article
abstract: (none)
keywords: O: fin
Sh:334
Hrushovski, E., & Shelah, S. (1991). Stability and omitting types. Israel J. Math., 74(2-3), 289–321. DOI: 10.1007/BF02775793 MR: 1135241
type: article
abstract: (none)
keywords: M: cla, (mod), (sta)
Sh:335
Judah, H. I., & Shelah, S. (1989). MA(\sigma-centered): Cohen reals, strong measure zero sets and strongly meager sets. Israel J. Math., 68(1), 1–17. DOI: 10.1007/BF02764965 MR: 1035877
type: article
abstract: (none)
keywords: S: for, S: str, (set), (leb), (meag)
Sh:336
Judah, H. I., & Shelah, S. (1991). Q-sets, Sierpiński sets, and rapid filters. Proc. Amer. Math. Soc., 111(3), 821–832. DOI: 10.2307/2048420 MR: 1045594
type: article
abstract: (none)
keywords: S: for, S: str, (set)
Sh:337
Judah, H. I., & Shelah, S. (1993). \boldsymbol{\Delta}^1_3-sets of reals. J. Symbolic Logic, 58(1), 72–80. DOI: 10.2307/2275325 MR: 1217177
type: article
abstract: (none)
keywords: S: for, S: str, S: dst, (set)
Sh:338
Judah, H. I., & Shelah, S. (1991). Forcing minimal degree of constructibility. J. Symbolic Logic, 56(3), 769–782. DOI: 10.2307/2275046 MR: 1129141
type: article
abstract: (none)
keywords: S: for, S: str, S: dst, (set)
Sh:339
Judah, H. I., Shelah, S., & Woodin, W. H. (1990). The Borel conjecture. Ann. Pure Appl. Logic, 50(3), 255–269. NB: A correction of the third section has appeared in 8.3.B of [Bartoszyński, Judah: Set theory. ISBN 1-56881-044-X] DOI: 10.1016/0168-0072(90)90058-A MR: 1086456
type: article
abstract: (none)
keywords: S: str, (set)
Sh:340
Shelah, S., & Stanley, L. J. (1995). A combinatorial forcing for coding the universe by a real when there are no sharps. J. Symbolic Logic, 60(1), 1–35. arXiv: math/9311204 DOI: 10.2307/2275507 MR: 1324499
type: article
abstract: Assuming 0^\sharp does not exist, we present a combinatorial approach to Jensen’s method of coding by a real. The forcing uses combinatorial consequences of fine structure (including the Covering Lemma, in various guises), but makes no direct appeal to fine structure itself.
keywords: S: for, (set)
Sh:341
Juhász, I., & Shelah, S. (1989). \pi (X)=\delta(X) for compact X. Topology Appl., 32(3), 289–294. NB: Proof in local case is a mistake DOI: 10.1016/0166-8641(89)90035-7 MR: 1007107
type: article
abstract: (none)
keywords: S: ico, (gt)
Sh:342
Hrushovski, E., & Shelah, S. (1989). A dichotomy theorem for regular types. Ann. Pure Appl. Logic, 45(2), 157–169. DOI: 10.1016/0168-0072(89)90059-6 MR: 1044122
type: article
abstract: (none)
keywords: M: cla, (mod), (sta)
Sh:343
Gurevich, Y., & Shelah, S. (1989). Time polynomial in input or output. J. Symbolic Logic, 54(3), 1083–1088. DOI: 10.2307/2274767 MR: 1011194
type: article
abstract: (none)
keywords: O: fin
Sh:344
Gitik, M., & Shelah, S. (1989). On certain indestructibility of strong cardinals and a question of Hajnal. Arch. Math. Logic, 28(1), 35–42. DOI: 10.1007/BF01624081 MR: 987765
type: article
abstract: (none)
keywords: S: for, (set), (large)
Sh:345
Shelah, S. (1990). Products of regular cardinals and cardinal invariants of products of Boolean algebras. Israel J. Math., 70(2), 129–187. DOI: 10.1007/BF02807866 MR: 1070264
type: article
abstract: (none)
keywords: S: pcf, (ba)
Sh:346
Komjáth, P., & Shelah, S. (1996). On Taylor’s problem. Acta Math. Hungar., 70(3), 217–225. arXiv: math/9402213 DOI: 10.1007/BF02188208 MR: 1374388
type: article
abstract: We describe some (countably many) classes K^{n,e} of finite graphs and prove that if \lambda^{\aleph_0}=\lambda then every \lambda^+-chromatic graph of cardinal \lambda^+ contains, for some n, e, all members of K^{n,e} as subgraphs. On the other hand, it is consistent for every regular infinite cardinal \kappa that there is a \kappa^+-chromatic graph on \kappa^+ that contains finite subgraphs only from K^{n,e}.
keywords: S: for, (set), (graph)
Sh:347
Shelah, S. (1990). Incompactness for chromatic numbers of graphs. In A tribute to Paul Erdős, Cambridge Univ. Press, Cambridge, pp. 361–371. MR: 1117029
type: article
abstract: (none)
keywords: S: ico, (set)
Sh:348
Bartoszyński, T., Judah, H. I., & Shelah, S. (1989). The cofinality of cardinal invariants related to measure and category. J. Symbolic Logic, 54(3), 719–726. DOI: 10.2307/2274736 MR: 1011163
type: article
abstract: (none)
keywords: S: for, S: str, (leb), (inv), (meag)
Sh:349
Shelah, S. (1989). A consistent counterexample in the theory of collectionwise Hausdorff spaces. Israel J. Math., 65(2), 219–224. DOI: 10.1007/BF02764862 MR: 998672
type: article
abstract: (none)
keywords: S: for, (gt)
Sh:350
Dror Farjoun, E., Orr, K. E., & Shelah, S. (1989). Bousfield localization as an algebraic closure of groups. Israel J. Math., 66(1-3), 143–153. DOI: 10.1007/BF02765889 MR: 1017158
type: article
abstract: (none)
keywords: O: top, (grp)
Sh:351
Shelah, S. (1991). Reflecting stationary sets and successors of singular cardinals. Arch. Math. Logic, 31(1), 25–53. DOI: 10.1007/BF01370693 MR: 1126352
type: article
abstract: (none)
keywords: S: ico, S: for, (set), (ref)
Sh:352
Eklof, P. C., Fuchs, L., & Shelah, S. (1990). Baer modules over domains. Trans. Amer. Math. Soc., 322(2), 547–560. DOI: 10.2307/2001714 MR: 974514
type: article
abstract: (none)
keywords: O: alg, (ab)
Sh:353
Shelah, S., & Steprāns, J. (1994). Homogeneous almost disjoint families. Algebra Universalis, 31(2), 196–203. DOI: 10.1007/BF01236517 MR: 1259349
type: article
abstract: (none)
keywords: S: for, S: str, (set)
Sh:354
Perles, M. A., & Shelah, S. (1990). A closed (n+1)-convex set in \mathbf R^2 is a union of n^6 convex sets. Israel J. Math., 70(3), 305–312. DOI: 10.1007/BF02801466 MR: 1074494
type: article
abstract: (none)
keywords: O: fin, (fc)
Sh:356
Milner, E. C., & Shelah, S. (1990). Graphs with no unfriendly partitions. In A tribute to Paul Erdős, Cambridge Univ. Press, Cambridge, pp. 373–384. DOI: 10.1017/cbo9780511983917.031 MR: 1117030
type: article
abstract: (none)
keywords: S: ico, (graph)
Sh:357
Gitik, M., & Shelah, S. (1989). Forcings with ideals and simple forcing notions. Israel J. Math., 68(2), 129–160. DOI: 10.1007/BF02772658 MR: 1035887
type: article
abstract: (none)
keywords: S: str, (set), (leb), (meag)
Sh:358
Judah, H. I., & Shelah, S. (1990). Around random algebra. Arch. Math. Logic, 30(3), 129–138. DOI: 10.1007/BF01621466 MR: 1080233
type: article
abstract: (none)
keywords: S: for, S: str, (set), (leb)
Sh:359
Bonnet, R., & Shelah, S. (1993). On HCO spaces. An uncountable compact T_2 space, different from \aleph_1+1, which is homeomorphic to each of its uncountable closed subspaces. Israel J. Math., 84(3), 289–332. DOI: 10.1007/BF02760945 MR: 1244672
type: article
abstract: (none)
keywords: M: non, (set), (ba), (diam)
Sh:360
Baldwin, J. T., & Shelah, S. (1991). The primal framework. II. Smoothness. Ann. Pure Appl. Logic, 55(1), 1–34. arXiv: math/9201246 DOI: 10.1016/0168-0072(91)90095-4 MR: 1134914
See [Sh:360a]
type: article
abstract: This is the second in a series of articles developing abstract classification theory for classes that have a notion of prime models over independent pairs and over chains. It deals with the problem of smoothness and establishing the existence and uniqueness of a ‘monster model’. We work here with a predicate for a canonically prime model.
keywords: M: nec, (mod)
Sh:361
Shelah, S., & Thomas, S. (1989). Homogeneity of infinite permutation groups. Arch. Math. Logic, 28(2), 143–147. DOI: 10.1007/BF01633987 MR: 996316
type: article
abstract: (none)
keywords: S: ico
Sh:362
Kolman, O., & Shelah, S. (1996). Categoricity of theories in L_{\kappa\omega}, when \kappa is a measurable cardinal. I. Fund. Math., 151(3), 209–240. arXiv: math/9602216 MR: 1424575
type: article
abstract: We assume a theory T in the logic L_{\kappa \omega} is categorical in a cardinal \lambda\geq \kappa, and \kappa is a measurable cardinal. Here we prove that the class of model of T of cardinality < \lambda (but \geq |T|+\kappa) has the amalgamation property; this is a step toward understanding the character of such classes of models.
keywords: M: nec, (mod), (nni), (cat)
Sh:364
Juhász, I., & Shelah, S. (1992). On partitioning the triples of a topological space. Topology Appl., 44(1-3), 203–208. DOI: 10.1016/0166-8641(92)90095-H MR: 1173259
type: article
abstract: (none)
keywords: S: ico, S: for, O: top, (pc), (gt)
Sh:366
Mekler, A. H., & Shelah, S. (1995). Almost free algebras. Israel J. Math., 89(1-3), 237–259. arXiv: math/9408213 DOI: 10.1007/BF02808203 MR: 1324464
type: article
abstract: The essentially non-free spectrum is the class of uncountable cardinals \kappa in which there is an essentially non-free algebra of cardinality \kappa which is almost free. In L, the essentially non-free spectrum of a variety is entirely determined by whether or not the construction principle holds. In ZFC may be more complicated. For some varieties, such as groups, abelian groups or any variety of modules over a non-left perfect ring, the essentially non-free spectrum contains not only \aleph_1 but \aleph_n for all n>0. The reason for this being true in ZFC (rather than under some special set theoretic hypotheses) is that these varieties satisfy stronger versions of the construction principle. We conjecture that the hierarchy of construction principles is strict, i.e., that for each n>0 there is a variety which satisfies the n-construction principle but not the n+1-construction principle. In this paper we will show that the 1-construction principle does not imply the 2-construction principle. We prove that, assuming the consistency of some large cardinal hypothesis, it is consistent that a variety has an essentially non-free almost free algebra of cardinality \aleph_n if and only if it satisfies the n-construction principle.
keywords: O: alg, (stal), (af)
Sh:367
Mekler, A. H., & Shelah, S. (1989). The consistency strength of “every stationary set reflects”. Israel J. Math., 67(3), 353–366. DOI: 10.1007/BF02764953 MR: 1029909
type: article
abstract: (none)
keywords: S: for, (set), (iter), (stal), (ref), (af), (pure(large))
Sh:368
Bartoszyński, T., Judah, H. I., & Shelah, S. (1993). The Cichoń diagram. J. Symbolic Logic, 58(2), 401–423. arXiv: math/9905122 DOI: 10.2307/2275212 MR: 1233917
type: article
abstract: (none)
keywords: S: for, S: str, (set)
Sh:369
Goldstern, M., Judah, H. I., & Shelah, S. (1991). A regular topological space having no closed subsets of cardinality \aleph_2. Proc. Amer. Math. Soc., 111(4), 1151–1159. DOI: 10.2307/2048582 MR: 1052572
type: article
abstract: We show in ZFC that there is a regular (even zerodimensional) topological space of size > \aleph_2 in which there are no closed sets of size \aleph_2. The proof starts by noticing that if \beta\omega does not work, then we can use a \diamondsuit.
keywords: S: ico, (gt)
Sh:370
Shelah, S., & Soukup, L. (1994). On the number of nonisomorphic subgraphs. Israel J. Math., 86(1-3), 349–371. arXiv: math/9401210 DOI: 10.1007/BF02773686 MR: 1276143
type: article
abstract: Let \mathcal K be the family of graphs on \omega_1 without cliques or independent subsets of size \omega_1. We prove that:

1) it is consistent with CH that every G\in{\mathcal K} has 2^{\omega_1} many pairwise non-isomorphic subgraphs,

2) the following proposition holds in L: (*) there is a G\in{\mathcal K} such that for each partition (A,B) of \omega_1 either G\cong G[A] or G\cong G[B],

3) the failure of (*) is consistent with ZFC.
keywords: S: ico, (nni), (graph)
Sh:372
Judah, H. I., Miller, A. W., & Shelah, S. (1992). Sacks forcing, Laver forcing, and Martin’s axiom. Arch. Math. Logic, 31(3), 145–161. DOI: 10.1007/BF01269943 MR: 1147737
type: article
abstract: (none)
keywords: S: for, S: str, (set)
Sh:373
Judah, H. I., Rosłanowski, A., & Shelah, S. (1994). Examples for Souslin forcing. Fund. Math., 144(1), 23–42. arXiv: math/9310224 DOI: 10.4064/fm-144-1-23-42 MR: 1271476
type: article
abstract: We give a model where there is a ccc Souslin forcing which does not satisfy the Knaster condition. Next, we present a model where there is a \sigma-linked not \sigma-centered Souslin forcing such that all its small subsets are \sigma-centered but Martin Axiom fails for this order. Furthermore, we construct a totally nonhomogeneous Souslin forcing and we build a Souslin forcing which is proper but not ccc that does not contain a perfect set of mutually incompatible conditions. Finally we show that ccc \Sigma^1_2-notions of forcing may not be indestructible ccc.
keywords: S: for, S: str, (set)
Sh:374
Judah, H. I., & Shelah, S. (1993). Adding dominating reals with the random algebra. Proc. Amer. Math. Soc., 119(1), 267–273. DOI: 10.2307/2159852 MR: 1152278
type: article
abstract: (none)
keywords: S: for, S: str, (set)
Sh:375
Mekler, A. H., & Shelah, S. (1993). Some compact logics—results in ZFC. Ann. Of Math. (2), 137(2), 221–248. arXiv: math/9301204 DOI: 10.2307/2946538 MR: 1207207
type: article
abstract: We show that if we enrich first order logic by allowing quantification over isomorphisms between definable ordered fields the resulting logic, L(Q_{\rm Of}), is fully compact. In this logic, we can give standard compactness proofs of various results. Next, we attempt to get compactness results for some other logics without recourse to \diamondsuit, i.e., all our results are in ZFC. We get the full result for the language where we quantify over automorphisms (isomorphisms) of ordered fields in Theorem 6.4. Unfortunately we are not able to show that the language with quantification over automorphisms of Boolean algebras is compact, but will have to settle for a close relative of that logic. This is theorem 5.1. In section 4 we prove we can construct models in which all relevant automorphism are somewhat definable: 4.1, 4.8 for BA, 4.13 for ordered fields. We also give a new proof of the compactness of another logic – the one which is obtained when a quantifier Q_{{\rm Brch}} is added to first order logic which says that a level tree (definitions will be given later) has an infinite branch. This logic was previously shown to be compact, but our proof yields a somewhat stronger result and provides a nice illustration of one of our methods.
keywords: M: non, (mod)
Sh:376
Shelah, S., & Soukup, L. (1995). Some remarks on a problem of J. D. Monk. Period. Math. Hungar., 30(2), 155–163. DOI: 10.1007/BF01876630 MR: 1326777
type: article
abstract: (none)
keywords: S: for, (ba)
Sh:377
Shelah, S., Tuuri, H., & Väänänen, J. A. (1993). On the number of automorphisms of uncountable models. J. Symbolic Logic, 58(4), 1402–1418. arXiv: math/9301205 DOI: 10.2307/2275150 MR: 1253929
type: article
abstract: Let s({\mathcal A}) denote the number of automorphisms of a model {\mathcal A} of power \omega_1. We derive a necessary and sufficient condition in terms of trees for the existence of an {\mathcal A} with \omega_1 < s({\mathcal A}) < 2^{\omega_1}. We study the sufficiency of some conditions for s({\mathcal A})=2^{\omega_1}. These conditions are analogous to conditions studied by D.Kueker in connection with countable models.
keywords: M: non, O: alg, (mod)
Sh:378
Jech, T. J., & Shelah, S. (1989). A note on canonical functions. Israel J. Math., 68(3), 376–380. arXiv: math/9201239 DOI: 10.1007/BF02764992 MR: 1039481
type: article
abstract: We construct a generic extension in which the \aleph_2 nd canonical function on \aleph_1 exists.
keywords: S: for, (set), (iter), (normal), (large)
Sh:379
Eklof, P. C., & Shelah, S. (1991). On Whitehead modules. J. Algebra, 142(2), 492–510. DOI: 10.1016/0021-8693(91)90321-X MR: 1127077
type: article
abstract: (none)
keywords: O: alg, (ab), (stal), (wh)
Sh:381
Shelah, S. (1991). Kaplansky test problem for R-modules. Israel J. Math., 74(1), 91–127. DOI: 10.1007/BF02777818 MR: 1135231
type: article
abstract: (none)
keywords: M: non, (ab)
Sh:382
Shelah, S., & Spencer, J. H. (1994). Can you feel the double jump? Random Structures Algorithms, 5(1), 191–204. arXiv: math/9401211 DOI: 10.1002/rsa.3240050118 MR: 1248186
type: article
abstract: Paul Erdos and Alfred Renyi considered the evolution of the random graph G(n,p) as p “evolved” from 0 to 1. At p=1/n a sudden and dramatic change takes place in G. When p=c/n with c<1 the random G consists of small components, the largest of size \Theta(\log n). But by p=c/n with c>1 many of the components have “congealed” into a “giant component” of size \Theta (n). Erdos and Renyi called this the double jump, the terms phase transition (from the analogy to percolation) and Big Bang have also been proferred. Now imagine an observer who can only see G through a logical fog. He may refer to graph theoretic properties A within a limited logical language. Will he be able to detect the double jump? The answer depends on the strength of the language. Our rough answer to this rough question is: the double jump is not detectible in the First Order Theory of Graphs but it is detectible in the Second Order Monadic Theory of Graphs.
keywords: M: odm, O: fin, (fmt), (graph), (mon)
Sh:383
Jech, T. J., & Shelah, S. (1993). Full reflection of stationary sets at regular cardinals. Amer. J. Math., 115(2), 435–453. arXiv: math/9204218 DOI: 10.2307/2374864 MR: 1216437
type: article
abstract: A stationary subset S of a regular uncountable cardinal \kappa reflects fully at regular cardinals if for every stationary set T\subseteq\kappa of higher order consisting of regular cardinals there exists an \alpha\in T such that S\cap\alpha is a stationary subset of \alpha. We prove that the Axiom of Full Reflection which states that every stationary set reflects fully at regular cardinals, together with the existence of n-Mahlo cardinals is equiconsistent with the existence of \Pi^1_n-indescribable cardinals. We also state the appropriate generalization for greatly Mahlo cardinals.
keywords: S: for, (set), (iter)
Sh:385
Jech, T. J., & Shelah, S. (1991). On a conjecture of Tarski on products of cardinals. Proc. Amer. Math. Soc., 112(4), 1117–1124. arXiv: math/9201247 DOI: 10.2307/2048662 MR: 1070525
type: article
abstract: We look at an old conjecture of A. Tarski on cardinal arithmetic and show that if a counterexample exists, then there exists one of length \omega_1 + \omega.
keywords: S: pcf, (set)
Sh:387
Jech, T. J., & Shelah, S. (1990). Full reflection of stationary sets below \aleph_\omega. J. Symbolic Logic, 55(2), 822–830. arXiv: math/9201242 DOI: 10.2307/2274667 MR: 1056391
type: article
abstract: It is consistent that for every n \ge 2, every stationary subset of \omega_n consisting of ordinals of cofinality \omega_k where k = 0 or k \le n -3 reflects fully in the set of ordinals of cofinality \omega_{n-1}. We also show that this result is best possible.
keywords: S: for, (set), (iter), (large), (ref)
Sh:388
Goldstern, M., & Shelah, S. (1990). Ramsey ultrafilters and the reaping number—Con(\mathfrak r<\mathfrak u). Ann. Pure Appl. Logic, 49(2), 121–142. DOI: 10.1016/0168-0072(90)90063-8 MR: 1077075
type: article
abstract: We show that the reaping number r is consistenly smaller than the smallest base of an ultrafilter. We use a forcing notion P_U that destroys a selected ultrafilter U and all ultrafilters below it, but preserves all Ramsey ultrafilters that are not below U in the Rudin-Keisler order.
keywords: S: for, S: str, (set), (inv)
Sh:389
Shelah, S., & Soukup, L. (1993). The existence of large \omega_1-homogeneous but not \omega-homogeneous permutation groups is consistent with ZFC+GCH. J. London Math. Soc. (2), 48(2), 193–203. DOI: 10.1112/jlms/s2-48.2.193 MR: 1231709
type: article
abstract: (none)
keywords: S: for, (set)
Sh:390
Kanamori, A., & Shelah, S. (1995). Complete quotient Boolean algebras. Trans. Amer. Math. Soc., 347(6), 1963–1979. arXiv: math/9401212 DOI: 10.2307/2154916 MR: 1282888
type: article
abstract: For I a proper, countably complete ideal on {\mathcal P}(X) for some set X, can the quotient Boolean algebra {\mathcal P}(X)/I be complete? This question was raised by Sikorski in 1949. By a simple projection argument as for measurable cardinals, it can be assumed that X is an uncountable cardinal \kappa, and that I is a \kappa-complete ideal on {\mathcal P}(\kappa ) containing all singletons. In this paper we provide consequences from and consistency results about completeness.
keywords: S: for, (set), (iter), (ba)
Sh:391
Hodges, W., Hodkinson, I. M., Lascar, D., & Shelah, S. (1993). The small index property for \omega-stable \omega-categorical structures and for the random graph. J. London Math. Soc. (2), 48(2), 204–218. DOI: 10.1112/jlms/s2-48.2.204 MR: 1231710
type: article
abstract: (none)
keywords: M: cla, O: alg, (stal), (auto), (sta)
Sh:392
Jech, T. J., & Shelah, S. (1991). A partition theorem for pairs of finite sets. J. Amer. Math. Soc., 4(4), 647–656. arXiv: math/9201248 DOI: 10.2307/2939283 MR: 1122043
type: article
abstract: Every partition of [[\omega_1]^{<\omega}]^2 into finitely many pieces has a cofinal homogeneous set. Furthermore, it is consistent that every directed partially ordered set satisfies the partition property if and only if it has finite character.
keywords: S: ico
Sh:393
Baldwin, J. T., & Shelah, S. (1995). Abstract classes with few models have “homogeneous-universal” models. J. Symbolic Logic, 60(1), 246–265. arXiv: math/9502231 DOI: 10.2307/2275520 MR: 1324512
type: article
abstract: This paper is concerned with a class K of models and an abstract notion of submodel \leq. Experience in first order model theory has shown the desirability of finding a ‘monster model’ to serve as a universal domain for K. In the original constructions of Jonsson and Fraisse, K was a universal class and ordinary substructure played the role of \leq. Working with a cardinal \lambda satisfying \lambda^{<\lambda}=\lambda guarantees appropriate downward Lowenheim-Skolem theorems; the existence and uniqueness of a homogeneous-universal model appears to depend centrally on the amalgamation property. We make this apparent dependence more precise in this paper. The major innovation of this paper is the introduction of weaker notion to replace the natural notion of ({\bf K},\leq)-homogeneous-universal model. Modulo a weak extension of ZFC (provable if V=L), we show that a class K obeying certain minimal restrictions satisfies a fundamental dichotomy: For arbitrarily large \lambda, either K has the maximal number of models in power \lambda or K has a unique chain homogenous-universal model of power \lambda. We show that in a class with amalgamation this dichotomy holds for the notion of K-homogeneous-universal model in the more normal sense.
keywords: M: nec, (mod)
Sh:394
Shelah, S. (1999). Categoricity for abstract classes with amalgamation. Ann. Pure Appl. Logic, 98(1-3), 261–294. arXiv: math/9809197 DOI: 10.1016/S0168-0072(98)00016-5 MR: 1696853
type: article
abstract: Let {\mathfrak K} be an abstract elementary class with amalgamation, and Lowenheim Skolem number LS({\mathfrak K}). We prove that for a suitable Hanf number \chi_0 if \chi_0 < \lambda_0\le \lambda_1, and {\mathfrak K} is categorical in \lambda^+_1 then it is categorical in \lambda_0.
keywords: M: nec, (mod), (nni), (cat), (aec)
Sh:396
Frankiewicz, R., Shelah, S., & Zbierski, P. (1993). On closed P-sets with ccc in the space \omega^*. J. Symbolic Logic, 58(4), 1171–1176. arXiv: math/9303207 DOI: 10.2307/2275135 MR: 1253914
type: article
abstract: It is proved that – consistently – there can be no ccc closed P-sets in the remainder space \omega^*.
keywords: S: for, S: str, (set)
Sh:397
Shelah, S. (1992). Factor = quotient, uncountable Boolean algebras, number of endomorphism and width. Math. Japon., 37(2), 385–400. arXiv: math/9201250 MR: 1159041
type: article
abstract: We prove that assuming suitable cardinal arithmetic, if B is a Boolean algebra every homomorphic image of which is isomorphic to a factor, then B has locally small density. We also prove that for an (infinite) Boolean algebra B, the number of subalgebras is not smaller than the number of endomorphisms, and other related inequalities. Lastly we deal with the obtainment of the supremum of the cardinalities of sets of pairwise incomparable elements of a Boolean algebra.
keywords: S: ico, (set), (ba), (inv(ba))
Sh:398
Mekler, A. H., & Shelah, S. (1993). The canary tree. Canad. Math. Bull., 36(2), 209–215. arXiv: math/9308210 DOI: 10.4153/CMB-1993-030-6 MR: 1222536
type: article
abstract: A canary tree is a tree of cardinality the continuum which has no uncountable branch, but gains a branch whenever a stationary set is destroyed (without adding reals). Canary trees are important in infinitary model theory. The existence of a canary tree is independent of ZFC + GCH.
keywords: S: for, (iter), (set-mod), (cont)
Sh:399
Goldstern, M., Judah, H. I., & Shelah, S. (1991). Saturated families. Proc. Amer. Math. Soc., 111(4), 1095–1104. DOI: 10.2307/2048577 MR: 1052573
type: article
abstract: (none)
keywords: S: ico, (set)
Sh:400a
Shelah, S. (1992). Cardinal arithmetic for skeptics. Bull. Amer. Math. Soc. (N.S.), 26(2), 197–210. arXiv: math/9201251 DOI: 10.1090/S0273-0979-1992-00261-6 MR: 1112424
type: article
abstract: We present a survey of some results of the pcf-theory and their applications to cardinal arithmetic. We review basics notions (in §1), briefly look at history in §2 (and some personal history in §3). We present main results on pcf in §5 and describe applications to cardinal arithmetic in §6. The limitations on independence proofs are discussed in §7, and in §8 we discuss the status of two axioms that arise in the new setting. Applications to other areas are found in §9.
keywords: S: pcf, O: odo
Sh:401
Shelah, S. (2004). Characterizing an \aleph_\epsilon-saturated model of superstable NDOP theories by its \mathbb L_{\infty,\aleph_\epsilon}-theory. Israel J. Math., 140, 61–111. arXiv: math/9609215 DOI: 10.1007/BF02786627 MR: 2054839
type: article
abstract: After the main gap theorem was proved (see [Sh:c]), in discussion, Harrington expressed a desire for a finer structure - of finitary character (when we have a structure theorem at all). I point out that the logic L_{\infty,\aleph_0}(d.q.) (d.q. stands for dimension quantifier) does not suffice: e.g., for T=Th(\lambda\times {}^\omega 2,E_n)_{n<\omega} where (\alpha,\eta)E_n(\beta,\nu) =: \eta|n=\nu|n and for S\subseteq {}^\omega 2 we define M_S = M|\{(\alpha,\eta):[\eta\in S\Rightarrow\alpha<\omega_1] and [\eta\in {}^\omega 2 \backslash S\Rightarrow\alpha<\omega]\}. Hence, it seems to me we should try L_{\infty,\aleph_\epsilon}(d.q.) (essentially, in {\mathfrak C} we can quantify over sets which are included in the algebraic closure of finite sets), and Harrington accepts this interpretation. Here the conjecture is proved for \aleph_\epsilon-saturated models. I.e., the main theorem is M\equiv_{{\mathcal L}_{\infty,\aleph_\epsilon}(d.q.)}N \Leftrightarrow M \cong N for \aleph_\epsilon-saturated models of a superstable countable (first order) theory T without dop.
keywords: M: cla, (mod)
Sh:402
Shelah, S. (1999). Borel Whitehead groups. Math. Japon., 50(1), 121–130. arXiv: math/9809198 MR: 1710476
type: article
abstract: We investigate the Whiteheadness of Borel abelian groups (\aleph_1-free, wlog, as otherwise this is trivial). We show that CH (and even WCH) implies any such abelian group is free, and always \aleph_2-free.
keywords: S: dst, O: alg, (stal), (wh)
Sh:403
Abraham, U., & Shelah, S. (1993). A \Delta^2_2 well-order of the reals and incompactness of L(Q^\mathrm{MM}). Ann. Pure Appl. Logic, 59(1), 1–32. arXiv: math/9812115 DOI: 10.1016/0168-0072(93)90228-6 MR: 1197203
type: article
abstract: A forcing poset of size 2^{2^{\aleph_1}} which adds no new reals is described and shown to provide a \Delta^2_2 definable well-order of the reals (in fact, any given relation of the reals may be so encoded in some generic extension). The encoding of this well-order is obtained by playing with products of Aronszajn trees: Some products are special while other are Suslin trees. The paper also deals with the Magidor-Malitz logic: it is consistent that this logic is highly non compact.
keywords: S: for, S: str, S: dst, (set)
Sh:404
Givant, S. R., & Shelah, S. (1994). Universal theories categorical in power and \kappa-generated models. Ann. Pure Appl. Logic, 69(1), 27–51. arXiv: math/9401213 DOI: 10.1016/0168-0072(94)90018-3 MR: 1301605
type: article
abstract: We investigate a notion called uniqueness in power \kappa that is akin to categoricity in power \kappa, but is based on the cardinality of the generating sets of models instead of on the cardinality of their universes. The notion is quite useful for formulating categoricity-like questions regarding powers below the cardinality of a theory. We prove, for (uncountable) universal theories T, that if T is \kappa-unique for one uncountable \kappa, then it is \kappa-unique for every uncountable \kappa; in particular, it is categorical in powers greater than the cardinality of T.
keywords: M: cla, (mod), (cat), (ua)
Sh:405
Shelah, S. (1994). Vive la différence. II. The Ax-Kochen isomorphism theorem. Israel J. Math., 85(1-3), 351–390. arXiv: math/9304207 DOI: 10.1007/BF02758648 MR: 1264351
type: article
abstract: We show in §1 that the Ax-Kochen isomorphism theorem requires the continuum hypothesis. Most of the applications of this theorem are insensitive to set theoretic considerations. (A probable exception is the work of Moloney.) In §2 we give an unrelated result on cuts in models of Peano arithmetic which answers a question on the ideal structure of countable ultraproducts of {\mathbb Z}. In §1 we also answer a question of Keisler and Schmerl regarding Scott complete ultrapowers of {\mathbb R}.
keywords: S: for, (set-mod), (up)
Sh:406
Fremlin, D. H., & Shelah, S. (1993). Pointwise compact and stable sets of measurable functions. J. Symbolic Logic, 58(2), 435–455. arXiv: math/9209218 DOI: 10.2307/2275214 MR: 1233919
See [Sh:406a]
type: article
abstract: In a series of papers, M.Talagrand, the second author and others investigated at length the properties and structure of pointwise compact sets of measurable functions. A number of problems, interesting in themselves and important for the theory of Pettis integration, were solved subject to various special axioms. It was left unclear just how far the special axioms were necessary. In particular, several results depended on the fact that it is consistent to suppose that every countable relatively pointwise compact set of Lebesgue measurable functions is ‘stable’ in Talagrand’s sense; the point being that stable sets are known to have a variety of properties not shared by all pointwise compact sets. In the present paper we present a model of set theory in which there is a countable relatively pointwise compact set of Lebesgue measurable functions which is not stable, and discuss the significance of this model in relation to the original questions. A feature of our model which may be of independent interest is the following: in it, there is a closed negligible set Q\subseteq [0,1]^2 such that whenever D\subseteq [0,1] has outer measure 1 then the set Q^{-1}[D]=\{x:(\exists y\in D)((x,y)\in Q)\} has inner measure 1.
keywords: S: for, S: str, (set), (leb), (creatures)
Sh:407
Shelah, S. (1992). CON(\mathfrak u>\mathfrak i). Arch. Math. Logic, 31(6), 433–443. DOI: 10.1007/BF01277485 MR: 1175937
type: article
abstract: (none)
keywords: S: for, S: str, (set), (inv)
Sh:408
Kojman, M., Perles, M. A., & Shelah, S. (1990). Sets in a Euclidean space which are not a countable union of convex subsets. Israel J. Math., 70(3), 313–342. DOI: 10.1007/BF02801467 MR: 1074495
type: article
abstract: (none)
keywords: S: str, O: top
Sh:409
Kojman, M., & Shelah, S. (1992). Nonexistence of universal orders in many cardinals. J. Symbolic Logic, 57(3), 875–891. arXiv: math/9209201 DOI: 10.2307/2275437 MR: 1187454
type: article
abstract: We give an example of a first order theory T with countable D(T) which cannot have a universal model at \aleph_1 without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove – again in ZFC – that for a large class of cardinals there is no universal linear order (e.g. in every \aleph_1< \lambda< 2^{\aleph_0}). In fact, what we show is that if there is a universal linear order at a regular \lambda and its existence is not a result of a trivial cardinal arithmetical reason, then \lambda “resembles” \aleph_1 – a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the non existence of a universal linear order, we show the non-existence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).
keywords: S: ico, (set), (app(pcf)), (univ)
Sh:410
Shelah, S. (1993). More on cardinal arithmetic. Arch. Math. Logic, 32(6), 399–428. arXiv: math/0406550 DOI: 10.1007/BF01270465 MR: 1245523
type: article
abstract: (none)
keywords: S: pcf, (set)
Sh:411
Lifsches, S., & Shelah, S. (1992). The monadic theory of (\omega_2,<) may be complicated. Arch. Math. Logic, 31(3), 207–213. DOI: 10.1007/BF01269949 MR: 1147743
type: article
abstract: (none)
keywords: S: for, (mod), (mon)
Sh:412
Gitik, M., & Shelah, S. (1993). More on simple forcing notions and forcings with ideals. Ann. Pure Appl. Logic, 59(3), 219–238. DOI: 10.1016/0168-0072(93)90094-T MR: 1213273
type: article
abstract: (none)
keywords: S: str, (set), (leb), (meag)
Sh:413
Shelah, S. (2003). More Jonsson algebras. Arch. Math. Logic, 42(1), 1–44. arXiv: math/9809199 DOI: 10.1007/s001530100119 MR: 1953112
type: article
abstract: We prove that on many inaccessible there is a Jonsson algebra, so e.g. the first regular Jonsson cardinal \lambda is \lambda\times\omega-Mahlo. We give further restrictions on successor of singulars which are Jonsson cardinals. E.g. there is a Jonsson algebra of cardinality \beth^+_\omega. Lastly, we give further information on guessing of clubs.
keywords: S: pcf, (set)
Sh:414
Komjáth, P., & Shelah, S. (1993). A consistent edge partition theorem for infinite graphs. Acta Math. Hungar., 61(1-2), 115–120. DOI: 10.1007/BF01872104 MR: 1200965
type: article
abstract: (none)
keywords: S: for, (set), (pc), (graph)
Sh:415
Koppelberg, S., & Shelah, S. (1995). Densities of ultraproducts of Boolean algebras. Canad. J. Math., 47(1), 132–145. arXiv: math/9404226 DOI: 10.4153/CJM-1995-007-0 MR: 1319693
type: article
abstract: We answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density \pi(A) resp. the topological density d(A) of a Boolean algebra A commutes with formation of ultraproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra.
keywords: S: for, (ba), (up)
Sh:416
Mekler, A. H., Shelah, S., & Väänänen, J. A. (1993). The Ehrenfeucht-Fraïssé-game of length \omega_1. Trans. Amer. Math. Soc., 339(2), 567–580. arXiv: math/9305204 DOI: 10.2307/2154287 MR: 1191613
type: article
abstract: Let ({\mathcal A}) and ({\mathcal B}) be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraı̈ssé-game of length \omega_1 of {\mathcal A} and {\mathcal B} which we denote by G_{\omega_1}({\mathcal A},{\mathcal B}). This game is like the ordinary Ehrenfeucht-Fraı̈ssé-game of L_{\omega\omega} except that there are \omega_1 moves. It is clear that G_{\omega_1}({\mathcal A},{\mathcal B}) is determined if \mathcal A and {\mathcal B} are of cardinality \leq\aleph_1. We prove the following results:

Theorem A: If V=L, then there are models \mathcal A and \mathcal B of cardinality \aleph_2 such that the game G_{\omega_1}({\mathcal A},{\mathcal B}) is non-determined.

Theorem B: If it is consistent that there is a measurable cardinal, then it is consistent that G_{\omega_1}({\mathcal A},{\mathcal B}) is determined for all \mathcal A and \mathcal B of cardinality \le\aleph_2.

Theorem C: For any \kappa\geq\aleph_3 there are \mathcal A and \mathcal B of cardinality \kappa such that the game G_{\omega_1}({\mathcal A},{\mathcal B}) is non-determined.
keywords: M: odm, (set-mod)
Sh:417
Mekler, A. H., Shelah, S., & Spinas, O. (1996). The essentially free spectrum of a variety. Israel J. Math., 93, 1–8. arXiv: math/9411234 DOI: 10.1007/BF02761091 MR: 1380631
type: article
abstract: We partially prove a conjecture from [MkSh:366] which says that the spectrum of almost free, essentially free, non-free algebras in a variety is either empty or consists of the class of all successor cardinals.
keywords: O: alg, (stal), (af)
Sh:418
Mekler, A. H., & Shelah, S. (1993). Every coseparable group may be free. Israel J. Math., 81(1-2), 161–178. arXiv: math/9305205 DOI: 10.1007/BF02761303 MR: 1231184
type: article
abstract: We show that if 2^{\aleph_0} Cohen reals are added to the universe, then for every reduced non-free torsion-free abelian group A of cardinality less than the continuum, there is a prime p so that {\rm Ext}_p(A, {\mathbb Z}) \neq 0. In particular if it is consistent that there is a supercompact cardinal, then it is consistent (even with weak CH) that every coseparable group is free. The use of some large cardinal hypothesis is needed.
keywords: S: for, (ab), (wh)
Sh:419
Shelah, S., & Stanley, L. J. (2000). Filters, Cohen sets and consistent extensions of the Erdős-Dushnik-Miller theorem. J. Symbolic Logic, 65(1), 259–271. arXiv: math/9709228 DOI: 10.2307/2586535 MR: 1782118
type: article
abstract: We present two different types of models where, for certain singular cardinals \lambda of uncountable cofinality, \lambda\rightarrow(\lambda,\omega+1)^2, although \lambda is not a strong limit cardinal. We announce, here, and will present in a subsequent paper, that, for example, consistently, \aleph_{\omega_1}\not\rightarrow (\aleph_{\omega_1},\omega+1)^2 and consistently, 2^{\aleph_0}\not\rightarrow (2^{\aleph_0},\omega+1)^2.
keywords: S: for, (set), (pc)
Sh:420
Shelah, S. (1993). Advances in cardinal arithmetic. In Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), Vol. 411, Kluwer Acad. Publ., Dordrecht, pp. 355–383. arXiv: 0708.1979 MR: 1261217
type: article
abstract: (none)
keywords: S: pcf, (set)
Sh:422
Eklof, P. C., & Shelah, S. (1993). On a conjecture regarding nonstandard uniserial modules. Trans. Amer. Math. Soc., 340(1), 337–351. arXiv: math/9308211 DOI: 10.2307/2154560 MR: 1159192
type: article
abstract: We consider the question of which valuation domains (of cardinality \aleph _1) have non-standard uniserial modules. We show that a criterion conjectured by Osofsky is independent of ZFC + GCH.
keywords: S: for, (ab), (iter)
Sh:424
Shelah, S. (1993). On CH + 2^{\aleph_1}\to(\alpha)^2_2 for \alpha<\omega_2. In Logic Colloquium ’90 (Helsinki, 1990), Vol. 2, Springer, Berlin, pp. 281–289. arXiv: math/9308212 MR: 1279847
type: article
abstract: We prove the consistency of “CH + 2^{\aleph_1} is arbitrarily large + 2^{\aleph_1}\not\rightarrow(\omega_1\times\omega)^2_2”. If fact, we can get 2^{\aleph_1}\not\rightarrow[\omega_1\times\omega]^2_{\aleph_0}. In addition to this theorem, we give generalizations to other cardinals.
keywords: S: for, (set), (pc)
Sh:425
Shelah, S., & Stanley, L. J. (1995). The combinatorics of combinatorial coding by a real. J. Symbolic Logic, 60(1), 36–57. arXiv: math/9402214 DOI: 10.2307/2275508 MR: 1324500
type: article
abstract: We lay the combinatorial foundations for [ShSt:340] by setting up and proving the essential properties of the coding apparatus for singular cardinals. We also prove another result concerning the coding apparatus for inaccessible cardinals.
keywords: S: for, (set)
Sh:426
Eklof, P. C., Mekler, A. H., & Shelah, S. (1993). On coherent systems of projections for \aleph_1-separable groups. Comm. Algebra, 21(1), 343–353. arXiv: math/9308213 DOI: 10.1080/00927879208824564 MR: 1194564
type: article
abstract: It is proved consistent with either CH or the negation of CH that there is an \aleph_1-separable group of cardinality \aleph_1 which does not have a coherent system of projections. It had previously been shown that it is consistent with \negCH that every \aleph_1-separable group of cardinality \aleph_1 does have a coherent system of projections.
keywords: S: for, (ab), (cont)
Sh:427
Shelah, S., & Steprāns, J. (1994). Somewhere trivial autohomeomorphisms. J. London Math. Soc. (2), 49(3), 569–580. arXiv: math/9308214 DOI: 10.1112/jlms/49.3.569 MR: 1271551
type: article
abstract: It is shown to be consistent that there is a non-trivial autohomeomorphism of \beta{\bf N} while all such autohomeomorphisms are trivial on some open set. The model used is one due to Velickovic in which, coincidentally, Martin’s Axiom also holds.
keywords: S: str, (set)
Sh:428
Hyttinen, T., Shelah, S., & Tuuri, H. (1993). Remarks on strong nonstructure theorems. Notre Dame J. Formal Logic, 34(2), 157–168. DOI: 10.1305/ndjfl/1093634649 MR: 1231281
type: article
abstract: (none)
keywords: M: non, (set-mod)
Sh:429
Shelah, S. (1991). Multi-dimensionality. Israel J. Math., 74(2-3), 281–288. DOI: 10.1007/BF02775792 MR: 1135240
type: article
abstract: (none)
keywords: M: cla, (mod), (sta)
Sh:430
Shelah, S. (1996). Further cardinal arithmetic. Israel J. Math., 95, 61–114. arXiv: math/9610226 DOI: 10.1007/BF02761035 MR: 1418289
type: article
abstract: We continue the investigations in the author’s book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of ({\mathcal S}_{\le\aleph_0}(\kappa),\subseteq) for \kappa real valued measurable (Section 3), densities of box products (Section 5,3), prove the equality cov(\lambda,\lambda,\theta^+,2)=pp(\lambda) in more cases even when cf(\lambda)=\aleph_0 (Section 1), deal with bounds of pp(\lambda) for \lambda limit of inaccessible (Section 4) and give proofs to various claims I was sure I had already written but did not find (Section 6).
keywords: S: pcf
Sh:431
Komjáth, P., & Shelah, S. (1994). A note on a set-mapping problem of Hajnal and Máté. Period. Math. Hungar., 28(1), 39–42. DOI: 10.1007/BF01876368 MR: 1310757
type: article
abstract: (none)
keywords: S: ico, S: for, (set)
Sh:432
Shelah, S., & Spencer, J. H. (1994). Random sparse unary predicates. Random Structures Algorithms, 5(3), 375–394. arXiv: math/9401214 DOI: 10.1002/rsa.3240050302 MR: 1277609
type: article
abstract: The main result is the following

Theorem: Let p=p(n) be such that p(n)\in[0,1] for all n and either p(n)\ll n^{-1} or for some positive integer k, n^{-1/k}\ll p(n)\ll n^{-1/(k+1)} or for all \epsilon>0, n^{-\epsilon}\ll p(n) and n^{-\epsilon}\ll 1-p(n) or for some positive integer k, n^{-1/k}\ll 1-p(n)\ll n^{-1/(k+1)} or 1-p(n)\ll n^{-1}. Then p(n) satisfies the Zero-One Law for circular unary predicates. Inversely, if p(n) falls into none of the above categories then it does not satisfy the Zero-One Law for circular unary predicates.
keywords: M: odm, O: fin, (fmt)
Sh:433
Magidor, M., & Shelah, S. (1998). Length of Boolean algebras and ultraproducts. Math. Japon., 48(2), 301–307. arXiv: math/9805145 MR: 1674385
type: article
abstract: We prove the consistency with ZFC of “the length of an ultraproduct of Boolean algebras is smaller than the ultraproduct of the lengths”. Similarly for some other cardinal invariants of Boolean algebras.
keywords: S: ico, S: for, (set), (ba), (large), (up)
Sh:434
Bartoszyński, T., Goldstern, M., Judah, H. I., & Shelah, S. (1993). All meager filters may be null. Proc. Amer. Math. Soc., 117(2), 515–521. arXiv: math/9301206 DOI: 10.2307/2159190 MR: 1111433
type: article
abstract: We show that it is consistent with ZFC that all filters which have the Baire property are Lebesgue measurable. We also show that the existence of a Sierpinski set implies that there exists a nonmeasurable filter which has the Baire property.
keywords: S: str, (set), (leb), (meag)
Sh:435
Shelah, S., & Łuczak, T. (1995). Convergence in homogeneous random graphs. Random Structures Algorithms, 6(4), 371–391. arXiv: math/9501221 DOI: 10.1002/rsa.3240060402 MR: 1368840
type: article
abstract: For a sequence \bar{p}=(p(1),p(2),\dots) let G(n,\bar{p}) denote the random graph with vertex set \{1,2,\dots,n\} in which two vertices i, j are adjacent with probability p(|i-j|), independently for each pair. We study how the convergence of probabilities of first order properties of G(n,\bar{p}), can be affected by the behaviour of \bar{p} and the strength of the language we use.
keywords: M: odm, O: fin, (fmt), (graph)
Sh:436
Bartoszyński, T., & Shelah, S. (1992). Intersection of <2^{\aleph_0} ultrafilters may have measure zero. Arch. Math. Logic, 31(4), 221–226. arXiv: math/9904068 DOI: 10.1007/BF01794979 MR: 1155033
type: article
abstract: (none)
keywords: S: for, S: str, (leb), (inv)
Sh:437
Burke, M. R., & Shelah, S. (1992). Linear liftings for noncomplete probability spaces. Israel J. Math., 79(2-3), 289–296. arXiv: math/9201252 DOI: 10.1007/BF02808221 MR: 1248919
type: article
abstract: We show that it is consistent with ZFC that L^\infty(Y,{\mathcal B},\nu) has no linear lifting for many non-complete probability spaces (Y,{\mathcal B},\nu), in particular for Y=[0,1]^A, {\mathcal B}= Borel subsets of Y, \nu= usual Radon measure on {\mathcal B}.
keywords: S: str, (set), (leb)
Sh:438
Goldstern, M., Judah, H. I., & Shelah, S. (1993). Strong measure zero sets without Cohen reals. J. Symbolic Logic, 58(4), 1323–1341. arXiv: math/9306214 DOI: 10.2307/2275146 MR: 1253925
type: article
abstract: If ZFC is consistent, then each of the following are consistent with ZFC + 2^{{\aleph_0}}=\aleph_2:

1.) X subseteq R is of strong measure zero iff |X| \leq \aleph_1 + there is a generalized Sierpinski set.

2.) The union of \aleph_1 many strong measure zero sets is a strong measure zero set + there is a strong measure zero set of size \aleph_2.
keywords: S: str, (set), (leb)
Sh:439
Bartoszyński, T., & Shelah, S. (1992). Closed measure zero sets. Ann. Pure Appl. Logic, 58(2), 93–110. arXiv: math/9905123 DOI: 10.1016/0168-0072(92)90001-G MR: 1186905
type: article
abstract: (none)
keywords: S: str, (set), (leb)
Sh:440
Comfort, W. W., Kato, A., & Shelah, S. (1993). Topological partition relations of the form \omega^\ast\to(Y)^1_2. In Papers on general topology and applications (Madison, WI, 1991), Vol. 704, New York Acad. Sci., New York, pp. 70–79. arXiv: math/9305206 DOI: 10.1111/j.1749-6632.1993.tb52510.x MR: 1277844
type: article
abstract: Theorem: The topological partition relation \omega^{*}\rightarrow(Y)^{1}_{2}

(a) fails for every space Y with |Y|\geq 2^{\rm \bf c};

(b) holds for Y discrete if and only if |Y|\leq c;

(c) holds for certain non-discrete P-spaces Y;

(d) fails for Y=\omega\cup\{p\} with p\in\omega^{*};

(e) fails for Y infinite and countably compact.
keywords: S: ico, O: top, (pc), (gt)
Sh:441
Eklof, P. C., Mekler, A. H., & Shelah, S. (1992). Uniformization and the diversity of Whitehead groups. Israel J. Math., 80(3), 301–321. arXiv: math/9204219 DOI: 10.1007/BF02808073 MR: 1202574
type: article
abstract: The connections between Whitehead groups and uniformization properties were investigated by the third author in [Sh:98]. In particular it was essentially shown there that there is a non-free Whitehead (respectively, \aleph_1-coseparable) group of cardinality \aleph_1 if and only if there is a ladder system on a stationary subset of \omega_1 which satisfies 2-uniformization (respectively, omega-uniformization). These techniques allowed also the proof of various independence and consistency results about Whitehead groups, for example that it is consistent that there is a non-free Whitehead group of cardinality \aleph_1 but no non-free \aleph_1-coseparable group. However, some natural questions remained open, among them the following two: (i) Is it consistent that the class of W-groups of cardinality \aleph_1 is exactly the class of strongly \aleph_1-free groups of cardinality \aleph_1? (ii) If every strongly \aleph_1-free group of cardinality \aleph_1 is a W-group, are they also all \aleph_1-coseparable? In this paper we use the techniques of uniformization to answer the first question in the negative and give a partial affirmative answer to the second question.
keywords: S: for, O: alg, (ab), (stal), (wh), (unif)
Sh:442
Eklof, P. C., Mekler, A. H., & Shelah, S. (1994). Hereditarily separable groups and monochromatic uniformization. Israel J. Math., 88(1-3), 213–235. arXiv: math/0406552 DOI: 10.1007/BF02937512 MR: 1303496
type: article
abstract: We give a combinatorial equivalent to the existence of a non-free hereditarily separable group of cardinality \aleph_1. This can be used, together with a known combinatorial equivalent of the existence of a non-free Whitehead group, to prove that it is consistent that every Whitehead group is free but not every hereditarily separable group is free. We also show that the fact that {\mathbb Z} is a p.i.d. with infinitely many primes is essential for this result.
keywords: S: for, O: alg, (ab), (stal), (wh), (unif)
Sh:443
Diestel, R., Shelah, S., & Steprāns, J. (1994). Dominating functions and graphs. J. London Math. Soc. (2), 49(1), 16–24. arXiv: math/9308215 DOI: 10.1112/jlms/49.1.16 MR: 1253008
type: article
abstract: A graph is called dominating if its vertices can be labelled with integers in such a way that for every function f:\omega\to\omega the graph contains a ray whose sequence of labels eventually exceeds f. We obtain a characterization of these graphs by producing a small family of dominating graphs with the property that every dominating graph must contain some member of the family.
keywords: S: ico, (set), (graph)
Sh:444
Huck, A., Niedermeyer, F., & Shelah, S. (1994). Large \kappa-preserving sets in infinite graphs. J. Graph Theory, 18(4), 413–426. DOI: 10.1002/jgt.3190180411 MR: 1277518
type: article
abstract: (none)
keywords: S: ico, (set), (graph)
Sh:445
Shelah, S. (1995). Every null-additive set is meager-additive. Israel J. Math., 89(1-3), 357–376. arXiv: math/9406228 DOI: 10.1007/BF02808209 MR: 1324470
type: article
abstract: We show that every null-additive set is meager-additive, where:

(1) a set X\subseteq 2^\omega is null-additive if for every Lebesgue null set A\subseteq 2^\omega, X+A is null too;

(2) we say that X\subseteq 2^\omega is meager-additive if for every A\subseteq 2^\omega which is meager also X+A is meager.
keywords: S: ico, S: str, (set), (leb), (meag)
Sh:447
Kojman, M., & Shelah, S. (1992). The universality spectrum of stable unsuperstable theories. Ann. Pure Appl. Logic, 58(1), 57–72. arXiv: math/9201253 DOI: 10.1016/0168-0072(92)90034-W MR: 1169786
type: article
abstract: It is shown that if T is stable unsuperstable, and \aleph_1< \lambda=cf(\lambda)< 2^{\aleph_0}, or 2^{\aleph_0} < \mu^+< \lambda=cf(\lambda)< \mu^{\aleph_0} then T has no universal model in cardinality \lambda, and if e.g. \aleph_\omega < 2^{\aleph_0} then T has no universal model in \aleph_\omega. These results are generalized to \kappa=cf(\kappa) < \kappa(T) in the place of \aleph_0. Also: if there is a universal model in \lambda>|T|, T stable and \kappa< \kappa(T) then there is a universal tree of height \kappa+1 in cardinality \lambda.
keywords: M: cla, M: non, S: ico, (app(pcf)), (univ), (set-mod)
Sh:448
Goldstern, M., & Shelah, S. (1993). Many simple cardinal invariants. Arch. Math. Logic, 32(3), 203–221. arXiv: math/9205208 DOI: 10.1007/BF01375552 MR: 1201650
type: article
abstract: For g < f in \omega^\omega we define c(f,g) be the least number of uniform trees with g-splitting needed to cover a uniform tree with f-splitting. We show that we can simultaneously force \aleph_1 many different values for different functions (f,g). In the language of Blass: There may be \aleph_1 many distinct uniform \bf\Pi^0_1 characteristics.
keywords: S: for, S: str, (set), (inv)
Sh:449
Kojman, M., & Shelah, S. (1993). \mu-complete Souslin trees on \mu^+. Arch. Math. Logic, 32(3), 195–201. arXiv: math/9306215 DOI: 10.1007/BF01375551 MR: 1201649
type: article
abstract: We prove that \mu=\mu^{<\mu}, 2^\mu=\mu^+ and “there is a non reflecting stationary subset of \mu^+ composed of ordinals of cofinality < \mu” imply that there is a \mu-complete Souslin tree on \mu^+.
keywords: S: ico, (set)
Sh:450
Melles, G., & Shelah, S. (1994). A saturated model of an unsuperstable theory of cardinality greater than its theory has the small index property. Proc. London Math. Soc. (3), 69(3), 449–463. arXiv: math/9308216 DOI: 10.1112/plms/s3-69.3.449 MR: 1289859
type: article
abstract: A model M of cardinality \lambda is said to have the small index property if for every G\subseteq Aut(M) such that [Aut(M):G]\leq\lambda there is an A\subseteq M with |A|< \lambda such that Aut_A(M)\subseteq G. We show that if M^* is a saturated model of an unsuperstable theory of cardinality > Th(M), then M^* has the small index property.
keywords: S: ico, (mod), (auto)
Sh:451
Lascar, D., & Shelah, S. (1993). Uncountable saturated structures have the small index property. Bull. London Math. Soc., 25(2), 125–131. DOI: 10.1112/blms/25.2.125 MR: 1204064
type: article
abstract: (none)
keywords: S: ico, (mod), (auto)
Sh:452
Melles, G., & Shelah, S. (1994). \mathrm{Aut}(M) has a large dense free subgroup for saturated M. Bull. London Math. Soc., 26(4), 339–344. arXiv: math/9304201 DOI: 10.1112/blms/26.4.339 MR: 1302066
type: article
abstract: We prove that for a stable theory T, if M is a saturated model of T of cardinality \lambda where \lambda>|T|, then Aut(M) has a dense free subgroup on 2^{\lambda} generators. This affirms a conjecture of Hodges.
keywords: M: odm, (mod), (auto)
Sh:453
Mekler, A. H., Schipperus, R. J., Shelah, S., & Truss, J. K. (1993). The random graph and automorphisms of the rational world. Bull. London Math. Soc., 25(4), 343–346. DOI: 10.1112/blms/25.4.343 MR: 1222726
type: article
abstract: (none)
keywords: M: odm, (auto)
Sh:454
Shelah, S. (1993). Number of open sets for a topology with a countable basis. Israel J. Math., 83(3), 369–374. arXiv: math/9308217 DOI: 10.1007/BF02784064 MR: 1239070
type: article
abstract: Let T be the family of open subsets of a topological space (not necessarily Hausdorff or even T_0). We prove that if T has a countable base and is not countable, then T has cardinality at least continuum.
keywords: S: ico, S: pcf, O: top, (inv), (gt)
Sh:454a
Shelah, S. (1994). Cardinalities of topologies with small base. Ann. Pure Appl. Logic, 68(1), 95–113. arXiv: math/9403219 DOI: 10.1016/0168-0072(94)90049-3 MR: 1278551
type: article
abstract: Let T be the family of open subsets of a topological space (not necessarily Hausdorff or even T_0). We prove that if T has a base of cardinality \leq \mu, \lambda\leq \mu< 2^\lambda, \lambda strong limit of cofinality \aleph_0, then T has cardinality \leq \mu or \geq 2^\lambda. This is our main conclusion. First we prove it under some set theoretic assumption, which is clear when \lambda=\mu; then we eliminate the assumption by a theorem on pcf from [Sh 460] motivated originally by this. Next we prove that the simplest examples are the basic ones; they occur in every example (for \lambda=\aleph_0 this fulfill a promise from [Sh 454]). The main result for the case \lambda=\aleph_0 was proved in [Sh 454].
keywords: S: pcf
Sh:455
Kojman, M., & Shelah, S. (1995). Universal abelian groups. Israel J. Math., 92(1-3), 113–124. arXiv: math/9409207 DOI: 10.1007/BF02762072 MR: 1357747
type: article
abstract: We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example:

Theorem: For n\ge 2, there is a purely universal separable p-group in \aleph_n if, and only if, {2^{\aleph_0}}\leq \aleph_n.
keywords: O: alg, (ab), (stal), (app(pcf)), (univ)
Sh:456
Shelah, S. (1996). Universal in (<\lambda)-stable abelian group. Math. Japon., 44(1), 1–9. arXiv: math/9509225 MR: 1402794
type: article
abstract: A characteristic result is that if 2^{\aleph_0}< \mu< \mu^+< \lambda= cf(\lambda)< \mu^{\aleph_0}, then among the separable reduced p-groups of cardinality \lambda which are (< \lambda)-stable there is no universal one.
keywords: S: ico, O: alg, (ab), (stal), (app(pcf)), (univ)
Sh:457
Shelah, S. (1993). The universality spectrum: consistency for more classes. In Combinatorics, Paul Erdős is eighty, Vol. 1, János Bolyai Math. Soc., Budapest, pp. 403–420. arXiv: math/9412229 MR: 1249724
type: article
abstract: We deal with consistency results for the existence of universal models in natural classes of models (more exactly–a somewhat weaker version). We apply a result on quite general family to T_{\rm feq} and to the class of triangle-free graphs.
keywords: S: ico, S: for, (graph), (univ), (set-mod)
Sh:458
Abraham, U., & Shelah, S. (1996). Martin’s axiom and \Delta^2_1 well-ordering of the reals. Arch. Math. Logic, 35(5-6), 287–298. arXiv: math/9408214 DOI: 10.1007/s001530050046 MR: 1420259
type: article
abstract: Assuming an inaccessible cardinal \kappa, there is a generic extension in which MA + 2^{\aleph_0} = \kappa holds and the reals have a \Delta^2_1 well-ordering.
keywords: S: for, S: dst, (set)
Sh:459
Baumgartner, J. E., Shelah, S., & Thomas, S. (1993). Maximal subgroups of infinite symmetric groups. Notre Dame J. Formal Logic, 34(1), 1–11. DOI: 10.1305/ndjfl/1093634559 MR: 1213842
type: article
abstract: (none)
keywords: S: ico
Sh:460
Shelah, S. (2000). The generalized continuum hypothesis revisited. Israel J. Math., 116, 285–321. arXiv: math/9809200 DOI: 10.1007/BF02773223 MR: 1759410
type: article
abstract: We argue that we solved Hilbert’s first problem positively (after reformulating it just to avoid the known consistency results) and give some applications. Let \lambda to the revised power of \kappa, denoted \lambda^{[\kappa]}, be the minimal cardinality of a family of subsets of \lambda each of cardinality \kappa such that any other subset of \lambda of cardinality \kappa is included in the union of <\kappa members of the family. The main theorem says that almost always this revised power is equal to \lambda. Our main result is The Revised GCH Theorem: Assume we fix an uncountable strong limit cardinal mu (i.e. \mu>\aleph_0, (\forall \theta<\mu)[2^\theta<\mu]); e.g. \mu=\beth_\omega. Then for every \lambda \geq \mu, for some \kappa<\mu, we have:

(a) \kappa \leq \theta < \mu \geq \lambda^{[\theta]}= \lambda and

(b) there is a family {P} of \lambda subsets of \lambda, each of cardinality < \mu, such that every subset of \lambda of cardinality \mu is equal to the union of < \kappa members of {P}.
keywords: S: pcf
Sh:461
Eklof, P. C., & Shelah, S. (1993). Explicitly nonstandard uniserial modules. J. Pure Appl. Algebra, 86(1), 35–50. arXiv: math/9301207 DOI: 10.1016/0022-4049(93)90151-I MR: 1213152
type: article
abstract: A new construction is given of non-standard uniserial modules over certain valuation domains; the construction resembles that of a special Aronszajn tree in set theory. A consequence is the proof of a sufficient condition for the existence of non-standard uniserial modules; this is a theorem of ZFC which complements an earlier independence result.
keywords: S: ico, (ab)
Sh:462
Shelah, S. (1997). \sigma-entangled linear orders and narrowness of products of Boolean algebras. Fund. Math., 153(3), 199–275. arXiv: math/9609216 DOI: 10.4064/fm-153-3-199-275 MR: 1467577
type: article
abstract: We investigate \sigma-entangled linear orders and narrowness of Boolean algebras. We show existence of \sigma-entangled linear orders in many cardinals, and we build Boolean algebras with neither large chains nor large pies. We study the behavior of these notions in ultraproducts.
keywords: S: ico, (set), (app(pcf)), (linear order)
Sh:463
Shelah, S. (1996). On the very weak 0-1 law for random graphs with orders. J. Logic Comput., 6(1), 137–159. arXiv: math/9507221 DOI: 10.1093/logcom/6.1.137 MR: 1376723
type: article
abstract: Let us draw a graph R on 0,1,...,n-1 by having an edge i,j with probability p_(|i-j|), where \sum_i p_i is finite and let M_n=(n,<,R). For a first order sentence \psi let a^n_\psi be the probability of "M_n satisfies \psi". We prove that the limit of a^n_\psi-a^{n+1}_\psi is 0, as n goes to infinity.
keywords: M: odm, O: fin, (fmt), (graph)
Sh:464
Baldwin, J. T., Laskowski, M. C., & Shelah, S. (1993). Forcing isomorphism. J. Symbolic Logic, 58(4), 1291–1301. arXiv: math/9301208 DOI: 10.2307/2275144 MR: 1253923
type: article
abstract: A forcing extension may create new isomorphisms between two models of a first order theory. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is classifiable if it is superstable and does not have either the dimensional order property or the omitting types order property. Shelah [Sh:c] showed that if a theory T is classifiable then each model of cardinality \lambda is described by a sentence of L_{\infty,\lambda}. In fact this sentence can be chosen in the L^*_{\lambda}. (L^*_{\lambda} is the result of enriching the language L_{\infty,\beth^+} by adding for each \mu<\lambda a quantifier saying the dimension of a dependence structure is greater than \mu.) The truth of such sentences will be preserved by any forcing that does not collapse cardinals \leq\lambda and that adds no new countable subsets of \lambda. Hence, if two models of a classifiable theory of power \lambda are non-isomorphic, they are non-isomorphic after a \lambda-complete forcing. Here we show that the hypothesis of the forcing adding no new countable subsets of \lambda cannot be eliminated. In particular, we show that non-isomorphism of models of a classifiable theory need not be preserved by ccc forcings.
keywords: M: cla, M: non, S: for, (set-mod), (sta)
Sh:465
Shelah, S., & Steprāns, J. (1993). Maximal chains in ^\omega\omega and ultrapowers of the integers. Arch. Math. Logic, 32(5), 305–319. arXiv: math/9204205 DOI: 10.1007/BF01409965 MR: 1223393
See [Sh:465a]
type: article
abstract: Various questions posed by P. Nyikos concerning ultrafilters on \omega and chains in the partial order (\omega,< ^*) are answered. The main tool is the oracle chain condition and variations of it. (Note: Corrections in [Sh:465a])
keywords: S: for, S: str
Sh:466
Jin, R., & Shelah, S. (1993). A model in which there are Jech-Kunen trees but there are no Kurepa trees. Israel J. Math., 84(1-2), 1–16. arXiv: math/9308218 DOI: 10.1007/BF02761687 MR: 1244655
type: article
abstract: By an \omega_1–tree we mean a tree of power \omega_1 and height \omega_1. We call an \omega_1–tree a Jech–Kunen tree if it has \kappa–many branches for some \kappa strictly between \omega_1 and 2^{\omega_1}. In this paper we construct the models of CH plus 2^{\omega_1}>\omega_2, in which there are Jech–Kunen trees and there are no Kurepa trees.
keywords: S: for, S: str, (set), (trees)
Sh:467
Shelah, S. (2002). Zero-one laws for graphs with edge probabilities decaying with distance. I. Fund. Math., 175(3), 195–239. arXiv: math/9606226 DOI: 10.4064/fm175-3-1 MR: 1969657
type: article
abstract: Let G_n be the random graph on [n]=\{1,\ldots,n\} with the possible edge \{i,j\} having probability being p_{|i-j|}= 1/|i-j|^\alpha, \alpha\in (0,1) irrational. We prove that the zero one law (for first order logic) holds. The paper is continued in [Sh:517]
keywords: M: odm, O: fin, (fmt), (graph)
Sh:468
Shelah, S., & Spinas, O. (1996). Gross spaces. Trans. Amer. Math. Soc., 348(10), 4257–4277. arXiv: math/9510215 DOI: 10.1090/S0002-9947-96-01658-3 MR: 1357403
type: article
abstract: A Gross space is a vector space E of infinite dimension over some field F, which is endowed with a symmetric bilinear form \Phi:E^2 \rightarrow F and has the property that every infinite dimensional subspace U\subseteq E satisfies dimU^\perp < dimE. Gross spaces over uncountable fields exist (in certain dimensions). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. Here we continue the investigation of Gross spaces. Among other things we show that if the cardinal invariant b equals \omega _1 a Gross space in dimension \omega _1 exists over every infinite field, and that it is consistent that Gross spaces exist over every infinite field but not over any finite field. We also generalize the notion of a Gross space and construct generalized Gross spaces in ZFC.
keywords: S: for, (stal), (app(pcf))
Sh:469
Jin, R., & Shelah, S. (1992). Planting Kurepa trees and killing Jech-Kunen trees in a model by using one inaccessible cardinal. Fund. Math., 141(3), 287–296. arXiv: math/9211214 DOI: 10.4064/fm-141-3-287-296 MR: 1199241
type: article
abstract: By an \omega_1–tree we mean a tree of power \omega_1 and height \omega_1. Under CH and 2^{\omega_1}>\omega_2 we call an \omega_1–tree a Jech–Kunen tree if it has \kappa many branches for some \kappa strictly between \omega_1 and 2^{\omega_1}. In this paper we prove that, assuming the existence of one inaccessible cardinal,

(1) it is consistent with CH plus 2^{\omega_1}>\omega_2 that there exist Kurepa trees and there are no Jech–Kunen trees,

(2) it is consistent with CH plus 2^{\omega_1}=\omega_4 that only Kurepa trees with \omega_3 many branches exist.
keywords: S: for, S: str, (set), (trees)
Sh:470
Rosłanowski, A., & Shelah, S. (1999). Norms on possibilities. I. Forcing with trees and creatures. Mem. Amer. Math. Soc., 141(671), xii+167. arXiv: math/9807172 DOI: 10.1090/memo/0671 MR: 1613600
type: article
abstract: In this paper we present a systematic study of the method of norms on possibilities of building forcing notions with keeping their properties under full control. This technique allows us to answer several open problems, but on our way to get the solutions we develop various ideas interesting per se. These include a new iterable condition for “not adding Cohen reals” (which has a flavour of preserving special properties of p-points), new intriguing properties of ultrafilters (weaker than being Ramsey but stronger than p–point) and some new applications of variants of the PP–property.
keywords: S: for, S: str, (set), (creatures)
Sh:471
Lifsches, S., & Shelah, S. (1997). Peano arithmetic may not be interpretable in the monadic theory of linear orders. J. Symbolic Logic, 62(3), 848–872. arXiv: math/9308219 DOI: 10.2307/2275575 MR: 1472126
type: article
abstract: Gurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in the monadic second-order theory of short chains (hence, in the monadic second-order theory of the real line). We show here that it is consistent that there is no interpretation even in the monadic second-order theory of all chains.
keywords: S: for, (mod), (iter), (mon)
Sh:472
Shelah, S. (2001). Categoricity of theories in L_{\kappa^\ast,\omega}, when \kappa^\ast is a measurable cardinal. II. Fund. Math., 170(1-2), 165–196. arXiv: math/9604241 DOI: 10.4064/fm170-1-10 MR: 1881375
type: article
abstract: We continue the work of [KlSh:362] and prove that for \lambdasuccessor, a \lambda-categorical theory T in L_{\kappa^*,\omega} is \mu-categorical for every \mu, \mu\leq\lambda which is above the (2^{LS(T)})^+-beth cardinal.
keywords: M: nec
Sh:473
Shelah, S. (1995). Possibly every real function is continuous on a non-meagre set. Publ. Inst. Math. (Beograd) (N.S.), 57(71), 47–60. arXiv: math/9511220 MR: 1387353
type: article
abstract: We prove consistency of the following sentence: “ZFC + every real function is continuous on a non-meagre set”, answering a question of Fremlin.
keywords: S: for, S: str, (set), (meag)
Sh:474
Hyttinen, T., & Shelah, S. (1994). Constructing strongly equivalent nonisomorphic models for unsuperstable theories. Part A. J. Symbolic Logic, 59(3), 984–996. arXiv: math/0406587 DOI: 10.2307/2275922 MR: 1295983
type: article
abstract: We study how equivalent nonisomorphic models an unsuperstable theory can have. We measure the equivalence by Ehrenfeucht-Fraisse games.
keywords: M: non, (mod)
Sh:475
Rosłanowski, A., & Shelah, S. (1996). More forcing notions imply diamond. Arch. Math. Logic, 35(5-6), 299–313. arXiv: math/9408215 DOI: 10.1007/s001530050047 MR: 1420260
type: article
abstract: We prove that the Sacks forcing collapses the continuum onto the dominating number {\mathfrak d}, answering the question of Carlson and Laver. Next we prove that if a proper forcing of the size at most continuum collapses \omega_2 then it forces \diamondsuit_{\omega_1}.
keywords: S: for, S: str, (set), (diam)
Sh:476
Jech, T. J., & Shelah, S. (1996). Possible PCF algebras. J. Symbolic Logic, 61(1), 313–317. arXiv: math/9412208 DOI: 10.2307/2275613 MR: 1380692
type: article
abstract: There exists a family \{B_{\alpha}\}_{\alpha<\omega_1} of sets of countable ordinals such that 1) \max B_{\alpha}=\alpha, 2) if \alpha\in B_{\beta} then B_{\alpha}\subseteq B_{\beta}, 3) if \lambda\leq \alpha and \lambda is a limit ordinal then B_{\alpha}\cap\lambda is not in the ideal generated by the B_{\beta}, 4) \beta< \alpha, and by the bounded subsets of \lambda, 5) there is a partition \{A_n\}_{n=0}^{\infty} of \omega_1 such that for every \alpha and every n, B_{\alpha}\cap A_n is finite.
keywords: S: for, (set)
Sh:477
Brendle, J., Judah, H. I., & Shelah, S. (1992). Combinatorial properties of Hechler forcing. Ann. Pure Appl. Logic, 58(3), 185–199. arXiv: math/9211202 DOI: 10.1016/0168-0072(92)90027-W MR: 1191940
type: article
abstract: Using a notion of rank for Hechler forcing we show:

1) assuming \omega_1^V = \omega_1^L, there is no real in V[d] which is eventually different from the reals in L[d], where d is Hechler over V;

2) adding one Hechler real makes the invariants on the left-hand side of Cichoń’s diagram equal \omega_1 and those on the right-hand side equal 2^\omega and produces a maximal almost disjoint family of subsets of \omega of size \omega_1;

3) there is no perfect set of random reals over V in V[r][d], where r is random over V and d Hechler over V[r].
keywords: S: for, S: str, (set)
Sh:478
Judah, H. I., & Shelah, S. (1994). Killing Luzin and Sierpiński sets. Proc. Amer. Math. Soc., 120(3), 917–920. arXiv: math/9401215 DOI: 10.2307/2160487 MR: 1164145
type: article
abstract: We will kill the old Luzin and Sierpinski sets in order to build a model where U(Meager) = U(Null)=\aleph_1 and there are neither Luzin nor Sierpinski sets. Thus we answer a question of J. Steprans, communicated by S. Todorcevic on route from Evans to MSRI.
keywords: S: for, S: str, (set), (leb), (meag)
Sh:479
Shelah, S. (1996). On Monk’s questions. Fund. Math., 151(1), 1–19. arXiv: math/9601218 MR: 1405517
type: article
abstract: Monk asks (problems 13, 15 in his list; \pi is the algebraic density): "For a Boolean algebra B, \aleph_0\le\theta\le\pi(B), does B have a subalgebra B' with \pi(B')=\theta?" If \theta is regular the answer is easily positive, we show that in general it may be negative, but for quite many singular cardinals - it is positive; the theorems are quite complementary. Next we deal with \pi\chi and we show that the \pi\chi of an ultraproduct of Boolean algebras is not necessarily the ultraproduct of the \pi\chi’s. We also prove that for infinite Boolean algebras A_i (i<\kappa) and a non-principal ultrafilter D on \kappa: if n_i<\aleph_0 for i<\kappa and \mu=\prod_{i<\kappa} n_i/D is regular, then \pi\chi(A)\ge \mu. Here A=\prod_{i<\kappa}A_i/D. By a theorem of Peterson the regularity of \mu is needed.
keywords: S: ico, (ba)
Sh:480
Shelah, S. (1994). How special are Cohen and random forcings, i.e. Boolean algebras of the family of subsets of reals modulo meagre or null. Israel J. Math., 88(1-3), 159–174. arXiv: math/9303208 DOI: 10.1007/BF02937509 MR: 1303493
type: article
abstract: The feeling that those two forcing notions -Cohen and Random-(equivalently the corresponding Boolean algebras Borel(R)/(meager sets), Borel(R)/(null sets)) are special, was probably old and widespread. A reasonable interpretation is to show them unique, or “minimal” or at least characteristic in a family of “nice forcing” like Borel. We shall interpret “nice” as Souslin as suggested by Judah Shelah [JdSh 292]. We divide the family of Souslin forcing to two, and expect that: among the first part, i.e. those adding some non-dominated real, Cohen is minimal (=is below every one), while among the rest random is quite characteristic even unique. Concerning the second class we have weak results, concerning the first class, our results look satisfactory. We have two main results: one (1.14) says that Cohen forcing is “minimal” in the first class, the other (1.10) says that all c.c.c. Souslin forcing have a property shared by Cohen forcing and Random real forcing, so it gives a weak answer to the problem on how special is random forcing, but says much on all c.c.c. Souslin forcing.
keywords: S: str, (set), (pure(for))
Sh:481
Shelah, S. (1996). Was Sierpiński right? III. Can continuum-c.c. times c.c.c. be continuum-c.c.? Ann. Pure Appl. Logic, 78(1-3), 259–269. arXiv: math/9509226 DOI: 10.1016/0168-0072(95)00036-4 MR: 1395402
type: article
abstract: We prove the consistency of: if B_1, B_2 are Boolean algebra satisfying the c.c.c. and the 2^{\aleph_0}-c.c. respectively then B_1 \times B_2 satisfies the 2^{\aleph_0}-c.c.
keywords: S: for, (set), (iter), (pc)
Sh:483
Louveau, A., Shelah, S., & Veličković, B. (1993). Borel partitions of infinite subtrees of a perfect tree. Ann. Pure Appl. Logic, 63(3), 271–281. arXiv: math/9301209 DOI: 10.1016/0168-0072(93)90151-3 MR: 1237234
type: article
abstract: A theorem of Galvin asserts that if the unordered pairs of reals are partitioned into finitely many Borel classes then there is a perfect set P such that all pairs from P lie in the same class. The generalization to n-tuples for n\geq 3 is false. Let us identify the reals with 2^\omega ordered by the lexicographical ordering and define for distinct x,y\in 2^\omega, D(x,y) to be the least n such that x(n)\neq y(n). Let the type of an increasing n-tuple \{x_0,\ldots x_{n-1}\}_< be the ordering <^* on \{0,\ldots,n-2\} defined by i<^*j iff D(x_i,x_{i+1})< D(x_j,x_{j+1}). Galvin proved that for any Borel coloring of triples of reals there is a perfect set P such that the color of any triple from P depends only on its type. Blass proved an analogous result is true for any n. As a corollary it follows that if the unordered n-tuples of reals are colored into finitely many Borel classes there is a perfect set P such that the n-tuples from P meet at most (n-1)! classes. We consider extensions of this result to partitions of infinite increasing sequences of reals. We show, that for any Borel or even analytic partition of all increasing sequences of reals there is a perfect set P such that all strongly increasing sequences from P lie in the same class.
keywords: S: str, (set)
Sh:484
Liu, K., & Shelah, S. (1997). Cofinalities of elementary substructures of structures on \aleph_\omega. Israel J. Math., 99, 189–205. arXiv: math/9604242 DOI: 10.1007/BF02760682 MR: 1469093
type: article
abstract: Let 0<n^*<\omega and f:X\to n^*+1 be a function where X\subseteq\omega\backslash (n^*+1) is infinite. Consider the following set S_f=\{x\subset\aleph_\omega: |x|\le\aleph_{n^*}\ \& \ (\forall n\in X)cf(x\cap\alpha_n)=\aleph_{f(n)}\}. The question, first posed by Baumgartner, is whether S_f is stationary in [\alpha_\omega]^{<\aleph_{n^*+1}}. By a standard result, the above question can also be rephrased as certain transfer property. Namely, S_f is stationary iff for any structure A=\langle\aleph_\omega, \ldots\rangle there’s a B\prec A such that |B|=\aleph_{n^*} and for all n\in X we have cf(B\cap\aleph_n)=\aleph_{f(n)}. In this paper, we are going to prove a few results concerning the above question.
keywords: S: pcf, (set)
Sh:485
Abraham, U., & Shelah, S. (2002). Coding with ladders a well ordering of the reals. J. Symbolic Logic, 67(2), 579–597. arXiv: math/0104195 DOI: 10.2178/jsl/1190150099 MR: 1905156
type: article
abstract: Any model of ZFC + GCH has a generic extension (made with a poset of size \aleph_2) in which the following hold: MA + 2^{\aleph_0}=\aleph_2+ there exists a \Delta^2_1-well ordering of the reals. The proof consists in iterating posets designed to change at will the guessing properties of ladder systems on \omega_1. Therefore, the study of such ladders is a main concern of this article.
keywords: S: for, S: str, S: dst, (set)
Sh:487
Goldstern, M., Repický, M., Shelah, S., & Spinas, O. (1995). On tree ideals. Proc. Amer. Math. Soc., 123(5), 1573–1581. arXiv: math/9311212 DOI: 10.2307/2161150 MR: 1233972
type: article
abstract: Let l^0 and m^0 be the ideals associated with Laver and Miller forcing, respectively. We show that {\bf add }(l^0) < {\bf cov}(l^0) and {\bf add }(m^0) < {\bf cov}(m^0) are consistent. We also show that both Laver and Miller forcing collapse the continuum to a cardinal \le {\bf h}.
keywords: S: for, S: str, (set)
Sh:488
Halbeisen, L. J., & Shelah, S. (1994). Consequences of arithmetic for set theory. J. Symbolic Logic, 59(1), 30–40. arXiv: math/9308220 DOI: 10.2307/2275247 MR: 1264961
type: article
abstract: In this paper, we consider certain cardinals in ZF (set theory without AC, the Axiom of Choice). In ZFC (set theory with AC), given any cardinals \mathcal{C} and \mathcal{D}, either \mathcal{C} \leq \mathcal{D} or \mathcal{D} \leq \mathcal{C}. However, in ZF this is no longer so. For a given infinite set A consider Seq (A), the set of all sequences of A without repetition. We compare |Seq (A)|, the cardinality of this set, to |{\mathcal {P}}(A)|, the cardinality of the power set of A.

What is provable about these two cardinals in ZF? The main result of this paper is that

ZF\vdash\forall A: |Seq(A)|\neq|{\mathcal{P}}(A)| and we show that this is the best possible result.

Furthermore, it is provable in ZF that if B is an infinite set, then |fin(B)|<|{\mathcal{P}}(B)|, even though the existence for some infinite set B^* of a function f from fin(B^*) onto {\mathcal{P}}(B^*) is consistent with ZF.
keywords: S: ico, (AC)
Sh:489
Laskowski, M. C., & Shelah, S. (1993). On the existence of atomic models. J. Symbolic Logic, 58(4), 1189–1194. arXiv: math/9301210 DOI: 10.2307/2275137 MR: 1253916
type: article
abstract: We give an example of a countable theory T such that for every cardinal \lambda\ge\aleph_2 there is a fully indiscernible set A of power \lambda such that the principal types are dense over A, yet there is no atomic model of T over A. In particular, T(A) is a theory of size \lambda where the principal types are dense, yet T(A) has no atomic model.
keywords: M: odm, (mod)
Sh:490
Bartoszyński, T., Rosłanowski, A., & Shelah, S. (1996). Adding one random real. J. Symbolic Logic, 61(1), 80–90. arXiv: math/9406229 DOI: 10.2307/2275599 MR: 1380678
type: article
abstract: We study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions maybe large while the covering for measure is small.
keywords: S: for, S: str, (set), (leb)
Sh:491
Gilchrist, M., & Shelah, S. (1996). Identities on cardinals less than \aleph_\omega. J. Symbolic Logic, 61(3), 780–787. arXiv: math/9505215 DOI: 10.2307/2275784 MR: 1412509
type: article
abstract: Let \kappa be an uncountable cardinal and the edges of a complete graph with \kappa vertices be colored with \aleph_0 colors. For \kappa>2^{\aleph_0} the Erdős-Rado theorem implies that there is an infinite monochromatic subgraph. However, if \kappa\leq 2^{\aleph_0}, then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when \kappa\leq 2^{\aleph_0}. In particular, we are concerned with the case \aleph_1\leq\kappa\leq\aleph_\omega
keywords: S: for, (set-mod)
Sh:492
Komjáth, P., & Shelah, S. (1995). Universal graphs without large cliques. J. Combin. Theory Ser. B, 63(1), 125–135. arXiv: math/9308221 DOI: 10.1006/jctb.1995.1008 MR: 1309360
type: article
abstract: We give some existence/nonexistence statements on universal graphs, which under GCH give a necessary and sufficient condition for the existence of a universal graph of size \lambda with no K(\kappa), namely, if either \kappa is finite or cf(\kappa)>cf(\lambda). (Here K(\kappa) denotes the complete graph on \kappa vertices.) The special case when \lambda^{<\kappa}=\lambda was first proved by F. Galvin. Next, we investigate the question that if there is no universal K(\kappa)-free graph of size \lambda then how many of these graphs embed all the other. It was known, that if \lambda^{<\lambda}= \lambda (e.g., if \lambda is regular and the GCH holds below \lambda), and \kappa=\omega, then this number is \lambda^+. We show that this holds for every \kappa\leq\lambda of countable cofinality. On the other hand, even for \kappa=\omega_1, and any regular \lambda\geq\omega_1 it is consistent that the GCH holds below \lambda, 2^{\lambda} is as large as we wish, and the above number is either \lambda^+ or 2^{\lambda}, so both extremes can actually occur.
keywords: S: ico, S: for, (set), (graph), (univ)
Sh:493
Jin, R., & Shelah, S. (1994). The strength of the isomorphism property. J. Symbolic Logic, 59(1), 292–301. arXiv: math/9401216 DOI: 10.2307/2275266 MR: 1264980
type: article
abstract: In §1 of this paper, we characterize the isomorphism property of nonstandard universes in terms of the realization of some second–order types in model theory. In §2, several applications are given. One of the applications answers a question of D. Ross about infinite Loeb measure spaces
keywords: M: odm
Sh:494
Shelah, S., & Spinas, O. (2000). The distributivity numbers of \mathcal P(\omega)/\mathrm{fin} and its square. Trans. Amer. Math. Soc., 352(5), 2023–2047. arXiv: math/9606227 DOI: 10.1090/S0002-9947-99-02270-9 MR: 1751223
type: article
abstract: We show that in a model obtained by forcing with a countable support iteration of length \omega_2 of Mathias forcing {\bf h}(2), the distributivity number of r.o.({\mathcal P}(\omega)/fin)^2, is \omega_1 but {\bf h}, the one of {\mathcal P}(\omega)/fin, is \omega_2.
keywords: S: str, (set), (inv)
Sh:495
Apter, A. W., & Shelah, S. (1997). On the strong equality between supercompactness and strong compactness. Trans. Amer. Math. Soc., 349(1), 103–128. arXiv: math/9502232 DOI: 10.1090/S0002-9947-97-01531-6 MR: 1333385
type: article
abstract: We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V \models ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension V[G]\models ZFC + GCH in which, (a) (preservation) for \kappa \le \lambda regular, if V \models ``\kappa is \lambda supercompact”, then V[G] \models ``\kappa is \lambda supercompact” and so that, (b) (equivalence) for \kappa \le \lambda regular, V[G] \models ``\kappa is \lambda strongly compact” iff V[G] \models ``\kappa is \lambda supercompact”, except possibly if \kappa is a measurable limit of cardinals which are \lambda supercompact.
keywords: S: for, (set), (iter), (pure(large))
Sh:496
Apter, A. W., & Shelah, S. (1997). Menas’ result is best possible. Trans. Amer. Math. Soc., 349(5), 2007–2034. arXiv: math/9512226 DOI: 10.1090/S0002-9947-97-01691-7 MR: 1370634
type: article
abstract: Generalizing some earlier techniques due to the second author, we show that Menas’ theorem which states that the least cardinal \kappa which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not 2^\kappa supercompact is best possible. Using these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpublished theorem due to the first author.
keywords: S: for, (set), (iter), (pure(large))
Sh:497
Shelah, S. (1997). Set theory without choice: not everything on cofinality is possible. Arch. Math. Logic, 36(2), 81–125. arXiv: math/9512227 DOI: 10.1007/s001530050057 MR: 1462202
type: article
abstract: We prove, in ZF+DC, if e.g. \mu=|\mathcal{H}(\mu)| then \mu ^+ is regular non measurable. This is in contrast with the results for \mu=\aleph_{\omega} on measurability — see Apter-Magidor [ApMg].
keywords: S: pcf, (set), (AC)
Sh:498
Jin, R., & Shelah, S. (1994). Essential Kurepa trees versus essential Jech-Kunen trees. Ann. Pure Appl. Logic, 69(1), 107–131. arXiv: math/9401217 DOI: 10.1016/0168-0072(94)90021-3 MR: 1301608
type: article
abstract: By an \omega_1–tree we mean a tree of size \omega_1 and height \omega_1. An \omega_1–tree is called a Kurepa tree if all its levels are countable and it has more than \omega_1 branches. An \omega_1–tree is called a Jech–Kunen tree if it has \kappa branches for some \kappa strictly between \omega_1 and 2^{\omega_1}. A Kurepa tree is called an essential Kurepa tree if it contains no Jech–Kunen subtrees. A Jech–Kunen tree is called an essential Jech–Kunen tree if it contains no Kurepa subtrees. In this paper we prove that (1) it is consistent with CH and 2^{\omega_1}>\omega_2 that there exist essential Kurepa trees and there are no essential Jech–Kunen trees, (2) it is consistent with CH and 2^{\omega_1}>\omega_2 plus the existence of a Kurepa tree with 2^{\omega_1} branches that there exist essential Jech–Kunen trees and there are no essential Kurepa trees. In the second result we require the existence of a Kurepa tree with 2^{\omega_1} branches in order to avoid triviality.
keywords: S: for, (set), (iter), (trees)
Sh:499
Kojman, M., & Shelah, S. (1995). Homogeneous families and their automorphism groups. J. London Math. Soc. (2), 52(2), 303–317. arXiv: math/9409205 DOI: 10.1112/jlms/52.2.303 MR: 1356144
type: article
abstract: A homogeneous family of subsets over a given set is one with a very “rich” automorphism group. We prove the existence of a bi-universal element in the class of homogeneous families over a given infinite set and give an explicit construction of 2^{\bf c} isomorphism types of homogeneous families over a countable set.
keywords: S: ico, (set), (auto)
Sh:500
Shelah, S. (1996). Toward classifying unstable theories. Ann. Pure Appl. Logic, 80(3), 229–255. arXiv: math/9508205 DOI: 10.1016/0168-0072(95)00066-6 MR: 1402297
type: article
abstract: The paper deals with two issues: the existence of universal models of a theory T and related properties when cardinal arithmetic does not give this existence offhand. In the first section we prove that simple theories (e.g., theories without the tree property, a class properly containing the stable theories) behaves “better” than theories with the strict order property, by criterion from [Sh:457]. In the second section we introduce properties SOP_n such that the strict order property implies SOP_{n+1}, which implies SOP_n, which in turn implies the tree property. Now SOP_4 already implies non-existence of universal models in cases where earlier the strict order property was needed, and SOP_3 implies maximality in the Keisler order, again improving an earlier result which had used the strict order property.
keywords: M: cla, (mod), (univ), (up)
Sh:501
Rosłanowski, A., & Shelah, S. (1996). Localizations of infinite subsets of \omega. Arch. Math. Logic, 35(5-6), 315–339. arXiv: math/9506222 DOI: 10.1007/s001530050048 MR: 1420261
type: article
abstract: In the present paper we are interested in properties of forcing notions which measure in a sense the distance between the ground model reals and the reals in the extension. We look at the ways the “new” reals can be aproximated by “old” reals. We consider localizations for infinite subsets of \omega. Though each member of [\omega]^\omega can be identified with its increasing enumeration, the (standard) localizations of the enumeration does not provide satisfactory information on successive points of the set. They give us “candidates” for the n-th point of the set but the same candidates can appear several times for distinct n. That led to a suggestion that we should consider disjoint subsets of \omega as sets of “candidates” for successive points of the localized set. We have two possibilities. Either we can demand that each set from the localization contains a limited number of members of the localized set or we can postulate that each intersection of that kind is large. Localizations of this kind are studied in section 1. In the second section we investigate localizations of infinite subsets of \omega by sets of integers from the ground model. These localizations might be thought as localizations by partitions of \omega into successive intervals.
keywords: S: for, S: str, (set), (creatures)
Sh:502
Komjáth, P., & Shelah, S. (1993). On uniformly antisymmetric functions. Real Anal. Exchange, 19(1), 218–225. arXiv: math/9308222 MR: 1268847
type: article
abstract: We show that there is always a uniformly antisymmetric f:A\to\{0,1\} if A\subset R is countable. We prove that the continuum hypothesis is equivalent to the statement that there is an f:R\to\omega with |S_x|\leq 1 for every x\in R. If the continuum is at least \aleph_n then there exists a point x such that S_x has at least 2^n-1 elements. We also show that there is a function f:Q\to\{0,1,2,3\} such that S_x is always finite, but no such function with finite range on R exists
keywords: S: ico
Sh:503
Shelah, S. (1994). The number of independent elements in the product of interval Boolean algebras. Math. Japon., 39(1), 1–5. arXiv: math/9312212 MR: 1261328
type: article
abstract: We prove that in the product of \kappa many Boolean algebras we cannot find an independent set of more than 2^\kappa elements solving a problem of Monk (earlier it was known that we cannot find more than 2^{2^\kappa} but can find 2^\kappa).
keywords: S: ico, (ba), (inv)
Sh:504
Koppelberg, S., & Shelah, S. (1996). Subalgebras of Cohen algebras need not be Cohen. In Logic: from foundations to applications (Staffordshire, 1993), Oxford Univ. Press, New York, pp. 261–275. arXiv: math/9610227 MR: 1428008
type: article
abstract: We give an example of a regular and complete subalgebra of a Cohen algebra which is not Cohen.
keywords: S: ico, (set), (ba), (pure(for))
Sh:505
Eklof, P. C., & Shelah, S. (1994). A combinatorial principle equivalent to the existence of non-free Whitehead groups. In Abelian group theory and related topics (Oberwolfach, 1993), Vol. 171, Amer. Math. Soc., Providence, RI, pp. 79–98. arXiv: math/9403220 DOI: 10.1090/conm/171/01765 MR: 1293134
type: article
abstract: As a consequence of identifying the principle described in the title, we prove that for any uncountable cardinal \lambda, if there is a \lambda-free Whitehead group of cardinality \lambda which is not free, then there are many “nice” Whitehead groups of cardinality \lambda which are not free.
keywords: S: ico, O: alg, (ab), (wh), (unif)
Sh:506
Shelah, S. (1997). The pcf theorem revisited. In The mathematics of Paul Erdős, II, Vol. 14, Springer, Berlin, pp. 420–459. arXiv: math/9502233 DOI: 10.1007/978-3-642-60406-5_36 MR: 1425231
type: article
abstract: The \textrm{pcf} theorem (of the possible cofinality theory) was proved for reduced products \prod_{i< \kappa} \lambda_i/I, where \kappa< \min_{i< \kappa} \lambda_i. Here we prove this theorem under weaker assumptions such as wsat(I)< \min_{i< \kappa} \lambda_i, where wsat(I) is the minimal \theta such that \kappa cannot be delivered to \theta sets \notin I (or even slightly weaker condition). We also look at the existence of exact upper bounds relative to < _I (< _I-eub) as well as cardinalities of reduced products and the cardinals T_D(\lambda). Finally we apply this to the problem of the depth of ultraproducts (and reduced products) of Boolean algebras
keywords: S: pcf, (set)
Sh:507
Goldstern, M., & Shelah, S. (1995). The bounded proper forcing axiom. J. Symbolic Logic, 60(1), 58–73. arXiv: math/9501222 DOI: 10.2307/2275509 MR: 1324501
type: article
abstract: The bounded proper forcing axiom BPFA is the statement that for any family of \aleph_1 many maximal antichains of a proper forcing notion, each of size \aleph_1, there is a directed set meeting all these antichains.

A regular cardinal \kappa is called {\Sigma}_1-reflecting, if for any regular cardinal \chi, for all formulas \varphi, “H(\chi)\models `\varphi’ ” implies “\exists\delta<\kappa, H(\delta)\models `\varphi’ ”

We show that BPFA is equivalent to the statement that two nonisomorphic models of size \aleph_1 cannot be made isomorphic by a proper forcing notion, and we show that the consistency strength of the bounded proper forcing axiom is exactly the existence of a \Sigma_1-reflecting cardinal (which is less than the existence of a Mahlo cardinal).

We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure.
keywords: S: for, (set), (iter)
Sh:508
Rosłanowski, A., & Shelah, S. (1997). Simple forcing notions and forcing axioms. J. Symbolic Logic, 62(4), 1297–1314. arXiv: math/9606228 DOI: 10.2307/2275644 MR: 1617945
type: article
abstract: In the present paper we are interested in simple forcing notions and Forcing Axioms. A starting point for our investigations was the article [JR1] in which several problems were posed. We answer some of those problems here.
keywords: S: for, S: str, (set)
Sh:509
Shelah, S. (2008). Vive la différence. III. Israel J. Math., 166, 61–96. arXiv: math/0112237 DOI: 10.1007/s11856-008-1020-3 MR: 2430425
type: article
abstract: We show that, consistently, there is an ultrafilter {\mathcal F} on \omega such that if N^\ell_n=(P^\ell_n\cup Q^\ell_n, P^\ell_n,Q^\ell_n,R^\ell_n) (for \ell=1,2, n<\omega), P^\ell_n \cup Q^\ell_n \subseteq\omega, and \prod\limits_{n<\omega} N^1_n/ {\mathcal F}\equiv\prod\limits_{n<\omega}N^2_n/{\mathcal F} are models of the canonical theory t^{\rm ind} of the strong independence property, then every isomorphism from \prod\limits_{n<\omega} N^1_n/{\mathcal F} onto \prod\limits_{n< \omega} N^2_n/{\mathcal F} is a product isomorphism.
keywords: M: odm, S: for
Sh:510
Shelah, S., & Steprāns, J. (1994). Decomposing Baire class 1 functions into continuous functions. Fund. Math., 145(2), 171–180. arXiv: math/9401218 MR: 1297403
type: article
abstract: Let \mathfrak{dec} be the least cardinal \kappa such that every function of first Baire class can be decomposed into \kappa continuous functions. Cichon, Morayne, Pawlikowski and Solecki proved that cov(Meager)\leq \mathfrak{dec}\leq \mathfrak{d} and asked whether these inequalities could, consistently, be strict. By cov(Meager) is meant the least number of closed nowhere dense sets required to cover the real line and by \mathfrak{d} is denoted the least cardinal of a dominating family in \omega^\omega. Steprans showed that it is consistent that cov(Meager)\neq \mathfrak{dec}. In this paper we show that the second inequality can also be made strict. The model where \mathfrak{dec} is different from \mathfrak{d} is the one obtained by adding \omega_2 Miller - sometimes known as super-perfect or rational-perfect - reals to a model of the Continuum Hypothesis. It is somewhat surprising that the model used to establish the consistency of the other inequality, cov(Meager)\neq\mathfrak{dec}, is a slight modification of the iteration of super-perfect forcing.
keywords: S: str, (set)
Sh:512
Balcerzak, M., Rosłanowski, A., & Shelah, S. (1998). Ideals without ccc. J. Symbolic Logic, 63(1), 128–148. arXiv: math/9610219 DOI: 10.2307/2586592 MR: 1610790
type: article
abstract: Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F\subseteq P(X) of size {\bf c}, consisting of Borel sets which are not in I. Condition (M) states that there is a function f:X\rightarrow X with f^{-1}[\{x\}]\notin I for each x\in X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B\notin I and a perfect set P\subseteq X for which the family \{ B+x: x\in P\} is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D)\Rightarrow (M)\Rightarrow (B)\Rightarrow not-ccc can hold. We build a \sigma-ideal on the Cantor group witnessing "(M) and not (D)" (Section 2). A modified version of that \sigma-ideal contains the whole space (Section 3). Some consistency results deriving (M) from (B) for "nicely" defined ideals are established (Section 4). We show that both ccc and (M) can fail (Theorems 1.3 and 4.2). Finally, some sharp versions of (M) for invariant ideals on Polish groups are investigated (Section 5).
keywords: S: for, S: str, (set)
Sh:513
Shelah, S. (2002). PCF and infinite free subsets in an algebra. Arch. Math. Logic, 41(4), 321–359. arXiv: math/9807177 DOI: 10.1007/s001530100101 MR: 1906504
type: article
abstract: We give another proof that for every \lambda\geq\beth_\omega for every large enough regular \kappa<\beth_\omega we have \lambda^{[\kappa]}=\lambda, dealing with sufficient conditions for replacing \beth_\omega by \aleph_\omega. In §2 we show that large pcf(\mathfrak{a}) implies existence of free sets. An example is that if pp(\aleph_\omega)>\aleph_{\omega_1} then for every algebra M of cardinality \aleph_\omega with countably many functions, for some a_n\in M (for n<\omega) we have a_n\notin c\ell_M(\{a_l: l\neq n, l<\omega\}). Then we present results complementary to those of section 2 (but not close enough): if IND(\mu,\sigma) (in every algebra with universe \lambda and \le\sigma functions there is an infinite independent subset) then for no distinct regular \lambda_i\in {\rm Reg}\setminus\mu^+ (for i<\kappa) does \prod_{i<\kappa}\lambda_i/[\kappa]^{\le\sigma} have true cofinality. We look at IND(\langle J^{bd}_{\kappa_n}: n<\omega\rangle) and more general version, and from assumptions as in §2 get results even for the non stationary ideal. Lastly, we deal with some other measurements of [\lambda]^{\ge \theta} and give an application by a construction of a Boolean Algebra.
keywords: S: pcf, (set)
Sh:514
Magidor, M., & Shelah, S. (1994). \mathrm{Bext}^2(G,T) can be nontrivial, even assuming GCH. In Abelian group theory and related topics (Oberwolfach, 1993), Vol. 171, Amer. Math. Soc., Providence, RI, pp. 287–294. arXiv: math/9405214 DOI: 10.1090/conm/171/01778 MR: 1293148
type: article
abstract: Using the consistency of some large cardinals we produce a model of Set Theory in which the generalized continuum hypothesis holds and for some torsion-free abelian group G of cardinality \aleph_{\omega+1} and for some torsion group T, Bext^2(G,T)\not=0.
keywords: S: for, (ab)
Sh:515
Shelah, S. (1997). A finite partition theorem with double exponential bound. In The mathematics of Paul Erdős, II, Vol. 14, Springer, Berlin, pp. 240–246. arXiv: math/9502234 DOI: 10.1007/978-3-642-60406-5_21 MR: 1425218
type: article
abstract: We prove that double exponentiation is an upper bound to Ramsey theorem for colouring of pairs when we want to predetermine the order of the differences of successive members of the homogeneous set.
keywords: O: fin, (pc), (fc)
Sh:516
Komjáth, P., & Shelah, S. (1996). Coloring finite subsets of uncountable sets. Proc. Amer. Math. Soc., 124(11), 3501–3505. arXiv: math/9505216 DOI: 10.1090/S0002-9939-96-03450-8 MR: 1342032
type: article
abstract: It is consistent for every 1\leq n< \omega that 2^\omega=\omega_n and there is a function F:[\omega_n]^{< \omega}\to\omega such that every finite set can be written at most 2^n-1 ways as the union of two distinct monocolored sets. If GCH holds, for every such coloring there is a finite set that can be written at least \sum^n_{i=1}{n+i\choose n}{n\choose i} ways as the union of two sets with the same color.
keywords: S: ico, (set)
Sh:517
Shelah, S. (2005). Zero-one laws for graphs with edge probabilities decaying with distance. II. Fund. Math., 185(3), 211–245. arXiv: math/0404239 DOI: 10.4064/fm185-3-2 MR: 2161404
type: article
abstract: (none)
keywords: M: odm, O: fin, (fmt)
Sh:518
Laskowski, M. C., & Shelah, S. (1996). Forcing isomorphism. II. J. Symbolic Logic, 61(4), 1305–1320. arXiv: math/0011169 DOI: 10.2307/2275818 MR: 1456109
type: article
abstract: If T has only countably many complete types, yet has a type of infinite multiplicity then there is a ccc forcing notion Q such that, in any Q–generic extension of the universe, there are non-isomorphic models M_1 and M_2 of T that can be forced isomorphic by a ccc forcing. We give examples showing that the hypothesis on the number of complete types is necessary and what happens if “ccc” is replaced other cardinal-preserving adjectives. We also give an example showing that membership in a pseudo-elementary class can be altered by very simple cardinal-preserving forcings.
keywords: S: for, (mod), (sta)
Sh:519
Göbel, R., & Shelah, S. (1995). On the existence of rigid \aleph_1-free abelian groups of cardinality \aleph_1. In Abelian groups and modules (Padova, 1994), Vol. 343, Kluwer Acad. Publ., Dordrecht, pp. 227–237. arXiv: math/0104194 MR: 1378201
type: article
abstract: An abelian group is said to be \aleph_1–free if all its countable subgroups are free. Our main result is:

If R is a ring with R^+ free and |R|<\lambda\leq 2^{\aleph_0}, then there exists an \aleph_1–free abelian group G of cardinality \lambda with {\rm End} G = R.

A corollary to this theorem is:

Indecomposable \aleph_1–free abelian groups of cardinality \aleph_1 do exist.
keywords: S: ico, (ab)
Sh:520
Eklof, P. C., Foreman, M. D., & Shelah, S. (1995). On invariants for \omega_1-separable groups. Trans. Amer. Math. Soc., 347(11), 4385–4402. arXiv: math/9501223 DOI: 10.2307/2155042 MR: 1316849
type: article
abstract: We study the classification of \omega_1-separable groups using Ehrenfeucht-Fraı̈ssé games and prove a strong classification result assuming PFA, and a strong non-structure theorem assuming \diamondsuit.
keywords: O: alg, (ab), (stal), (diam)
Sh:521
Shelah, S. (1996). If there is an exactly \lambda-free abelian group then there is an exactly \lambda-separable one in \lambda. J. Symbolic Logic, 61(4), 1261–1278. arXiv: math/9503226 DOI: 10.2307/2275815 MR: 1456106
type: article
abstract: We give a solution stated in the title to problem 3 of part 1 of the problems listed in the book of Eklof and Mekler [EM],(p.453). There, in pp. 241-242, this is discussed and proved in some cases. The existence of strongly \lambda-free ones was proved earlier by the criteria in [Sh:161] in [MkSh:251]. We can apply a similar proof to a large class of other varieties in particular to the variety of (non-commutative) groups.
keywords: S: ico, O: alg, (ab), (stal), (af)
Sh:522
Shelah, S. (1999). Borel sets with large squares. Fund. Math., 159(1), 1–50. arXiv: math/9802134 MR: 1669643
type: article
abstract: For a cardinal \mu we give a sufficient condition (*)_\mu (involving ranks measuring existence of independent sets) for:

[(**)_\mu] if a Borel set B\subseteq R\times R contains a \mu-square (i.e. a set of the form A \times A, |A|=\mu) then it contains a 2^{\aleph_0}-square and even a perfect square,

and also for

[(***)_\mu] if \psi\in L_{\omega_1,\omega} has a model of cardinality \mu then it has a model of cardinality continuum generated in a nice, absolute way.

Assuming MA+ 2^{\aleph_0}>\mu for transparency, those three conditions ((*)_\mu, (**)_\mu and (***)_\mu) are equivalent, and by this we get e.g. (\forall\alpha<\omega_1)(2^{\aleph_0}\geq \aleph_\alpha \Rightarrow \neg (**)_{\aleph_\alpha}), and also \min\{\mu:(*)_\mu\}, if <2^{\aleph_0}, has cofinality \aleph_1. We deal also with Borel rectangles and related model theoretic problems.
keywords: S: for, S: dst, (mod), (inf-log)
Sh:523
Shelah, S. (1997). Existence of almost free abelian groups and reflection of stationary set. Math. Japon., 45(1), 1–14. arXiv: math/9606229 MR: 1434949
type: article
abstract: §2: We answer a question of Mekler Eklof on the closure operations of the incompactness spectrum. We answer a question of Foreman and Magidor on reflection of stationary subsets of {\mathcal S}_{< \aleph_2}(\lambda) = \{ a \subseteq \lambda: |a| < \aleph_2 \}]. §3 - NPT is not transitive. We prove NPT(\lambda,\mu) + NPT(\mu,\kappa) \not\Rightarrow NPT(\lambda,\kappa)
keywords: S: ico, O: alg, (ab), (stal), (app(pcf)), (ref), (af)
Sh:524
Shelah, S., & Thomas, S. (1997). The cofinality spectrum of the infinite symmetric group. J. Symbolic Logic, 62(3), 902–916. arXiv: math/9412230 DOI: 10.2307/2275578 MR: 1472129
type: article
abstract: A group G that is not finitely generated can be written as the union of a chain of proper subgroups. The cofinality spectrum of G, written CF(S), is the set of regular cardinals \lambda such that G can be expressed as the union of a chain of \lambda proper subgroups. The cofinality of G, written c(G), is the least element of CF(G). We show that it is consistent that CF(S) is quite a bizarre set of cardinals. For example, we prove

Theorem (A): Let T be any subset of \omega\setminus \{0\}. Then it is consistent that \aleph_n \in CF(S) if and only if n\in T.

One might suspect that it is consistent that CF(S) is an arbitrarily prescribed set of regular uncountable cardinals, subject only to the above mentioned constraint. This is not the case.

Theorem (B): If \aleph_n \in CF(S) for all n\in \omega\setminus \{0\}, then \aleph_{\omega+1} \in CF(S).
keywords: S: for, (iter), (stal), (app(pcf))
Sh:525
Gurevich, Y., Immerman, N., & Shelah, S. (1994). McColm’s conjecture [positive elementary inductions]. In Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, pp. 10–19. arXiv: math/9411235 DOI: 10.1109/LICS.1994.316091
type: article
abstract: Gregory McColm conjectured that positive elementary inductions are bounded in a class K of finite structures if every (FO + LFP) formula is equivalent to a first-order formula in K. Here (FO + LFP) is the extension of first-order logic with the least fixed point operator. We disprove the conjecture. Our main results are two model-theoretic constructions, one deterministic and the other randomized, each of which refutes McColm’s conjecture.
keywords: M: odm, O: fin, (fmt)
Sh:526
Gurevich, Y., & Shelah, S. (1996). On finite rigid structures. J. Symbolic Logic, 61(2), 549–562. arXiv: math/9411236 DOI: 10.2307/2275675 MR: 1394614
type: article
abstract: The main result of this paper is a probabilistic construction of finite rigid structures. It yields a finitely axiomatizable class of finite rigid structures where no L^\omega_{\infty,\omega} formula with counting quantifiers defines a linear order.
keywords: M: smt, O: fin, (fmt)
Sh:527
Lifsches, S., & Shelah, S. (1999). Random graphs in the monadic theory of order. Arch. Math. Logic, 38(4-5), 273–312. arXiv: math/9701219 DOI: 10.1007/s001530050129 MR: 1697962
type: article
abstract: We continue the works of Gurevich-Shelah and Lifsches-Shelah by showing that it is consistent with ZFC that the first-order theory of random graphs is not interpretable in the monadic theory of all chains. It is provable from ZFC that the theory of random graphs is not interpretable in the monadic second order theory of short chains (hence, in the monadic theory of the real line).
keywords: M: smt, O: fin, (mod), (mon)
Sh:528
Baldwin, J. T., & Shelah, S. (1997). Randomness and semigenericity. Trans. Amer. Math. Soc., 349(4), 1359–1376. arXiv: math/9607226 DOI: 10.1090/S0002-9947-97-01869-2 MR: 1407480
type: article
abstract: Let L contain only the equality symbol and let L^+ be an arbitrary finite symmetric relational language containing L. Suppose probabilities are defined on finite L^+ structures with "edge probability" n^{-\alpha}. By T^\alpha, the almost sure theory of random L^+-structures we mean the collection of L^+-sentences which have limit probability 1. T_\alpha denotes the theory of the generic structures for K_\alpha, (the collection of finite graphs G with \delta_{\alpha}(G)=|G|-\alpha\cdot |\text{edges of $ G$}| hereditarily nonnegative.)

THEOREM: T_\alpha, the almost sure theory of random L^+-structures is the same as the theory T_\alpha of the K_\alpha-generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.
keywords: M: cla, O: fin, (mod), (fmt)
Sh:529
Hyttinen, T., & Shelah, S. (1995). Constructing strongly equivalent nonisomorphic models for unsuperstable theories. Part B. J. Symbolic Logic, 60(4), 1260–1272. arXiv: math/9202205 DOI: 10.2307/2275887 MR: 1367209
type: article
abstract: We study how equivalent nonisomorphic models of unsuperstable theories can be. We measure the equivalence by Ehrenfeucht-Fraisse games. This paper continues [HySh:474].
keywords: M: non, (mod)
Sh:530
Cummings, J., & Shelah, S. (1995). A model in which every Boolean algebra has many subalgebras. J. Symbolic Logic, 60(3), 992–1004. arXiv: math/9509227 DOI: 10.2307/2275769 MR: 1349006
type: article
abstract: We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subset A such that 2^{|A|} = 2^{|B|}. This implies in particular that B has 2^{|B|} subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk. The paper is intended to be accessible as far as possible to a general audience, in particular we have confined the more technical material to a “black box” at the end. The proof involves a variation on Foreman and Woodin’s model in which GCH fails everywhere.
keywords: S: for, (set)
Sh:531
Shelah, S., & Spinas, O. (1998). The distributivity numbers of finite products of {\mathcal P}(\omega)/\mathrm{fin}. Fund. Math., 158(1), 81–93. arXiv: math/9801151 MR: 1641157
type: article
abstract: Generalizing [ShSi:494], for every n<\omega we construct a ZFC-model where the distributivity number of r.o.({\mathcal P}(\omega)/\textrm{fin})^{n+1}, {\bf h}(n+1), is smaller than the one of r.o.({\mathcal P}(\omega)/\textrm{fin})^{n}. This answers an old problem of Balcar, Pelant and Simon. We also show that Laver and Miller forcing collapse the continuum to {\bf h}(n) for every n< \omega, hence by the first result, consistently they collapse it below {\bf h}(n)
keywords: S: str, (set), (inv)
Sh:533
Blass, A. R., Gurevich, Y., & Shelah, S. (1999). Choiceless polynomial time. Ann. Pure Appl. Logic, 100(1-3), 141–187. arXiv: math/9705225 DOI: 10.1016/S0168-0072(99)00005-6 MR: 1711992
See [Sh:533a]
type: article
abstract: Turing machines define polynomial time (PTime) on strings but cannot deal with structures like graphs directly, and there is no known, easily computable string encoding of isomorphism classes of structures. Is there a computation model whose machines do not distinguish between isomorphic structures and compute exactly PTime properties? This question can be recast as follows: Does there exist a logic that captures polynomial time (without presuming the presence of a linear order)? Earlier, one of us conjectured the negative answer. The problem motivated a quest for stronger and stronger PTime logics. All these logics avoid arbitrary choice. Here we attempt to capture the choiceless fragment of PTime. Our computation model is a version of abstract state machines (formerly called evolving algebras). The idea is to replace arbitrary choice with parallel execution. The resulting logic is more expressive than other PTime logics in the literature. A more difficult theorem shows that the logic does not capture all PTime.
keywords: M: odm, O: fin, (fmt)
Sh:534
Rosłanowski, A., & Shelah, S. (1998). Cardinal invariants of ultraproducts of Boolean algebras. Fund. Math., 155(2), 101–151. arXiv: math/9703218 MR: 1606511
type: article
abstract: We deal with some of problems posed by Monk and related to cardinal invariant of ultraproducts of Boolean algebras. We also introduce and investigate some new cardinal invariants.
keywords: S: ico, (ba), (up), (inv(ba))
Sh:535
Eisworth, T., & Shelah, S. (2005). Successors of singular cardinals and coloring theorems. I. Arch. Math. Logic, 44(5), 597–618. arXiv: math/9808138 DOI: 10.1007/s00153-004-0258-7 MR: 2210148
type: article
abstract: We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [Sh:413], but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.
keywords: S: pcf, (set)
Sh:536
Gurevich, Y., & Shelah, S. (2003). Spectra of Monadic Second-Order Formulas with One Unary Function. In 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings., pp. 291–300. arXiv: math/0404150 DOI: 10.1109/LICS.2003.1210069
type: article
abstract: We establish the eventual periodicity of the spectrum of any monadic second-order formula where: (i) all relation symbols, except equality, are unary, and (ii) there is only one function symbol and that symbol is unary.
keywords: M: odm, O: fin, (fmt)
Sh:537
Abraham, U., & Shelah, S. (2001). Lusin sequences under CH and under Martin’s axiom. Fund. Math., 169(2), 97–103. arXiv: math/9807178 DOI: 10.4064/fm169-2-1 MR: 1852375
type: article
abstract: Assuming the continuum hypothesis there is an inseparable sequence of length \omega_1 that contains no Lusin subsequence, while if Martin’s Axiom and the negation of CH is assumed then every inseparable sequence (of length \omega_1) is a union of countably many Lusin subsequences.
keywords: S: str, (set)
Sh:539
Lifsches, S., & Shelah, S. (1996). Uniformization, choice functions and well orders in the class of trees. J. Symbolic Logic, 61(4), 1206–1227. arXiv: math/9404227 DOI: 10.2307/2275812 MR: 1456103
type: article
abstract: The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have a definable choice function (by a monadic formula with parameters)? A natural dichotomy arises where the trees that fall in the first class don’t have a definable choice function and the trees in the second class have even a definable well ordering of their elements. This has a close connection to the uniformization problem.
keywords: M: odm, (set-mod), (mon)
Sh:540
Brendle, J., & Shelah, S. (1996). Evasion and prediction. II. J. London Math. Soc. (2), 53(1), 19–27. arXiv: math/9407207 DOI: 10.1112/jlms/53.1.19 MR: 1362683
type: article
abstract: A subgroup G\leq {\mathbb Z}^\omega exhibits the Specker phenomenon if every homomorphism G \to {\mathbb Z} maps almost all unit vectors to 0. We give several combinatorial characterizations of the cardinal \mathfrak{se}, the size of the smallest G\leq {\mathbb Z}^\omega exhibiting the Specker phenomenon. We also prove the consistency of {\bf b}< {\bf e}, where {\bf b} is the unbounding number and {\bf e} the evasion number. Our results answer several questions addressed by Blass.
keywords: S: str, (set), (inv)
Sh:541
Cummings, J., & Shelah, S. (1995). Cardinal invariants above the continuum. Ann. Pure Appl. Logic, 75(3), 251–268. arXiv: math/9509228 DOI: 10.1016/0168-0072(95)00003-Y MR: 1355135
type: article
abstract: We prove some consistency results about {\bf b}(\lambda) and {\bf d}(\lambda), which are natural generalisations of the cardinal invariants of the continuum {\bf b} and {\bf d}. We also define invariants {\bf b}_{\rm cl}(\lambda) and {\bf d}_{\rm cl}(\lambda), and prove that almost always {\bf b}(\lambda) = {\bf b}_{\rm cl}(\lambda) and {\bf d}(\lambda)={\bf d}_{\rm cl}(\lambda)
keywords: S: for, (set)
Sh:542
Shelah, S. (1996). Large normal ideals concentrating on a fixed small cardinality. Arch. Math. Logic, 35(5-6), 341–347. arXiv: math/9406219 DOI: 10.1007/s001530050049 MR: 1420262
type: article
abstract: A property of a filter, a kind of large cardinal property, suffices for the proof in Liu Shelah [LiSh:484] and is proved consistent as required there. A natural property which looks better, not only is not obtained here, but is shown to be false. On earlier related theorems see Gitik Shelah [GiSh310].
keywords: S: for, (set), (normal), (large)
Sh:543
Fuchino, S., Shelah, S., & Soukup, L. (1994). On a theorem of Shapiro. Math. Japon., 40(2), 199–206. arXiv: math/9405215 MR: 1297233
type: article
abstract: We show that a theorem of Leonid B. Shapiro which was proved under MA, is actually independent from ZFC. We also give a direct proof of the Boolean algebra version of the theorem under MA(Cohen).
keywords: S: ico, (ba)
Sh:544
Fuchino, S., Shelah, S., & Soukup, L. (1997). Sticks and clubs. Ann. Pure Appl. Logic, 90(1-3), 57–77. arXiv: math/9804153 DOI: 10.1016/S0168-0072(97)00030-4 MR: 1489304
type: article
abstract: We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side-by-side product of partial orders which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with \neg CH and Martin’s Axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of \omega_1 together with \neg CH and Martin’s Axiom for countable p.o.-sets.
keywords: S: for, (set)
Sh:545
Džamonja, M., & Shelah, S. (1996). Saturated filters at successors of singular, weak reflection and yet another weak club principle. Ann. Pure Appl. Logic, 79(3), 289–316. arXiv: math/9601219 DOI: 10.1016/0168-0072(95)00040-2 MR: 1395679
type: article
abstract: Suppose that \lambda is the successor of a singular cardinal \mu whose cofinality is an uncountable cardinal \kappa. We give a sufficient condition that the club filter of \lambda concentrating on the points of cofinality \kappa is not \lambda^+-saturated. The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection
keywords: S: pcf, (set), (ref)
Sh:546
Shelah, S. (2000). Was Sierpiński right? IV. J. Symbolic Logic, 65(3), 1031–1054. arXiv: math/9712282 DOI: 10.2307/2586687 MR: 1791363
type: article
abstract: We prove for any \mu=\mu^{<\mu}<\theta<\lambda,\lambda large enough (just strongly inaccessible Mahlo) the consistency of 2^\mu=\lambda\rightarrow [\theta]^2_3 and even 2^\mu=\lambda\rightarrow [\theta]^2_{\sigma,2} for \sigma<\mu. The new point is that possibly \theta>\mu^+.
keywords: S: for, (set), (iter), (pc)
Sh:547
Göbel, R., & Shelah, S. (1998). Endomorphism rings of modules whose cardinality is cofinal to omega. In Abelian groups, module theory, and topology (Padua, 1997), Vol. 201, Dekker, New York, pp. 235–248. arXiv: math/0011186 MR: 1651170
type: article
abstract: The main result is Theorem: Let A be an R-algebra, \mu,\lambda be cardinals such that |A|\leq\mu=\mu^{\aleph_0}<\lambda\leq 2^\mu. If A is \aleph_0-cotorsion-free or A is countably free, respectively, then there exists an \aleph_0-cotorsion-free or a separable (reduced, torsion-free) R-module G respectively of cardinality |G|=\lambda with {\rm End}_R G=A\oplus{\rm Fin} G.
keywords: S: ico, (ab)
Sh:548
Shelah, S. (1996). Very weak zero one law for random graphs with order and random binary functions. Random Structures Algorithms, 9(4), 351–358. arXiv: math/9606230 DOI: 10.1002/(SICI)1098-2418(199612)9:4<351::AID-RSA1>3.3.CO;2-D MR: 1605415
type: article
abstract: Let G_<(n,p) denote the usual random graph G(n,p) on a totally ordered set of n vertices. We will fix p=\frac{1}{2} for definiteness. Let L^< denote the first order language with predicates equality (x=y), adjacency (x\sim y) and less than (x<y). For any sentence A in L^< let f_A(n) denote the probability that the random G_<(n,p) has property A. It is known Compton, Henson and Shelah [CHSh:245] that there are A for which f_A(n) does not converge. Here we show what is called a very weak zero-one law (from [Sh 463]):

THEOREM: For every A in language L^<, \lim_{n\rightarrow\infty}(f_A(n+1)-f_A(n))=0.
keywords: M: odm, O: fin, (fmt), (graph)
Sh:549
Fuchino, S., Koppelberg, S., & Shelah, S. (1996). Partial orderings with the weak Freese-Nation property. Ann. Pure Appl. Logic, 80(1), 35–54. arXiv: math/9508220 DOI: 10.1016/0168-0072(95)00047-X MR: 1395682
type: article
abstract: A partial ordering P is said to have the weak Freese-Nation property (WFN) if there is a mapping f:P\longrightarrow [P]^{\leq\aleph_0} such that, for any a, b\in P, if a\leq b then there exists c\in f(a)\cap f(b) such that a\leq c\leq b. In this note, we study the WFN and some of its generalizations. Some features of the class of BAs with the WFN seem to be quite sensitive to additional axioms of set theory: e.g., under CH, every ccc cBA has this property while, under {\bf b}\geq\aleph_2, there exists no cBA with the WFN.
keywords: S: ico, (ba)
Sh:551
Shelah, S. (1996). In the random graph G(n,p), p=n^{-a}: if \psi has probability O(n^{-\epsilon}) for every \epsilon>0 then it has probability O(e^{-n^\epsilon}) for some \epsilon>0. Ann. Pure Appl. Logic, 82(1), 97–102. arXiv: math/9512228 DOI: 10.1016/0168-0072(95)00071-2 MR: 1416640
type: article
abstract: Shelah Spencer [ShSp:304] proved the 0-1 law for the random graphs G(n,p_n), p_n=n^{-\alpha}, \alpha\in (0,1) irrational (set of nodes in [n]=\{1,\ldots,n\}, the edges are drawn independently, probability of edge is p_n). One may wonder what can we say on sentences \psi for which Prob(G(n,p_n)\models\psi) converge to zero, Lynch asked the question and did the analysis, getting (for every \psi):

EITHER [(\alpha)] Prob[G(n,p_n)\models\psi]=cn^{-\beta} + O(n^{-\beta-\varepsilon}) for some \varepsilon such that \beta>\varepsilon>0

OR [(\beta)] Prob(G(n,p_n)\models\psi)= O(n^{-\varepsilon}) for every \varepsilon>0.

Lynch conjectured that in case (\beta) we have

[(\beta^+)] Prob(G(n,p_n)\models\psi)= O(e^{-n^\varepsilon}) for some \varepsilon>0. We prove it here.
keywords: M: odm, O: fin, (fmt)
Sh:552
Shelah, S. (1997). Non-existence of universals for classes like reduced torsion free abelian groups under embeddings which are not necessarily pure. In Advances in algebra and model theory (Essen, 1994; Dresden, 1995), Vol. 9, Gordon; Breach, Amsterdam, pp. 229–286. arXiv: math/9609217 MR: 1683540
type: article
abstract: We consider a class K of structures e.g. trees with \omega+1 levels, metric spaces and mainly, classes of Abelian groups like the one mentioned in the title and the class of reduced separable (Abelian) p-groups. We say M\in K is universal for K if any member N of K of cardinality not bigger than the cardinality of M can be embedded into M. This is a natural, often raised, problem. We try to draw consequences of cardinal arithmetic to non–existence of universal members for such natural classes.
keywords: O: alg, (ab), (app(pcf)), (univ)
Sh:553
Shafir, O., & Shelah, S. (2000). More on entangled orders. J. Symbolic Logic, 65(4), 1823–1832. arXiv: math/9711220 DOI: 10.2307/2695077 MR: 1812182
type: article
abstract: This paper grew as a continuation of [Sh462] but in the present form it can serve as a motivation for it as well. We deal with the same notions, and use just one simple lemma from there. Originally entangledness was introduced in order to get narrow Boolean algebras and examples of the nonmultiplicativity of c.c-ness. These applications became marginal when the hope to extract new such objects or strong colourings were not materialized, but after the pcf constructions which made their debut in [Sh:g] it seems that this notion gained independence. Generally we aim at characterizing the existence strong and weak entangled orders in cardinal arithmetic terms. In [Sh462] necessary conditions were shown for strong entangledness which in a previous version was erroneously proved to be equivalent to plain entangledness. In section 1 we give a forcing counterexample to this equivalence and in section 2 we get those results for entangledness (certainly the most interesting case). In §3 we get weaker results for positively entangledness, especially when supplemented with the existence of a separating point. An antipodal case is defined and completely characterized. Lastly we outline a forcing example showing that these two subcases of positive entangledness comprise no dichotomy.
keywords: S: ico, S: for, (set)
Sh:554
Goldstern, M., & Shelah, S. (1997). A partial order where all monotone maps are definable. Fund. Math., 152(3), 255–265. arXiv: math/9707202 DOI: 10.4064/fm-152-3-255-265 MR: 1444716
type: article
abstract: We show the consistency of “There is a p.o. of size continuum on which all monotone maps are first order definable”. The continuum can be aleph_1 or larger, and we may even have Martin’s axiom.
keywords: M: non, S: ico, (ua)
Sh:556
Fuchino, S., Koppelberg, S., & Shelah, S. (1996). A game on partial orderings. Topology Appl., 74(1-3), 141–148. arXiv: math/9505212 DOI: 10.1016/S0166-8641(96)00051-X MR: 1425933
type: article
abstract: We study the determinacy of the game G_\kappa(A) introduced in [FKSh:549] for uncountable regular \kappa and several classes of partial orderings A. Among trees or Boolean algebras, we can always find an A such that G_\kappa(A) is undetermined. For the class of linear orders, the existence of such A depends on the size of \kappa^{< \kappa}. In particular we obtain a characterization of \kappa^{< \kappa}=\kappa in terms of determinacy of the game G_\kappa(L) for linear orders L.
keywords: S: ico
Sh:557
Niedermeyer, F., Shelah, S., & Steffens, K. (2006). The f-factor problem for graphs and the hereditary property. Arch. Math. Logic, 45(6), 665–672. arXiv: math/0404179 DOI: 10.1007/s00153-006-0009-z MR: 2252248
type: article
abstract: If P is a hereditary property then we show that, for the existence of a perfect f-factor, P is a sufficient condition for countable graphs and yields a sufficient condition for graphs of size \aleph_1. Further we give two examples of a hereditary property which is even necessary for the existence of a perfect f-factor. We also discuss the \aleph_2-case.
keywords: S: ico, (ba), (inf-log)
Sh:558
Geschke, S., & Shelah, S. (2008). The number of openly generated Boolean algebras. J. Symbolic Logic, 73(1), 151–164. arXiv: math/0702600 DOI: 10.2178/jsl/1208358746 MR: 2387936
type: article
abstract: This article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly \sigma-filtered Boolean algebras.

We show that for every uncountable regular cardinal \kappa there are 2^\kappa pairwise non-isomorphic openly generated Boolean algebras of size \kappa>\aleph_1 provided there is an almost free non-free abelian group of size \kappa. The openly generated Boolean algebras constructed here are almost free.

Moreover, for every infinite regular cardinal \kappa we construct 2^\kappa pairwise non-isomorphic Boolean algebras of size \kappa that are tightly \sigma-filtered and c.c.c.

These two results contrast nicely with Koppelberg’s theorem hat for every uncountable regular cardinal \kappa there are only 2^{<\kappa} isomorphism types of projective Boolean algebras of size \kappa.
keywords: M: non, S: ico, (ba), (nni)
Sh:559
Eklof, P. C., & Shelah, S. (1996). New nonfree Whitehead groups by coloring. In Abelian groups and modules (Colorado Springs, CO, 1995), Vol. 182, Dekker, New York, pp. 15–22. MR: 1415620
See [Sh:559a]
type: article
abstract: We show that it is consistent that there is a strongly \aleph_{1}-free \aleph_{1}-coseparable group of cardinality \aleph_{1} which is not \aleph_{1}-separable.
keywords: S: for, O: alg, (ab), (iter), (stal), (wh), (unif)
Sh:560
Laskowski, M. C., & Shelah, S. (2001). The Karp complexity of unstable classes. Arch. Math. Logic, 40(2), 69–88. arXiv: math/0011167 DOI: 10.1007/s001530000047 MR: 1816478
type: article
abstract: A class {\bf K} of structures is controlled if, for all cardinals \lambda, the relation of L_{\infty,\lambda}-equivalence partitions {\bf K} into a set of equivalence classes (as opposed to a proper class). We prove that the class of doubly transitive linear orders is controlled, while any pseudo-elementary class with the \omega-independence property is not controlled.
keywords: M: cla, M: non, (mod), (inf-log)
Sh:561
Shelah, S., & Zapletal, J. (1997). Embeddings of Cohen algebras. Adv. Math., 126(2), 93–118. arXiv: math/9502230 DOI: 10.1006/aima.1996.1597 MR: 1442306
type: article
abstract: Complete Boolean algebras proved to be an important tool in topology and set theory. Two of the most prominent examples are B(\kappa), the algebra of Borel sets modulo measure zero ideal in the generalized Cantor space \{0,1\}^\kappa equipped with product measure, and C(\kappa), the algebra of regular open sets in the space \{0,1\}^\kappa, for \kappa an infinite cardinal. C(\kappa) is much easier to analyse than B(\kappa): C(\kappa) has a dense subset of size \kappa, while the density of B(\kappa) depends on the cardinal characteristics of the real line; and the definition of C(\kappa) is simpler. Indeed, C(\kappa) seems to have the simplest definition among all algebras of its size. In the Main Theorem of this paper we show that in a certain precise sense, C(\aleph_1) has the simplest structure among all algebras of its size, too.

MAIN THEOREM: If ZFC is consistent then so is ZFC + 2^{\aleph_0}=\aleph_2 +“for every complete Boolean algebra B of uniform density \aleph_1, C(\aleph_1) is isomorphic to a complete subalgebra of B”.
keywords: S: ico, S: for, (set), (app(pcf))
Sh:562
Džamonja, M., & Shelah, S. (1995). On squares, outside guessing of clubs and I_{<f}[\lambda]. Fund. Math., 148(2), 165–198. arXiv: math/9510216 DOI: 10.4064/fm-148-2-165-198 MR: 1360144
type: article
abstract: Suppose that \lambda=\mu^+. We consider two aspects of the square property on subsets of \lambda. First, we have results which show e.g. that for \aleph_0\le\kappa=cf (\kappa)< \mu, the equality cf([\mu]^{\le\kappa},\subseteq)=\mu is a sufficient condition for the set of elements of \lambda whose cofinality is bounded by \kappa, to be split into the union of \mu sets with squares. Secondly, we introduce a certain weak version of the square property and prove that if \mu is a strong limit, then this weak square property holds on \lambda without any additional assumptions
keywords: S: pcf, (set)
Sh:563
Jin, R., & Shelah, S. (1997). Can a small forcing create Kurepa trees. Ann. Pure Appl. Logic, 85(1), 47–68. arXiv: math/9504220 DOI: 10.1016/S0168-0072(96)00018-8 MR: 1443275
type: article
abstract: In the paper we probe the possibilities of creating a Kurepa tree in a generic extension of a model of CH plus no Kurepa trees by an \omega_1-preserving forcing notion of size at most \omega_1. In the first section we show that in the Levy model obtained by collapsing all cardinals between \omega_1 and a strongly inaccessible cardinal by forcing with a countable support Levy collapsing order many \omega_1-preserving forcing notions of size at most \omega_1 including all \omega-proper forcing notions and some proper but not \omega-proper forcing notions of size at most \omega_1 do not create Kurepa trees. In the second section we construct a model of CH plus no Kurepa trees, in which there is an \omega-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions.
keywords: S: for, (set), (trees)
Sh:564
Shelah, S. (1996). Finite canonization. Comment. Math. Univ. Carolin., 37(3), 445–456. arXiv: math/9509229 MR: 1426909
type: article
abstract: The canonization theorem says that for given m,n for some m^* (the first one is called ER(n;m)) we have: for every function f with domain [{1,\ldots,m^*}]^n, for some A\in [{1,\ldots,m^*}]^m, the question of when the equality f({i_1,\ldots,i_n})=f({j_1,\ldots,j_n}) (where i_1<\ldots<i_n and j_1 <\ldots< j_n are from A) holds has the simplest answer: for some v \subseteq \{1,\ldots,n\} the equality holds iff (\forall\ell\in v)(i_\ell = j_\ell).

In this paper we improve the bound on ER(n,m) so that fixing n the number of exponentiation needed to calculate ER(n,m) is best possible.
keywords: O: fin, (pc), (fc)
Sh:565
Jech, T. J., & Shelah, S. (1996). On countably closed complete Boolean algebras. J. Symbolic Logic, 61(4), 1380–1386. arXiv: math/9502203 DOI: 10.2307/2275822 MR: 1456113
type: article
abstract: It is unprovable that every complete subalgebra of a countably closed complete Boolean algebra is countably closed.
keywords: S: for, (set), (ba), (pure(for))
Sh:566
Jech, T. J., & Shelah, S. (1996). A complete Boolean algebra that has no proper atomless complete subalgebra. J. Algebra, 182(3), 748–755. arXiv: math/9501206 DOI: 10.1006/jabr.1996.0199 MR: 1398120
type: article
abstract: There exists a complete atomless Boolean algebra that has no proper atomless complete subalgebra.
keywords: S: for, (set), (ba), (pure(for))
Sh:567
Baldwin, J. T., & Shelah, S. (1998). DOP and FCP in generic structures. J. Symbolic Logic, 63(2), 427–438. arXiv: math/9607228 DOI: 10.2307/2586841 MR: 1625876
type: article
abstract: Spencer and Shelah [ShSp:304] constructed for each irrational \alpha between 0 and 1 the theory T^\alpha as the almost sure theory of random graphs with edge probability n^{-\alpha}. In [BlSh:528] we proved that this was the same theory as the theory T_\alpha built by constructing a generic model in Baldwin and Shi. In this paper we explore some of the more subtle model theoretic properties of this theory. We show that T^\alpha has the dimensional order property and does not have the finite cover property.
keywords: M: cla, O: fin, (mod), (fmt), (sta)
Sh:568
Göbel, R., & Shelah, S. (2001). Some nasty reflexive groups. Math. Z., 237(3), 547–559. arXiv: math/0003164 DOI: 10.1007/PL00004879 MR: 1845337
type: article
abstract: In Almost Free Modules, Set-theoretic Methods, p. 455, Problem 12, Eklof and Mekler raised the question about the existence of dual abelian groups G which are not isomorphic to {\mathbb Z} \oplus G. Recall that G is a dual group if G \cong D^* for some group D with D^*={\rm Hom}(D,{\mathbb Z}). The existence of such groups is not obvious because dual groups are subgroups of cartesian products {\mathbb Z}^D and therefore have very many homomorphisms into {\mathbb Z}. If \pi is such a homomorphism arising from a projection of the cartesian product, then D^*\cong{\rm ker}\pi \oplus {\mathbb Z}. In all “classical cases” of groups D of infinite rank it turns out that D^*\cong{\rm ker}\pi. Is this always the case? Also note that reflexive groups G in the sense of H. Bass are dual groups because by definition the evaluation map \sigma:G\longrightarrow G^{**} is an isomorphism, hence G is the dual of G^*. Assuming the diamond axiom for \aleph_1 we will construct a reflexive torsion-free abelian group of cardinality \aleph_1 which is not isomorphic to {\mathbb Z}\oplus G. The result is formulated for modules over countable principal ideal domains which are not field.
keywords: M: non, (ab)
Sh:569
Shami, Z., & Shelah, S. (1999). Rigid \aleph_\epsilon-saturated models of superstable theories. Fund. Math., 162(1), 37–46. arXiv: math/9908158 MR: 1734816
type: article
abstract: We look naturally at models with: no two dimensions are equal, so if such a model is not rigid it has an automorphism (non trivial) then it maps every regular type to one not orthogonal to it; here comes the main point: if some \aleph_\epsilon saturated model of T has such an automorphism and NDOP then every one has an automorphism; by the analysis from [Sh 401] to be completed: this automorphism share this property, imitating [Sh-c X] also in other cardinlas there are rigid models even when teh model is not with all dimensions distinct (use levels of the tree decomposition); generally if T has an \aleph_\epsilon saturated rigid model then it is strongly deep (every type has depth infinity (enough has depth >0)) for them we have NDOP when one side comes from this type, then use a decomposion theorem with zero and two successors
keywords: M: cla, (mod)
Sh:570
Baldwin, J. T., Grossberg, R. P., & Shelah, S. (1999). Transfering saturation, the finite cover property, and stability. J. Symbolic Logic, 64(2), 678–684. arXiv: math/9511205 DOI: 10.2307/2586492 MR: 1777778
type: article
abstract: Saturation is (\mu,\kappa)-transferable in T if and only if there is an expansion T_1 of T with |T_1|=|T| such that if M is a \mu-saturated model of T_1 and |M|\geq\kappa then the reduct M\restriction L(T) is \kappa-saturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively those where saturation is (\aleph_0,\lambda)-transferable or (\kappa(T),\lambda)-transferable for all \lambda. Further if for some \mu\geq |T|, 2^\mu>\mu^+, stability is equivalent to for all \mu\geq |T|, saturation is (\mu,2^\mu)-transferable.
keywords: M: cla, (mod), (sta)
Sh:571
Cummings, J., Džamonja, M., & Shelah, S. (1995). A consistency result on weak reflection. Fund. Math., 148(1), 91–100. arXiv: math/9504221 DOI: 10.4064/fm-148-1-91-100 MR: 1354940
type: article
abstract: In this paper we study the notion of strong non-reflection, and its contrapositive weak reflection. We say \theta strongly non-reflects at \lambda iff there is a function F:\theta\longrightarrow\lambda such that for all \alpha<\theta with cf(\alpha)=\lambda there is C club in \alpha such that F\restriction C is strictly increasing. We prove that it is consistent to have a cardinal \theta such that strong non-reflection and weak reflection each hold on an unbounded set of cardinals less than \theta.
keywords: S: for, (set), (large), (ref)
Sh:572
Shelah, S. (1997). Colouring and non-productivity of \aleph_2-c.c. Ann. Pure Appl. Logic, 84(2), 153–174. arXiv: math/9609218 DOI: 10.1016/S0168-0072(96)00020-6 MR: 1437644
type: article
abstract: We prove that colouring of pairs from \aleph_2 with strong properties exists. The easiest to state (and quite a well known problem) it solves: there are two topological spaces with cellularity \aleph_1 whose product has cellularity \aleph_2; equivalently we can speak on cellularity of Boolean algebras or on Boolean algebras satisfying the \aleph_2-c.c. whose product fails the \aleph_2-c.c. We also deal more with guessing of clubs.
keywords: S: ico, S: pcf, (set)
Sh:573
Lifsches, S., & Shelah, S. (1998). Uniformization and Skolem functions in the class of trees. J. Symbolic Logic, 63(1), 103–127. arXiv: math/9412231 DOI: 10.2307/2586591 MR: 1610786
type: article
abstract: The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have definable Skolem functions (by a monadic formula with parameters)? This continues [LiSh539] where the question was asked only with respect to choice functions. Here we define a subclass of the class of tame trees (trees with a definable choice function) and prove that this is exactly the class (actually set) of trees with definable Skolem functions.
keywords: M: odm, (mod), (mon)
Sh:574
Džamonja, M., & Shelah, S. (1999). Similar but not the same: various versions of \clubsuit do not coincide. J. Symbolic Logic, 64(1), 180–198. arXiv: math/9710215 DOI: 10.2307/2586758 MR: 1683902
type: article
abstract: We consider various versions of the \clubsuit principle. This principle is a known consequence of \diamondsuit. It is well known that \diamondsuit is not sensitive to minor changes in its definition, e.g. changing the guessing requirement form “guessing exactly” to “guessing modulo a finite set”. We show however, that this is not true for \clubsuit. We consider some other variants of \clubsuit as well.
keywords: S: for, (set), (iter)
Sh:575
Shelah, S. (2000). Cellularity of free products of Boolean algebras (or topologies). Fund. Math., 166(1-2), 153–208. arXiv: math/9508221 MR: 1804709
type: article
abstract: We answer Problem 1 of Monk if there are Boolean algebras B_1,B_2 such that c(B_i)\leq\lambda_i but c(B_1\times B_2)> \lambda_1+\lambda_2 where \lambda_1=\mu is singular and \mu>\lambda_2=\theta>cf(\mu)
keywords: S: ico, S: pcf, (set), (ba), (app(pcf))
Sh:576
Shelah, S. (2001). Categoricity of an abstract elementary class in two successive cardinals. Israel J. Math., 126, 29–128. arXiv: math/9805146 DOI: 10.1007/BF02784150 MR: 1882033
type: article
abstract: We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models or existence of large cardinals). We prove (assuming a weak version of GCH around \lambda) that if \mathfrak{K} is categorical in \lambda,\lambda^+, LS(\mathfrak{K}) \le\lambda and 1\le I(\lambda^{++},\mathfrak{K})< 2^{\lambda^{++}} then \mathfrak{K} has a model in \lambda^{+++}.
keywords: M: nec, M: non, (mod), (nni), (cat), (wd)
Sh:577
Gitik, M., & Shelah, S. (1997). Less saturated ideals. Proc. Amer. Math. Soc., 125(5), 1523–1530. arXiv: math/9503203 DOI: 10.1090/S0002-9939-97-03702-7 MR: 1363421
type: article
abstract: We prove the following:

(1) If \kappa is weakly inaccessible then NS_\kappa is not \kappa^+-saturated.

(2) If \kappa is weakly inaccessible and \theta<\kappa is regular then NS^\theta_\kappa is not \kappa^+-saturated.

(3) If \kappa is singular then NS^{cf(\kappa)}_{\kappa^+} is not \kappa^{++}-saturated.

Combining this with previous results of Shelah, one obtains the following: (A) If \kappa>\aleph_1 then NS_\kappa is not \kappa^+-saturated. (B) If \theta^+<\kappa then NS^\theta_\kappa is not \kappa^+-saturated.
keywords: S: ico, (set), (normal)
Sh:578
Milner, E. C., & Shelah, S. (1998). A tree-arrowing graph. In Set theory (Curaçao, 1995; Barcelona, 1996), Kluwer Acad. Publ., Dordrecht, pp. 175–182. arXiv: math/9708210 MR: 1602000
type: article
abstract: We answer a variant of a question of Rödl and Voigt by showing that, for a given infinite cardinal \lambda, there is a graph G of cardinality \kappa=(2^\lambda)^+ such that for any colouring of the edges of G with \lambda colours, there is an induced copy of the \kappa-tree in G in the set theoretic sense with all edges having the same colour.
keywords: S: ico, (set), (pc), (graph)
Sh:579
Göbel, R., & Shelah, S. (1996). GCH implies existence of many rigid almost free abelian groups. In Abelian groups and modules (Colorado Springs, CO, 1995), Vol. 182, Dekker, New York, pp. 253–271. arXiv: math/0011185 MR: 1415638
type: article
abstract: We begin with the existence of groups with trivial duals for cardinals \aleph_n (n\in \omega). Then we derive results about strongly \aleph_n-free abelian groups of cardinality \aleph_n (n\in\omega) with prescribed free, countable endomorphism ring. Finally we use combinatorial results of [Sh:108], [Sh:141] to give similar answers for cardinals >\aleph_\omega. As in Magidor and Shelah [MgSh:204], a paper concerned with the existence of \kappa-free, non-free abelian groups of cardinality \kappa, the induction argument breaks down at \aleph_\omega. Recall that \aleph_\omega is the first singular cardinal and such groups of cardinality \aleph_\omega do not exist by the well-known Singular Compactness Theorem (see [Sh:52]).
keywords: M: non, (ab)
Sh:580
Shelah, S. (2000). Strong covering without squares. Fund. Math., 166(1-2), 87–107. arXiv: math/9604243 MR: 1804706
type: article
abstract: We continue [Sh:b, Ch XIII] and [Sh:410]. Let W be an inner model of ZFC. Let \kappa be a cardinal in V. We say that \kappa-covering holds between V and W iff for all X\in V with X\subseteq ON and V\models|X|<\kappa, there exists Y\in W such that X\subseteq Y\subseteq ON and V\models |Y|<\kappa. Strong \kappa-covering holds between V and W iff for every structure {\mathcal M} \in V for some countable first-order language whose underlying set is some ordinal \lambda, and every X \in V with X\subseteq\lambda and V\models |X|<\kappa, there is Y\in W such that X\subseteq Y \prec M and V\models |Y|<\kappa.

We prove that if \kappa is V-regular, \kappa^+_V=\kappa^+_W, and we have both \kappa-covering and \kappa^+-covering between W and V, then strong \kappa-covering holds. Next we show that we can drop the assumption of \kappa^+-covering at the expense of assuming some more absoluteness of cardinals and cofinalities between W and V, and that we can drop the assumption that \kappa^+_W =\kappa^+_V and weaken the \kappa^+-covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).
keywords: S: pcf, (set)
Sh:582
Gitik, M., & Shelah, S. (2001). More on real-valued measurable cardinals and forcing with ideals. Israel J. Math., 124, 221–242. arXiv: math/9507208 DOI: 10.1007/BF02772619 MR: 1856516
type: article
abstract: (1) It is shown that if c is real-valued measurable then the Maharam type of (c, {\mathcal P}(c),\sigma) is 2^c. This answers a question of D. Fremlin.

(2) A different construction of a model with a real-valued measurable cardinal is given from that of R. Solovay. This answers a question of D. Fremlin.

(3) The forcing with a \kappa-complete ideal over a set X, |X|\geq\kappa cannot be isomorphic to Random\timesCohen or Cohen\timesRandom. The result for X=\kappa was proved in [GiSh:357] but as was pointed out to us by M. Burke the application of it in [GiSh:412] requires dealing with any X.
keywords: S: str, (set), (leb), (meag)
Sh:583
Gilchrist, M., & Shelah, S. (1997). The consistency of ZFC + 2^{\aleph_0}>\aleph_\omega+\mathcal I(\aleph_2)=\mathcal I(\aleph_\omega). J. Symbolic Logic, 62(4), 1151–1160. arXiv: math/9603219 DOI: 10.2307/2275632 MR: 1617993
type: article
abstract: An \omega-coloring is a pair \langle f,B\rangle where f:[B]^{2}\longrightarrow\omega. The set B is the field of f and denoted Fld(f). Let f,g be \omega-colorings. We say that f realizes the coloring g if there is a one-one function k:Fld(g)\longrightarrow Fld(f) such that for all \{x,y\}, \{u,v\}\in dom(g) we have f(\{k(x),k(y)\})\neq f(\{k(u),k(v)\}) \Rightarrow g(\{x,y\})\neq g(\{u,v\}). We write f\sim g if f realizes g and g realizes f. We call the \sim-classes of \omega-colorings with finite fields identities. We say that an identity I is of size r if |Fld(f)|=r for some/all f\in I. For a cardinal \kappa and f:[\kappa]^2\longrightarrow\omega we define {\mathcal I}(f) to be the collection of identities realized by f and {\mathcal I }(\kappa) to be \bigcap\{{\mathcal I}(f)| f:[\kappa]^2\longrightarrow\omega\}.

We show that, if ZFC is consistent then ZFC + 2^{\aleph_0}>\aleph_\omega + {\mathcal I}(\aleph_2)={\mathcal I}(\aleph_\omega) is consistent.
keywords: S: for, (set-mod)
Sh:584
Saxl, J., Shelah, S., & Thomas, S. (1996). Infinite products of finite simple groups. Trans. Amer. Math. Soc., 348(11), 4611–4641. arXiv: math/9605202 DOI: 10.1090/S0002-9947-96-01746-1 MR: 1376555
type: article
abstract: We classify the sequences \langle S_{n}: n\in N\rangle of finite simple nonabelian groups such that \prod_n S_n has uncountable cofinality.
keywords: O: alg, (stal), (auto)
Sh:585
Rabus, M., & Shelah, S. (2000). Covering a function on the plane by two continuous functions on an uncountable square—the consistency. Ann. Pure Appl. Logic, 103(1-3), 229–240. arXiv: math/9706223 DOI: 10.1016/S0168-0072(98)00053-0 MR: 1756147
type: article
abstract: It is consistent that for every function f:{\mathbb R}\times {\mathbb R}\rightarrow {\mathbb R} there is an uncountable set A\subseteq {\mathbb R} and two continuous functions f_0,f_1:D(A)\rightarrow {\mathbb R} such that f(\alpha,\beta)\in \{f_0(\alpha,\beta),f_1(\alpha,\beta)\} for every (\alpha,\beta) \in A^2, \alpha\not =\beta.
keywords: S: for, S: str, (set), (meag)
Sh:586
Shelah, S. (1998). A polarized partition relation and failure of GCH at singular strong limit. Fund. Math., 155(2), 153–160. arXiv: math/9706224 MR: 1606515
type: article
abstract: The main result is that for \lambda strong limit singular failing the continuum hypothesis (i.e. 2^\lambda> \lambda^+), a polarized partition theorem holds.
keywords: S: ico, (set), (pc), (app(pcf))
Sh:587
Shelah, S. (2003). Not collapsing cardinals \leq\kappa in (<\kappa)-support iterations. Israel J. Math., 136, 29–115. arXiv: math/9707225 DOI: 10.1007/BF02807192 MR: 1998104
type: article
abstract: We deal with the problem of preserving various versions of completeness in (<\kappa)–support iterations of forcing notions, generalizing the case “S–complete proper is preserved by CS iterations for a stationary co-stationary S\subseteq\omega_1”. We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinal \kappa and a stationary set S\subseteq\kappa with fat complement we can have uniformization for \langle A_\delta:\delta\in S'\rangle, A_\delta \subseteq\delta=\sup A_\delta, cf(\delta)=otp(A_\delta) and a stationary non-reflecting set S'\subseteq S.
keywords: S: for, (set), (unif)
Sh:588
Shelah, S. (2013). Large weight does not yield an irreducible base. Period. Math. Hungar., 66(2), 131–137. arXiv: 1007.2693 DOI: 10.1007/s10998-013-1031-7 MR: 3090811
type: article
abstract: Answering a question of Juhász, Soukup and Szentmiklóssy we show that it is consistent that some first countable space of uncountable weight does not contain an uncountable subspace which has an irreducible base.
keywords: S: for, (set), (pc)
Sh:589
Shelah, S. (2000). Applications of PCF theory. J. Symbolic Logic, 65(4), 1624–1674. arXiv: math/9804155 DOI: 10.2307/2695067 MR: 1812172
type: article
abstract: We deal with several pcf problems; we characterize another version of exponentiation: number of \kappa-branches in a tree with \lambda nodes, deal with existence of independent sets in stable theories, possible cardinalities of ultraproducts and the depth of ultraproducts of Boolean Algebras. (Claim 1.1 was revised June 2004.)

Also, we give cardinal invariants for each \lambda with a pcf restriction and investigate further T_D(f). The sections can be read independently, although there are some minor dependencies.
keywords: S: pcf, (set), (ba), (app(pcf)), (up)
Sh:590
Shelah, S. (2000). On a problem of Steve Kalikow. Fund. Math., 166(1-2), 137–151. arXiv: math/9705226 MR: 1804708
type: article
abstract: The Kalikow problem for a pair (\lambda,\kappa) of cardinal numbers, \lambda >\kappa (in particular \kappa=2) is whether we can map the family of \omega–sequences from \lambda to the family of \omega–sequences from \kappa in a very continuous manner. Namely, we demand that for \eta,\nu\in\lambda^\omega we have:

\eta,\nu are almost equal if and only if their images are.

We show consistency of the negative answer e.g. for \aleph_\omega but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.
keywords: S: for, S: str, (set), (cont), (large)
Sh:591
Göbel, R., & Shelah, S. (1998). Indecomposable almost free modules—the local case. Canad. J. Math., 50(4), 719–738. arXiv: math/0011182 DOI: 10.4153/CJM-1998-039-7 MR: 1638607
type: article
abstract: Let R be a countable, principal ideal domain which is not a field and A be a countable R-algebra which is free as an R-module. Then we will construct an \aleph_1-free R-module G of rank \aleph_1 with endomorphism algebra End_RG=A. Clearly the result does not hold for fields. Recall that an R-module is \aleph_1-free if all its countable submodules are free, a condition closely related to Pontryagin’s theorem. This result has many consequences, depending on the algebra A in use. For instance, if we choose A=R, then clearly G is an indecomposable ‘almost free’ module. The existence of such modules was unknown for rings with only finitely many primes like R={\mathbb Z}_{(p)}, the integers localized at some prime p. The result complements a classical realization theorem of Corner’s showing that any such algebra is an endomorphism algebra of some torsion-free, reduced R-module G of countable rank. Its proof is based on new combinatorial-algebraic techniques related with what we call rigid tree-elements coming from a module generated over a forest of trees.
keywords: M: non, (ab), (af)
Sh:592
Shelah, S. (2000). Covering of the null ideal may have countable cofinality. Fund. Math., 166(1-2), 109–136. arXiv: math/9810181 MR: 1804707
type: article
abstract: We prove that it is consistent that the covering of the ideal of measure zero sets has countable cofinality.
keywords: S: str, (set), (leb), (inv), (pure(for))
Sh:593
Fuchino, S., Mildenberger, H., Shelah, S., & Vojtáš, P. (1999). On absolutely divergent series. Fund. Math., 160(3), 255–268. arXiv: math/9903114 MR: 1708990
type: article
abstract: We show that in the \aleph_2-stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(\omega)/fin. This complements Vojtas’ result, that under cf(c)=p the two algebras are isomorphic.
keywords: S: for, S: str, (set), (inv)
Sh:594
Shelah, S. (1998). There may be no nowhere dense ultrafilter. In Logic Colloquium ’95 (Haifa), Vol. 11, Springer, Berlin, pp. 305–324. arXiv: math/9611221 DOI: 10.1007/978-3-662-22108-2_17 MR: 1690694
type: article
abstract: We show the consistency of ZFC + "there is no NWD-ultrafilter on \omega", which means: for every non principle ultrafilter D on the set of natural numbers, there is a function f from the set of natural numbers to the reals, such that for some nowhere dense set A of reals, the set \{n: f(n)\in A\} is not in D. This answers a question of van Douwen, which was put in more general context by Baumgartner
keywords: S: for, S: str, (set)
Sh:595
Shelah, S. (2000). Embedding Cohen algebras using pcf theory. Fund. Math., 166(1-2), 83–86. arXiv: math/9508201 MR: 1804705
type: article
abstract: Using a theorem from pcf theory, we show that for any singular cardinal \nu, the product of the Cohen forcing notions on \kappa, \kappa<\nu adds a generic for the Cohen forcing notion on \nu^+. This solves Problem 5.1 in Miller’s list (attributed to Rene David and Sy Friedman).
keywords: S: pcf, (set), (app(pcf)), (pure(for))
Sh:596
Cummings, J., & Shelah, S. (1999). Some independence results on reflection. J. London Math. Soc. (2), 59(1), 37–49. arXiv: math/9703219 DOI: 10.1112/S0024610798006863 MR: 1688487
type: article
abstract: We prove that there is a certain degree of independence between stationary reflection phenomena at different cofinalities; e.g. it is consistent that every stationary subset of S_1^3 reflects at a point of cofinality \aleph_2 while every stationary subset of S^3_0 has a non-reflecting stationary subset
keywords: S: for, (set), (large), (ref)
Sh:597
Gitik, M., & Shelah, S. (1998). On densities of box products. Topology Appl., 88(3), 219–237. arXiv: math/9603206 DOI: 10.1016/S0166-8641(97)00176-4 MR: 1632081
type: article
abstract: We construct two universes V_1, V_2 satisfying the following: GCH below \aleph_\omega, 2^{\aleph_\omega}=\aleph_{\omega+2} and the topological density of the space {}^{\aleph_\omega} 2 with \aleph_0 box product topology d_{<\aleph_1}(\aleph_\omega) is \aleph_{\omega+1} in V_1 and \aleph_{\omega+2} in V_2. Further related results are discussed as well.
keywords: S: for, S: pcf, (set), (large)
Sh:598
Abraham, U., & Shelah, S. (2004). Ladder gaps over stationary sets. J. Symbolic Logic, 69(2), 518–532. arXiv: math/0404151 DOI: 10.2178/jsl/1082418541 MR: 2058187
type: article
abstract: For a stationary set S\subseteq \omega_1 and a ladder system C over S, a new type of gaps called C-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some stationary set S, for every ladder C over S, every gap contains a subgap that is C-Hausdorff. But for every ladder E over \omega_1\setminus S there exists a gap with no subgap that is E-Hausdorff. A new type of chain condition, called polarized chain condition, is introduced. We prove that the iteration with finite support of polarized c.c.c posets is again a polarized c.c.c poset.
keywords: S: pcf, (set)
Sh:599
Rosłanowski, A., & Shelah, S. (2000). More on cardinal invariants of Boolean algebras. Ann. Pure Appl. Logic, 103(1-3), 1–37. arXiv: math/9808056 DOI: 10.1016/S0168-0072(98)00066-9 MR: 1756140
type: article
abstract: We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that irr(B_0\times B_1)=\max\{irr(B_0),irr(B_1)\}. We prove consistency of the statement “there is a Boolean algebra B such that irr(B)< s(B\otimes B)” and we force a superatomic Boolean algebra B_* such that s(B_*)=inc(B_*)=\kappa, irr(B_*)=Id(B_*)=\kappa^+ and Sub(B_*)=2^{\kappa^+}. Next we force a superatomic algebra B_0 such that irr(B_0)< inc(B_0) and a superatomic algebra B_1 such that t(B_1)> {\rm Aut}(B_1). Finally we show that consistently there is a Boolean algebra B of size \lambda such that there is no free sequence in B of length \lambda, there is an ultrafilter of tightness \lambda (so t(B)=\lambda) and \lambda\notin{\rm Depth}_{\rm Hs}(B).
keywords: S: for, (set), (inv(ba))
Sh:601
Kuhlmann, F.-V., Kuhlmann, S., & Shelah, S. (1997). Exponentiation in power series fields. Proc. Amer. Math. Soc., 125(11), 3177–3183. arXiv: math/9608214 DOI: 10.1090/S0002-9939-97-03964-6 MR: 1402868
type: article
abstract: We prove that for no nontrivial ordered abelian group G, the ordered power series field R((G)) admits an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non-surjective logarithm. For an arbitrary ordered field k, no exponential on k((G)) is compatible, that is, induces an exponential on k through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable.
keywords: S: ico, (stal)
Sh:602
Hyttinen, T., & Shelah, S. (1999). Constructing strongly equivalent nonisomorphic models for unsuperstable theories. Part C. J. Symbolic Logic, 64(2), 634–642. arXiv: math/9709229 DOI: 10.2307/2586489 MR: 1777775
type: article
abstract: In this paper we prove a strong nonstructure theorem for \kappa(T)-saturated models of a stable theory T with dop.
keywords: M: non, (mod)
Sh:603
Shelah, S. (2002). Few non minimal types on non structure. In In the Scope of Logic, Methodology and Philosophy of Science . Volume One of the 11th International Congress of Logic, Methodology and Philosophy of Science, Cracow, August 1999, Vol. 1, Springer Netherlands, pp. 29–53. arXiv: math/9906023
type: article
abstract: We pay two debts from [Sh:576]. The main demands little knowledge from [Sh:576], just quoting a model theoretic consequence of the weak diamond. We assume that \mathfrak{K} has amalgamation in \lambda, and that the minimal types are not dense to get many non-isomorphic models in \lambda^+. For this also pcf considerations are relevant. The minor debt was the use in one point of [Sh:576] of \lambda \ne \aleph_0, it is minor as for this case by [Sh:88] we “usually” know more.
keywords: M: nec, (nni), (app(pcf)), (set-mod)
Sh:604
Shelah, S. (2005). The pair (\aleph_n,\aleph_0) may fail \aleph_0-compactness. In Logic Colloquium ’01, Vol. 20, Assoc. Symbol. Logic, Urbana, IL, pp. 402–433. arXiv: math/0404240 MR: 2143906
type: article
abstract: Let P be a distinguished unary predicate and K=\{M: M a model of cardinality \aleph_n with P^M of cardinality \aleph_0\}. We prove that consistently for n=4, for some countable first order theory T we have: T has no model in K whereas every finite subset of T has a model in K. We then show how we prove it also for n=2, too.
keywords: S: for, (set-mod)
Sh:605
Shelah, S., & Truss, J. K. (1999). On distinguishing quotients of symmetric groups. Ann. Pure Appl. Logic, 97(1-3), 47–83. arXiv: math/9805147 DOI: 10.1016/S0168-0072(98)00023-2 MR: 1682068
type: article
abstract: A study is carried out of the elementary theory of quotients of symmetric groups in a similar spirit to [Sh:24]. Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S(\mu) on an infinite cardinal \mu are all of the form S_\kappa(\mu)= the subgroup consisting of elements whose support has cardinality <\kappa for some \kappa\le\mu^+. A many-sorted structure {\mathcal M}_{\kappa\lambda\mu} is defined which, it is shown, encapsulates the first order properties of the group S_\lambda(\mu)/S_\kappa(\mu). Specifically, these two structures are (uniformly) bi-interpretable, where the interpretation of {\mathcal M}_{\kappa\lambda\mu} in S_\lambda(\mu)/S_\kappa(\mu) is in the usual sense, but in the other direction is in a weaker sense, which is nevertheless sufficient to transfer elementary equivalence. By considering separately the cases cf(\kappa) > 2^{\aleph_0}, cf(\kappa)\le 2^{\aleph_0}<\kappa, \aleph_0<\kappa< 2^{\aleph_0}, and \kappa = \aleph_0, we make a further analysis of the first order theory of S_\lambda(\mu)/S_\kappa(\mu), introducing many-sorted second order structures {\mathcal N}^2_{\kappa \lambda \mu}, all of whose sorts have cardinality at most 2^{\aleph_0}.
keywords: O: alg, (mod)
Sh:606
Shelah, S. (1999). On T_3-topological space omitting many cardinals. Period. Math. Hungar., 38(1-2), 87–98. arXiv: math/9811177 DOI: 10.1023/A:1004707417470 MR: 1721480
type: article
abstract: We prove that for every (infinite cardinal) \lambda there is a T_3-space X with clopen basis, 2^\mu points where \mu = 2^\lambda, such that every closed subspace of cardinality <|X| has cardinality <\lambda.
keywords: S: ico, (gt)
Sh:607
Bartoszyński, T., & Shelah, S. (2001). Strongly meager sets do not form an ideal. J. Math. Log., 1(1), 1–34. arXiv: math/9805148 DOI: 10.1142/S0219061301000028 MR: 1838340
type: article
abstract: A set X \subseteq {\bf R} is strongly meager if for every measure zero set H, X+H \neq {\bf R}. Let SM denote the collection of strongly meager sets. We show that assuming CH, SM is not an ideal.
keywords: S: str, (set)
Sh:608
Shelah, S., & Stanley, L. J. (2001). Forcing many positive polarized partition relations between a cardinal and its powerset. J. Symbolic Logic, 66(3), 1359–1370. arXiv: math/9710216 DOI: 10.2307/2695112 MR: 1856747
type: article
abstract: Corrected in [918]. We present a forcing for blowing up 2^\lambda and making “many positive polarized partition relations” (in a sense made precise in (c) of our main theorem) hold in the interval [\lambda,\ 2^\lambda]. This generalizes results of [276], Section 1, and the forcing is a ‘‘many cardinals” version of the forcing there.
keywords: S: for, (set), (pc)
Sh:609
Kojman, M., & Shelah, S. (1998). A ZFC Dowker space in \aleph_{\omega+1}: an application of PCF theory to topology. Proc. Amer. Math. Soc., 126(8), 2459–2465. arXiv: math/9512202 DOI: 10.1090/S0002-9939-98-04884-9 MR: 1605988
type: article
abstract: A Dowker space is a normal Hausdorff topological space whose product with the unit interval is not normal. Using pcf theory we construct a Dowker space of cardinality \aleph_{\omega+1}.
keywords: S: pcf, O: top, (gt), (app(pcf))
Sh:610
Shelah, S., & Zapletal, J. (1999). Canonical models for \aleph_1-combinatorics. Ann. Pure Appl. Logic, 98(1-3), 217–259. arXiv: math/9806166 DOI: 10.1016/S0168-0072(98)00022-0 MR: 1696852
type: article
abstract: We define the property of \Pi_2-compactness of a statement \phi of set theory, meaning roughly that the hard core of the impact of \phi on combinatorics of \aleph_1 can be isolated in a canonical model for the statement \phi. We show that the following statements are \Pi_2-compact: “dominating number=\aleph_1,” “cofinality of the meager ideal=\aleph_1”, “cofinality of the null ideal=\aleph_1”, existence of various types of Souslin trees and variations on uniformity of measure and category =\aleph_1. Several important new metamathematical patterns among classical statements of set theory are pointed out.
keywords: S: for, S: str, S: dst, (set)
Sh:611
Rosen, E., Shelah, S., & Weinstein, S. (1997). k-universal finite graphs. In Logic and random structures (New Brunswick, NJ, 1995), Vol. 33, Amer. Math. Soc., Providence, RI, pp. 65–77. arXiv: math/9604244 MR: 1465469
type: article
abstract: This paper investigates the class of k-universal finite graphs, a local analog of the class of universal graphs, which arises naturally in the study of finite variable logics. The main results of the paper, which are due to Shelah, establish that the class of k-universal graphs is not definable by an infinite disjunction of first-order existential sentences with a finite number of variables and that there exist k-universal graphs with no k-extendible induced subgraphs.
keywords: M: odm, O: fin, (fmt), (graph), (univ)
Sh:612
Juhász, I., & Shelah, S. (1998). On the cardinality and weight spectra of compact spaces. II. Fund. Math., 155(1), 91–94. arXiv: math/9703220 MR: 1487990
type: article
abstract: Let B(\kappa,\lambda) be the subalgebra of {\mathcal P}(\kappa) generated by [\kappa]^{\le\lambda}. It is shown that if B is any homomorphic image of B(\kappa,\lambda) then either |B|< 2^\lambda or |B|=|B|^\lambda, moreover if X is the Stone space of B then either |X|\le 2^{2^\lambda} or |X|=|B|=|B|^\lambda. This implies the existence of 0-dimensional compact T_2 spaces whose cardinality and weight spectra omit lots of singular cardinals of “small” cofinality.
keywords: S: ico, (gt)
Sh:613
Jin, R., & Shelah, S. (1998). Compactness of Loeb spaces. J. Symbolic Logic, 63(4), 1371–1392. arXiv: math/9604211 DOI: 10.2307/2586655 MR: 1665731
type: article
abstract: In this paper we show that the compactness of a Loeb space depends on its cardinality, the nonstandard universe it belongs to and the underlying model of set theory we live in. In §1 we prove that Loeb spaces are compact under various assumptions, and in §2 we prove that Loeb spaces are not compact under various other assumptions. The results in §1 and §2 give a quite complete answer to a question of D. Ross.
keywords: M: odm
Sh:614
Džamonja, M., & Shelah, S. (2004). On the existence of universal models. Arch. Math. Logic, 43(7), 901–936. arXiv: math/9805149 DOI: 10.1007/s00153-004-0235-1 MR: 2096141
type: article
abstract: Suppose that \lambda=\lambda^{<\lambda}\ge\aleph_0, and we are considering a theory T. We give a criterion on T which is sufficient for the consistent existence of \lambda^{++} universal models of T of size \lambda^+ for models of T of size \le\lambda^+, and is meaningful when 2^{\lambda^+}>\lambda^{++}. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories.

Having in mind possible aplpications in analysis, we further observe that for such \lambda, for any fixed \mu>\lambda^+ regular with \mu=\mu^{\lambda^+}, it is consistent that 2^\lambda=\mu and there is no normed vector space over {\mathbb Q} of size <\mu which is universal for normed vector spaces over {\mathbb Q} of dimension \lambda^+ under the notion of embedding h which specifies (a,b) such that \|h(x)\|/\|x\|\in (a,b) for all x.
keywords: S: for, (set), (iter), (univ)
Sh:615
Kuhlmann, F.-V., Kuhlmann, S., & Shelah, S. (2003). Functorial equations for lexicographic products. Proc. Amer. Math. Soc., 131(10), 2969–2976. arXiv: math/0107206 DOI: 10.1090/S0002-9939-03-06830-8 MR: 1993201
type: article
abstract: We generalize the main result of [KKSh:601] concerning the convex embeddings of a chain \Gamma in a lexicographic power \Delta^\Gamma. For a fixed nonempty chain \Delta, we derive necessary and sufficient conditions for the existence of nonempty solutions \Gamma to each of the lexicographic functional equations (\Delta^\Gamma)^{\leq 0}\simeq \Gamma, (\Delta^\Gamma)\simeq \Gamma, and (\Delta^\Gamma)^{<0}\simeq\Gamma.
keywords: S: ico, (stal)
Sh:616
Bartoszyński, T., Rosłanowski, A., & Shelah, S. (2000). After all, there are some inequalities which are provable in ZFC. J. Symbolic Logic, 65(2), 803–816. arXiv: math/9711222 DOI: 10.2307/2586571 MR: 1771087
type: article
abstract: We address ZFC inequalities between some cardinal invariants of the continuum, which turned to be true in spite of strong expectations given by [RoSh:470].
keywords: S: ico, S: str, (set), (pure(for))
Sh:617
Eklof, P. C., Huisgen-Zimmermann, B., & Shelah, S. (1997). Torsion modules, lattices and p-points. Bull. London Math. Soc., 29(5), 547–555. arXiv: math/9703221 DOI: 10.1112/S0024609397003329 MR: 1458714
type: article
abstract: Answering a long-standing question in the theory of torsion modules, we show that weakly productively bounded domains are necessarily productively bounded. Moreover, we prove a twin result for the ideal lattice L of a domain equating weak and strong global intersection conditions for families (X_i)_{i\in I} of subsets of L with the property that \bigcap_{i\in I} A_i\ne 0 whenever A_i\in X_i. Finally, we show that, for domains with Krull dimension (and countably generated extensions thereof), these lattice-theoretic conditions are equivalent to productive boundedness.
keywords: O: top, (ab)
Sh:618
Hamkins, J. D., & Shelah, S. (1998). Superdestructibility: a dual to Laver’s indestructibility. J. Symbolic Logic, 63(2), 549–554. arXiv: math/9612227 DOI: 10.2307/2586848 MR: 1625927
type: article
abstract: After small forcing, any <\kappa-closed forcing will destroy the supercompactness, even the strong compactness, of \kappa.
keywords: S: for, (set), (pure(large))
Sh:619
Shelah, S. (2003). The null ideal restricted to some non-null set may be \aleph_1-saturated. Fund. Math., 179(2), 97–129. arXiv: math/9705213 DOI: 10.4064/fm179-2-1 MR: 2029228
type: article
abstract: Our main result is that possibly some non-null set of reals cannot be divided to uncountably many non-null sets. We deal also with a non-null set of reals, the graph of any function from it is null and deal with our iterations somewhat more generally.
keywords: S: for, S: str, (set), (leb)
Sh:620
Shelah, S. (1999). Special subsets of ^{\mathrm{cf}(\mu)}\mu, Boolean algebras and Maharam measure algebras. Topology Appl., 99(2-3), 135–235. arXiv: math/9804156 DOI: 10.1016/S0166-8641(99)00138-8 MR: 1728851
type: article
abstract: The original theme of the paper is the existence proof of “there is \bar{\eta}=\langle\eta_\alpha:\alpha<\lambda\rangle which is a (\lambda,J)-sequence for \bar{I}=\langle I_i:i< \delta\rangle, a sequence of ideals. This can be thought of as in a generalization to Luzin sets and Sierpinski sets, but for the product \prod_{i< \delta} Dom(I_i), the existence proofs are related to pcf . The second theme is when does a Boolean algebra B has free caliber \lambda (i.e. if X\subseteq B and |X|=\lambda, then for some Y\subseteq X with |Y|=\lambda and Y is independent). We consider it for B being a Maharam measure algebra, or B a (small) product of free Boolean algebras, and \kappa-cc Boolean algebras. A central case \lambda= (\beth_\omega)^+ or more generally, \lambda=\mu^+ for \mu strong limit singular of “small” cofinality. A second one is \mu=\mu^{<\kappa}<\lambda< 2^\mu; the main case is \lambda regular but we also have things to say on the singular case. Lastly, we deal with ultraproducts of Boolean algebras in relation to irr(-) and s(-) Length, etc.
keywords: S: pcf, (set), (ba), (leb), (inv(ba))
Sh:621
Eklof, P. C., & Shelah, S. (2001). A non-reflexive Whitehead group. J. Pure Appl. Algebra, 156(2-3), 199–214. arXiv: math/9908157 DOI: 10.1016/S0022-4049(99)00152-8 MR: 1808823
type: article
abstract: It is proved, via an iterated forcing construction with finite support, that it is consistent that there is a strongly \aleph_1-free Whitehead group of cardinality \aleph_1 which is strongly non-reflexive. It is also proved consistent that there is a group A satisfying Ext(A, Z) is torsion and Hom(A,Z)=0.
keywords: O: alg, (ab), (stal), (wh)
Sh:622
Shelah, S. (2001). Non-existence of universal members in classes of abelian groups. J. Group Theory, 4(2), 169–191. arXiv: math/9808139 DOI: 10.1515/jgth.2001.014 MR: 1812323
type: article
abstract: We prove that if \mu^+<\lambda=cf(\lambda)<\mu^{\aleph_0}, then there is no universal reduced torsion free abelian group. Similarly if \aleph_0<\lambda< 2^{\aleph_0}. We also prove that if 2^{\aleph_0}<\mu^+<\lambda=cf(\lambda)< \mu^{\aleph_0}, then there is no universal reduced separable abelian p-group in \lambda. (Note: both results fail if \lambda = \lambda^{\aleph_0} or if \lambda is strong limit, cf(\mu)=\aleph_0<\mu).
keywords: S: ico, O: alg, (ab), (app(pcf)), (univ)
Sh:623
Baldwin, J. T., & Shelah, S. (2000). On the classifiability of cellular automata. Theoret. Comput. Sci., 230(1-2), 117–129. arXiv: math/9801152 DOI: 10.1016/S0304-3975(99)00042-0 MR: 1725633
type: article
abstract: Based on computer simulations Wolfram presented in several papers conjectured classifications of cellular automata into 4 types. He distinguishes the 4 classes of cellular automata by the evolution of the pattern generated by applying a cellular automaton to a finite input. Wolfram’s qualitative classification is based on the examination of a large number of simulations. In addition to this classification based on the rate of growth, he conjectured a similar classification according to the eventual pattern. We consider here one formalization of his rate of growth suggestion. After completing our major results (based only on Wolfram’s work), we investigated other contributions to the area and we report the relation of some them to our discoveries.
keywords: O: fin
Sh:624
Shelah, S. (1999). On full Suslin trees. Colloq. Math., 79(1), 1–7. arXiv: math/9608215 DOI: 10.4064/cm-79-1-1-7 MR: 1665618
type: article
abstract: In this note we answer a question of Kunen (15.13 [Mi91]) showing that it is consistent that there are full Souslin trees
keywords: S: for, (set), (trees)
Sh:625
Eklof, P. C., & Shelah, S. (1998). The Kaplansky test problems for \aleph_1-separable groups. Proc. Amer. Math. Soc., 126(7), 1901–1907. arXiv: math/9709230 DOI: 10.1090/S0002-9939-98-04749-2 MR: 1485469
type: article
abstract: We answer a long-standing open question by proving in ordinary set theory, ZFC, that the Kaplansky test problems have negative answers for \aleph_1-separable abelian groups of cardinality \aleph_1. In fact, there is an \aleph_1-separable abelian group M such that M is isomorphic to M\oplus M\oplus M but not to M\oplus M.
keywords: M: non, (ab)
Sh:626
Jin, R., & Shelah, S. (1999). Possible size of an ultrapower of \omega. Arch. Math. Logic, 38(1), 61–77. arXiv: math/9801153 DOI: 10.1007/s001530050115 MR: 1667288
type: article
abstract: Let \omega be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In §1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of \omega, whose cardinality is (1) a singular strong limit cardinal, (2) a strongly inaccessible cardinal. This answers two questions in [CK], modulo the assumption of supercompactness. In §2 we construct several \lambda-Archimedean ultrapowers of \omega under some large cardinal assumptions. For example, we show that, assuming the consistency of a measurable cardinal, there may exist a \lambda-Archimedean ultrapower of \omega for some uncountable cardinal \lambda. This answers a question in [KS], modulo the assumption of measurability.
keywords: S: for, (set), (up)
Sh:627
Shelah, S. (1998). Erdős and Rényi conjecture. J. Combin. Theory Ser. A, 82(2), 179–185. arXiv: math/9707226 DOI: 10.1006/jcta.1997.2845 MR: 1620869
type: article
abstract: Affirming a conjecture of Erdös and Rényi we prove that for any (real number) c_1>0 for some c_2>0, if a graph G has no c_1(\log n) nodes on which the graph is complete or edgeless (i.e. G exemplifies |G|\not\rightarrow (c_1\log n)^2_2) then G has at least 2^{c_2n} non-isomorphic (induced) subgraphs.
keywords: O: fin, (nni), (graph), (fc)
Sh:628
Rosłanowski, A., & Shelah, S. (1997). Norms on possibilities. II. More ccc ideals on 2^\omega. J. Appl. Anal., 3(1), 103–127. arXiv: math/9703222 DOI: 10.1515/JAA.1997.103 MR: 1618851
type: article
abstract: We use the method of norms on possibilities to answer a question of Kunen and construct a ccc \sigma–ideal on 2^\omega with various closure properties and distinct from the ideal of null sets, the ideal of meager sets and their intersection.
keywords: S: for, S: str, (set), (creatures)
Sh:629
Hyttinen, T., & Shelah, S. (2000). Strong splitting in stable homogeneous models. Ann. Pure Appl. Logic, 103(1-3), 201–228. arXiv: math/9911229 DOI: 10.1016/S0168-0072(99)00044-5 MR: 1756146
type: article
abstract: In this paper we study elementary submodels of a stable homogeneous structure. We improve the independence relation defined in [Hy1]. We apply this to prove a structure theorem. We also solve a question from [Hy3]: We show that dop and sdop are essentially equivalent, where the negation of dop is a property used to get structure theorems and sdop implies nonstructure.
keywords: M: cla, (mod)
Sh:630
Shelah, S. (2004). Properness without elementaricity. J. Appl. Anal., 10(2), 169–289. arXiv: math/9712283 DOI: 10.1515/JAA.2004.169 MR: 2115943
type: article
abstract: We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary sub-models of some (\mathcal H(\chi),\in). This leads to forcing notions which are “reasonably" definable. We present two specific properties materializing this intuition: nep (non-elementary properness) and snep (Souslin non-elementary properness) and also the older Souslin proper. For this we consider candidates (countable models to which the definition applies). A major theme here is “preservation by iteration”, but we also show a dichotomy: if such forcing notions preserve the positiveness of the set of old reals for some naturally defined c.c.c. ideal, then they preserve the positiveness of any old positive set hence preservation by composition of two follows. Last but not least, we prove that (among such forcing notions) the only one commuting with Cohen is Cohen itself; in other words, any other such forcing notion make the set of old reals to a meager set. In the end we present some open problems in this area.
keywords: S: str, (set), (iter)
Sh:631
Rabus, M., & Shelah, S. (1999). Topological density of ccc Boolean algebras—every cardinality occurs. Proc. Amer. Math. Soc., 127(9), 2573–2581. arXiv: math/9709231 DOI: 10.1090/S0002-9939-99-04813-3 MR: 1486748
type: article
abstract: For every uncountable cardinal \mu there is a ccc Boolean algebra whose topological density is \mu.
keywords: S: ico, O: top, (set), (ba)
Sh:632
Hyttinen, T., & Shelah, S. (1998). On the number of elementary submodels of an unsuperstable homogeneous structure. MLQ Math. Log. Q., 44(3), 354–358. arXiv: math/9702228 DOI: 10.1002/malq.19980440307 MR: 1645490
type: article
abstract: We show that if M is a stable unsuperstable homogeneous structure, then for most \kappa< |M|, the number of elementary submodels of M of power \kappa is 2^\kappa.
keywords: M: cla, M: non, (mod), (nni)
Sh:633
Goldstern, M., & Shelah, S. (1998). Order polynomially complete lattices must be large. Algebra Universalis, 39(3-4), 197–209. arXiv: math/9707203 DOI: 10.1007/s000120050075 MR: 1636999
type: article
abstract: If L is an order-polynomially complete lattice, then the cardinality of L must be a strongly inaccessible cardinal
keywords: S: ico, O: alg, O: fin, (ua)
Sh:634
Shelah, S. (2000). Choiceless Polynomial Time Logic: Inability to express. In P. G. Clote & H. Schwichtenberg, eds., Computer Science Logic, 14th International Workshop, CSL 2000, Annual Conference of the EACSL, Fischbachau, Germany, August 21–26, 2000, Proceedings, Vol. 1862, Springer, pp. 72–125. arXiv: math/9807179
type: article
abstract: We prove for the logic CPTime (the logic from the title) a sufficient condition for two models to be equivalent for any set of sentences which is “small” (certainly any finite set ), parallel to the Ehrenfeucht Fraïssé games. This enables us to show that sentences cannot express some properties in the logic CPTime and prove 0-1 laws for it.
keywords: M: smt, O: fin, (fmt)
Sh:635
Shelah, S., & Villaveces, A. (1999). Toward categoricity for classes with no maximal models. Ann. Pure Appl. Logic, 97(1-3), 1–25. arXiv: math/9707227 DOI: 10.1016/S0168-0072(98)00015-3 MR: 1682066
type: article
abstract: We provide here the first steps toward Classification Theory of Abstract Elementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some \lambda greater than its Löwenheim-Skolem number. We study the degree to which amalgamation may be recovered, the behaviour of non \mu-splitting types. Most importantly, the existence of saturated models in a strong enough sense is proved, as a first step toward a complete solution to the Los Conjecture for these classes.
keywords: M: nec, (mod), (cat)
Sh:636
Shelah, S. (1998). The lifting problem with the full ideal. J. Appl. Anal., 4(1), 1–17. arXiv: math/9712284 DOI: 10.1515/JAA.1998.1 MR: 1648938
type: article
abstract: We prove in ZFC that for \mu\geq\aleph_2 there is a \sigma–ideal I on \mu and a Boolean \sigma–subalgebra B of the family of subsets of \mu which includes I such that the natural homomorphism from B onto B/I cannot be lifted.
keywords: S: ico, (set)
Sh:638
Shelah, S. (2021). More on weak diamond. Acta Math. Hungar., 165(1), 1–27. arXiv: math/9807180 DOI: 10.1007/s10474-021-01182-2 MR: 4323582
type: article
abstract: We deal with the combinatorial principle Weak Diamond, showing that we always either a local version is not saturated or we can increase the number of colours. Then we point out a model theoretic consequence of Weak Diamond.
keywords: S: ico, (set), (wd)
Sh:639
Shelah, S. (2000). On quantification with a finite universe. J. Symbolic Logic, 65(3), 1055–1075. arXiv: math/9809201 DOI: 10.2307/2586688 MR: 1791364
type: article
abstract: We consider a finite universe {\mathcal U} (more exactly - a family \mathfrak{U} of them). Can second order quantifier Q_K, where for each {\mathcal U} this means quantifying over a family of n(K)-place relations closed under permuting {\mathcal U}. We define some natural orders and shed some light on the classification problem of those quantifiers.
keywords: M: odm, O: fin, (fmt)
Sh:640
Błaszczyk, A., & Shelah, S. (2001). Regular subalgebras of complete Boolean algebras. J. Symbolic Logic, 66(2), 792–800. arXiv: math/9712285 DOI: 10.2307/2695044 MR: 1833478
type: article
abstract: It is shown that there exists a complete, atomless, \sigma-centered Boolean algebra, which does not contain any regular countable subalgebra if and only if there exist a nowhere dense ultrafilter. Therefore the existence of such algebras is undecidable in ZFC.
keywords: S: for, S: str, (set), (pure(for))
Sh:641
Shelah, S. (2001). Constructing Boolean algebras for cardinal invariants. Algebra Universalis, 45(4), 353–373. arXiv: math/9712286 DOI: 10.1007/s000120050219 MR: 1816973
type: article
abstract: We construct Boolean Algebras answering questions of Monk on cardinal invariants. The results are proved in ZFC (rather than giving consistency results). We deal with the existence of superatomic Boolean Algebras with “few automorphisms", with entangled sequences of linear orders, and with semi-ZFC examples of the non-attainment of the spread (and hL, hd).
keywords: S: ico, (ba), (app(pcf)), (auto), (linear order), (inv(ba))
Sh:642
Brendle, J., & Shelah, S. (1999). Ultrafilters on \omega—their ideals and their cardinal characteristics. Trans. Amer. Math. Soc., 351(7), 2643–2674. arXiv: math/9710217 DOI: 10.1090/S0002-9947-99-02257-6 MR: 1686797
type: article
abstract: For a free ultrafilter \mathcal U on \omega we study several cardinal characteristics which describe part of the combinatorial structure of \mathcal U. We provide various consistency results; e.g. we show how to force simultaneously many characters and many \pi–characters. We also investigate two ideals on the Baire space \omega^\omega naturally related to \mathcal U and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter.
keywords: S: str, (set), (inv)
Sh:643
Shelah, S., & Spasojević, Z. (2002). Cardinal invariants \mathfrak b_\kappa and \mathfrak t_\kappa. Publ. Inst. Math. (Beograd) (N.S.), 72(86), 1–9. arXiv: math/0003141 DOI: 10.2298/PIM0272001S MR: 1997605
type: article
abstract: This paper studies cardinal invariants \mathfrak{b}_\kappa and \mathfrak{t}_\kappa, the natural generalizations of the invariants \mathfrak{b} and \mathfrak{t} to a regular cardinal \kappa.
keywords: S: ico, S: for, (set), (iter)
Sh:644
Shelah, S., & Väisänen, P. (2000). On inverse \gamma-systems and the number of L_{\infty\lambda}-equivalent, non-isomorphic models for \lambda singular. J. Symbolic Logic, 65(1), 272–284. arXiv: math/9807181 DOI: 10.2307/2586536 MR: 1782119
type: article
abstract: Suppose \lambda is a singular cardinal of uncountable cofinality \kappa. For a model M of cardinality \lambda, let No(M) denote the number of isomorphism types of models N of cardinality \lambda which are L_{\infty\lambda}-equivalent to M. In [Sh:189] inverse \kappa-systems A of abelian groups and their certain kind of quotient limits Gr(A)/Fact(A) were considered. It was proved that for every cardinal \mu there exists an inverse \kappa-system A such that A consists of abelian groups having cardinality at most \mu^\kappa and card(Gr(A)/Fact(A))=\mu. In [Sh:228] a strict connection between inverse \kappa-systems and possible values of No was proved. In this paper we show: for every nonzero \mu\leq\lambda^\kappa there is an inverse \kappa-system A of abelian groups having cardinality <\lambda such that card(Gr(A)/Fact(A))= \mu (under the assumptions 2^\kappa<\lambda and \theta^{<\kappa}<\lambda for all \theta<\lambda when \mu>\lambda), with the obvious new consequence concerning the possible value of No. Specifically, the case No(M)=\lambda is possible when \theta^\kappa<\lambda for every \theta<\lambda.
keywords: M: non, O: alg, (nni), (inf-log), (set-mod)
Sh:645
Komjáth, P., & Shelah, S. (2000). Two consistency results on set mappings. J. Symbolic Logic, 65(1), 333–338. arXiv: math/9807182 DOI: 10.2307/2586540 MR: 1782123
type: article
abstract: It is consistent that there is a set mapping from the four-tuples of \omega_n into the finite subsets with no free subsets of size t_n for some natural number t_n. For any n<\omega it is consistent that there is a set mapping from the pairs of \omega_n into the finite subsets with no infinite free sets.
keywords: S: ico, S: for, (set)
Sh:646
Shelah, S., & Väisänen, P. (2001). On the number of L_{\infty\omega_1}-equivalent non-isomorphic models. Trans. Amer. Math. Soc., 353(5), 1781–1817. arXiv: math/9908160 DOI: 10.1090/S0002-9947-00-02604-0 MR: 1707477
type: article
abstract: We prove that if ZF is consistent then ZFC + GCH is consistent with the following statement: There is for every k < \omega a model of cardinality \aleph_1 which is L_{\infty,\omega_1}-equivalent to exactly k non-isomorphic models of cardinality \aleph_1. In order to get this result we introduce ladder systems and colourings different from the "standard" counterparts, and prove the following purely combinatorial result: For each prime number p and positive integer m it is consistent with ZFC + GCH that there is a “good” ladder system having exactly p^m pairwise nonequivalent colourings.
keywords: M: non, (nni), (inf-log), (set-mod)
Sh:647
Göbel, R., & Shelah, S. (2000). Cotorsion theories and splitters. Trans. Amer. Math. Soc., 352(11), 5357–5379. arXiv: math/9910159 DOI: 10.1090/S0002-9947-00-02475-2 MR: 1661246
type: article
abstract: Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is {\rm Ext}_R(G,G)=0 holds and follow Schultz to call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples for splitters. Are there others? Answering an open problem by Schultz we will show that there are more splitters, in fact we are able to prescribe their endomorphism R-algebras with a free R-module structure. As a byproduct we are able to answer a problem of Salce showing that all rational cotorsion theories have enough injectives and enough projectives.
keywords: O: alg, (ab), (stal)
Sh:648
Shelah, S., & Villaveces, A. (2021). The Hart-Shelah example, in stronger logics. Ann. Pure Appl. Logic, 172(6), 102958, 23. arXiv: math/0404258 DOI: 10.1016/j.apal.2021.102958 MR: 4216281
type: article
abstract: We generalize the Hart-Shelah example from [HaSh:323] to higher infinitary logics. We build, for each natural number k\geq 2 and for each infinite cardinal \lambda, a sentence \psi_k^\lambda of the logic L_{(2^\lambda)^+,\omega} that (modulo mild set theoretical hypotheses around \lambda and assuming 2^\lambda < \lambda^{+k}) is categorical in \lambda^+,\dots,\lambda^{+k-1} but not in \beth_{k+1}(\lambda)^+ (or beyond); we study the dimensional encoding of combinatorics involved in the construction of this sentence and study various model-theoretic properties of the resulting abstract elementary class {\mathcal K}^*(\lambda,k)=(Mod(\psi_k^\lambda),\prec_{(2^\lambda)^+,\omega}) in the finite interval of cardinals \lambda,\lambda^+,\dots,\lambda^{+k}.
keywords: M: nec, (mod)
Sh:649
Kojman, M., & Shelah, S. (1999). Regressive Ramsey numbers are Ackermannian. J. Combin. Theory Ser. A, 86(1), 177–181. arXiv: math/9805150 DOI: 10.1006/jcta.1998.2906 MR: 1682971
type: article
abstract: We give an elementary proof of the fact that regressive Ramsey numbers are Ackermannian. This fact was first proved by Kanamori and McAloon with mathematical logic techniques.
keywords: O: fin, (pc), (fc)
Sh:650
Göbel, R., & Shelah, S. (2004). Uniquely transitive torsion-free abelian groups. In Rings, modules, algebras, and abelian groups, Vol. 236, Dekker, New York, pp. 271–290. arXiv: math/0404259 MR: 2050717
type: article
abstract: We will answer a question raised by Farjoun concerning the existence of torsion-free abelian groups G such that for any ordered pair of pure elements there is a unique automorphism mapping the first element onto the second one. We will show the existence of many such groups in the constructible universe L.
keywords: M: non, (ab)
Sh:651
Rosłanowski, A., & Shelah, S. (2001). Forcing for hL and hd. Colloq. Math., 88(2), 273–310. arXiv: math/9808104 DOI: 10.4064/cm88-2-9 MR: 1852911
type: article
abstract: The present paper addresses the problem of attainment of the supremums in various equivalent definitions of hereditary density {\rm hd} and hereditary Lindelöf degree {\rm hL} of Boolean algebras. We partially answer two problems of J. Donald Monk, Problems 50, 54, showing consistency of different attainment behaviour and proving that (for the considered variants) this is the best result we can expect.
keywords: S: for, (set), (ba), (inv(ba))
Sh:652
Shelah, S. (2002). More constructions for Boolean algebras. Arch. Math. Logic, 41(5), 401–441. arXiv: math/9605235 DOI: 10.1007/s001530100099 MR: 1918108
type: article
abstract: We address a number of problems on Boolean Algebras. For example, we construct, in ZFC, for any BA B, and cardinal \kappa BAs B_1,B_2 extending B such that the depth of the free product of B_1,B_2 over B is strictly larger than the depths of B_1 and of B_2 than \kappa. We give a condition (for \lambda, \mu and \theta) which implies that for some BA A_\theta there are B_1=B^1_{\lambda,\mu,\theta} and B_2B^2_{\lambda,\mu,\theta} such that Depth(B_t)\leq\mu and Depth(B_1\oplus_{A_\theta} B_1) \geq \lambda. We then investigate for a fixed A, the existence of such B_1,B_2 giving sufficient and necessary conditions, involving consistency results. Further we prove that e.g. if B is a BA of cardinality \lambda, \lambda\ge\mu and \lambda,\mu are strong limit singular of the same cofinality, then B has a homomorphic image of cardinality \mu (and with \mu ultrafilters). Next we show that for a BA B, if d(B)^\kappa<|B| then ind(B)>\kappa or Depth(B)\geq\log(|B|). Finally we prove that if \square_\lambda holds and \lambda=\lambda^{\aleph_0} then for some BAs B_n, Depth(B_n)\leq\lambda but for any uniform ultrafilter D on \omega, \prod_{n<\omega} B_n/D has depth \ge\lambda^+.
keywords: S: ico, (set), (ba), (app(pcf))
Sh:653
Ciesielski, K. C., & Shelah, S. (1999). A model with no magic set. J. Symbolic Logic, 64(4), 1467–1490. arXiv: math/9801154 DOI: 10.2307/2586790 MR: 1780064
type: article
abstract: We will prove that there exists a model of ZFC+“\mathfrak{c}=\omega_2” in which every M\subseteq R of cardinality less than continuum \mathfrak{c} is meager, and such that for every X\subseteq R of cardinality \mathfrak{c} there exists a continuous function f:R\to R with f[X]=[0,1]. In particular in this model there is no magic set, i.e., a set M\subseteq R such that the equation f[M]=g[M] implies f=g for every continuous nowhere constant functions f,g:R\to R.
keywords: S: for, S: str, (set), (creatures)
Sh:654
Just, W., Shelah, S., & Thomas, S. (1999). The automorphism tower problem revisited. Adv. Math., 148(2), 243–265. arXiv: math/0003120 DOI: 10.1006/aima.1999.1852 MR: 1736959
type: article
abstract: It is known that the automorphism towers of infinite centreless groups of cardinality \kappa terminate in less than \left( 2^{\kappa} \right)^{+} steps. In this paper, we show that ZFC cannot settle the question of whether such automorphism towers actually terminate in less than 2^{\kappa} steps.
keywords: S: for, (stal), (auto), (grp)
Sh:655
Rosłanowski, A., & Shelah, S. (2001). Iteration of \lambda-complete forcing notions not collapsing \lambda^+. Int. J. Math. Math. Sci., 28(2), 63–82. arXiv: math/9906024 DOI: 10.1155/S016117120102018X MR: 1885053
type: article
abstract: We look for a parallel to the notion of “proper forcing" among \lambda-complete forcing notions not collapsing \lambda^+. We suggest such a definition and prove that it is preserved by suitable iterations.
keywords: S: for, (set)
Sh:657
Shelah, S., & Väänänen, J. A. (2000). Stationary sets and infinitary logic. J. Symbolic Logic, 65(3), 1311–1320. arXiv: math/9706225 DOI: 10.2307/2586701 MR: 1791377
type: article
abstract: Let K^0_\lambda be the class of structures \langle\lambda,< ,A\rangle, where A\subseteq\lambda is disjoint from a club, and let K^1_\lambda be the class of structures \langle\lambda,< ,A\rangle, where A\subseteq\lambda contains a club. We prove that if \lambda=\lambda^{< \kappa} is regular, then no sentence of L_{\lambda^+\kappa} separates K^0_\lambda and K^1_\lambda. On the other hand, we prove that if \lambda=\mu^+, \mu=\mu^{< \mu}, and a forcing axiom holds (and \aleph_1^L=\aleph_1 if \mu=\aleph_0), then there is a sentence of L_{\lambda\lambda} which separates K^0_\lambda and K^1_\lambda.
keywords: S: for, S: str, (inf-log), (set-mod)
Sh:658
Bartoszyński, T., & Shelah, S. (2002). Strongly meager and strong measure zero sets. Arch. Math. Logic, 41(3), 245–250. arXiv: math/9907137 DOI: 10.1007/s001530000068 MR: 1901186
type: article
abstract: In this paper we present several consistency results concerning the existence of large strong measure zero and strongly meager sets.
keywords: S: str, (set)
Sh:659
Džamonja, M., & Shelah, S. (2003). Universal graphs at the successor of a singular cardinal. J. Symbolic Logic, 68(2), 366–388. arXiv: math/0102043 DOI: 10.2178/jsl/1052669056 MR: 1976583
type: article
abstract: The paper is concerned with the existence of a universal graph at the successor of a strong limit singular \mu of cofinality \aleph_0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such \mu there are \mu^{++} graphs on \mu^+ that taken jointly are universal for the graphs on \mu^+, while 2^{\mu^+}>>\mu^{++}. The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal \kappa exists. The result on the existence of universal graphs is obtained as a specific application of a more general method.
keywords: S: for, (set), (iter), (graph), (univ)
Sh:660
Shelah, S. Covering numbers associated with trees branching into a countably generated set of possibilities. Real Anal. Exchange, 24(1), 205–213. arXiv: math/9711223 MR: 1691746
type: article
abstract: This paper is concerned with certain generalizations of meagreness and their combinatorial equivalents. The simplest example, and the one which motivated further study in this area, comes about by considering the following definition: a set X\subseteq R is said to be Q-nowhere dense if and only if for every rational q there exists and integer k such that the interval whose endpoints are q and q+1/k is disjoint from X. A set which is the union of countably many Q-nowhere dense sets will be called Q-very meagre.

Steprans considered the least number of Q-meagre sets required to cover the real line and denoted by d_1. He showed that there is a continuous function H — first constructed by Lebesgue — such that the least number of smooth functions into which H can be decomposed is equal to d_1. This paper will further study d_1 and some of its generalizations. As well, an equivalence will be established between Q-meagreness and certain combinatorial properties of trees. This will lead to new cardinal invariants and various independence results about these will then be established.
keywords: S: for, S: str, (set)
Sh:661
Kolman, O., & Shelah, S. (1998). A result related to the problem CN of Fremlin. J. Appl. Anal., 4(2), 161–165. arXiv: math/9712287 DOI: 10.1515/JAA.1998.161 MR: 1667030
type: article
abstract: We show that the set of injective functions from any uncountable cardinal less than the continuum into the real numbers is of second category in the box product topology.
keywords: S: ico, (set)
Sh:662
Halko, A., & Shelah, S. (2001). On strong measure zero subsets of ^\kappa2. Fund. Math., 170(3), 219–229. arXiv: math/9710218 DOI: 10.4064/fm170-3-1 MR: 1880900
type: article
abstract: This paper answers three questions posed by the first author. In Theorem 2.6 we show that the family of strong measure zero subsets of {}^{\omega_1}2 is 2^{\aleph_1}-additive under GMA and CH. In Theorem 3.1 we prove that the generalized Borel conjecture is false in {}^{\omega_1}2 assuming ZFC+CH. Next, in Theorem 4.2, we show that the family of subsets of {}^{\omega_1}2 with the property of Baire is not closed under the Souslin operation.
keywords: S: for, (set), (ps-dst)
Sh:663
Shelah, S., & Spinas, O. (1999). On tightness and depth in superatomic Boolean algebras. Proc. Amer. Math. Soc., 127(12), 3475–3480. arXiv: math/9802135 DOI: 10.1090/S0002-9939-99-04944-8 MR: 1610793
type: article
abstract: We introduce a large cardinal property which is consistent with L and show that for every superatomic Boolean algebra B and every cardinal \lambda with the large cardinal property, if tightness^+(B)\geq\lambda^+, then depth(B)\geq\lambda. This improves a theorem of Dow and Monk.
keywords: S: for, (ba), (inv(ba))
Sh:664
Shelah, S. (2001). Strong dichotomy of cardinality. Results Math., 39(1-2), 131–154. arXiv: math/9807183 DOI: 10.1007/BF03322680 MR: 1817405
type: article
abstract: A usual dichotomy is that in many cases, reasonably definable sets, satisfy the CH, i.e. if they are uncountable they have cardinality continuum. A strong dichotomy is when: if the cardinality is infinite it is continuum as in [Sh:273]. We are interested in such phenomena when \lambda=\aleph_0 is replaced by \lambda regular uncountable and also by \lambda=\beth_\omega or more generally by strong limit of cofinality \aleph_0.
keywords: S: ods, (stal), (ps-dst)
Sh:665
Shelah, S., & Steprāns, J. (2001). The covering numbers of Mycielski ideals are all equal. J. Symbolic Logic, 66(2), 707–718. arXiv: math/9712288 DOI: 10.2307/2695039 MR: 1833473
type: article
abstract: The Mycielski ideal M_k is defined to consist of all sets A\subseteq k^\omega such that \{f\restriction X: f\in A\}\neq k^X for all X\in [\omega]^{\aleph_0}. It will be shown that the covering numbers for these ideals are all equal. However, the covering numbers of the closely associated Roslanowski ideals will be shown to be consistently different.
keywords: S: for, S: str, (set), (creatures)
Sh:666
Shelah, S. (2000). On what I do not understand (and have something to say). I. Fund. Math., 166(1-2), 1–82. arXiv: math/9906113 MR: 1804704
type: article
abstract: This is a non-standard paper, containing some problems I have in various degrees been interested in, sometimes with discussion on what I have to say; why they seem interesting, sometimes how I have tried to solve them, failed tries, anecdote and opinion, so the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (see \ldots means: see the references there and possibly the paper itself).

The base were lectures in Rutgers Fall ’97.
keywords: S: ods, (set), (stal)
Sh:667
Shelah, S. (2003). Successor of singulars: combinatorics and not collapsing cardinals \le\kappa in (<\kappa)-support iterations. Israel J. Math., 134, 127–155. arXiv: math/9808140 DOI: 10.1007/BF02787405 MR: 1972177
type: article
abstract: We deal with (< \kappa)-supported iterated forcing notions which are (E_0,E_1)-complete, have in mind problems on Whitehead groups, uniformizations and the general problem. We deal mainly with the successor of a singular case. This continues [Sh:587]. We also deal with complimentary combinatorial results.
keywords: S: ico, S: for, (set), (unif)
Sh:668
Shelah, S. (2004). Anti-homogeneous partitions of a topological space. Sci. Math. Jpn., 59(2), 203–255. arXiv: math/9906025 MR: 2062196
type: article
abstract: We prove the consistency (modulo supercompact) of a negative answer to Arhangelskii’s problem (some Hausdorff compact space cannot be partitioned to two sets not containing a closed copy of Cantor discontinuum). In this model we have CH. Without CH we get consistency results using a pcf assumption, close relatives of which are necessary for such results.
keywords: S: for, S: pcf, (set), (gt)
Sh:669
Shelah, S. (2006). Non-Cohen oracle C.C.C. J. Appl. Anal., 12(1), 1–17. arXiv: math/0303294 DOI: 10.1515/JAA.2006.1 MR: 2243849
type: article
abstract: The oracle c.c.c. is closed related to Cohen forcing. During an iteration we can “omit a type”; i.e. preserve “the intersection of a given family of Borel sets of reals is empty” provided that Cohen forcing satisfies it. We generalize this to other cases. In §1 we replace Cohen by “nicely” definable c.c.c., do the parallel of the oracle c.c.c. and end with a criterion for extracting a subforcing (not a complete subforcing) of a given nicely one and satisfying the oracle.
keywords: S: for, (set), (iter), (cont)
Sh:671
Jech, T. J., & Shelah, S. (2000). On reflection of stationary sets in \mathcal P_\kappa\lambda. Trans. Amer. Math. Soc., 352(6), 2507–2515. arXiv: math/9801078 DOI: 10.1090/S0002-9947-99-02448-4 MR: 1650097
type: article
abstract: Let \kappa be an inaccessible cardinal, and let E_0=\{x\in{\mathcal P}_\kappa\kappa^+:cf(\lambda_x)=cf(\kappa_x)\} and E_1=\{x\in{\mathcal P}_\kappa\kappa^+:\kappa_x is regular and \lambda_x =\kappa_x^+\}.

It is consistent that the set E_1 is stationary and that every stationary subset of E_0 reflects at almost every a\in E_1.
keywords: S: for, (set), (iter), (normal), (ref), (pure(large))
Sh:672
Rosłanowski, A., & Shelah, S. (2004). Sweet & sour and other flavours of ccc forcing notions. Arch. Math. Logic, 43(5), 583–663. arXiv: math/9909115 DOI: 10.1007/s00153-004-0213-7 MR: 2076408
type: article
abstract: We continue developing the general theory of forcing notions built with the use of norms on possibilities, this time concentrating on ccc forcing notions and classifying them.
keywords: S: for, S: str, (set), (pure(for))
Sh:673
Kojman, M., & Shelah, S. (2000). The PCF trichotomy theorem does not hold for short sequences. Arch. Math. Logic, 39(3), 213–218. arXiv: math/9712289 DOI: 10.1007/s001530050143 MR: 1758508
type: article
abstract: We show that the assumption \lambda> \kappa^+ in the Trichotomy Theorem cannot be relaxed to \lambda> \kappa.
keywords: S: for, S: pcf, (set)
Sh:674
Balogh, Z. T., Davis, S. W., Just, W., Shelah, S., & Szeptycki, P. J. (2000). Strongly almost disjoint sets and weakly uniform bases. Trans. Amer. Math. Soc., 352(11), 4971–4987. arXiv: math/9803167 DOI: 10.1090/S0002-9947-00-02599-X MR: 1707497
type: article
abstract: A combinatorial principle CECA is formulated and its equivalence with GCH+ certain weakenings of \Box_\lambda for singular \lambda is proved. CECA is used to show that certain “almost point-< \tau” families can be refined to point-< \tau families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of “every first countable T_1-space with a weakly uniform base has a point-countable base.”
keywords: S: ico, S: pcf, (set)
Sh:675
Shelah, S. (1997). On Ciesielski’s problems. J. Appl. Anal., 3(2), 191–209. arXiv: math/9801155 DOI: 10.1515/JAA.1997.191 MR: 1619548
type: article
abstract: We discuss some problems posed by Ciesielski. For example we show that, consistently, d_c is a singular cardinal and e_c<d_c. Next we prove that the Martin Axiom for \sigma–centered forcing notions implies that for every function f:R^2\longrightarrow R there are functions g_n,h_n:R\longrightarrow R, n<\omega, such that f(x,y)=\sum_{n=0}^{\infty} g_n(x)h_n(y). Finally, we deal with countably continuous functions and we show that in the Cohen model they are exactly the functions f with the property that (\forall U\in [R]^{\aleph_1})(\exists U^*\in [U]^{\aleph_1}) (f\restriction U^* is continuous).
keywords: S: ico, (set), (app(pcf))
Sh:676
Hyttinen, T., & Shelah, S. (2001). Main gap for locally saturated elementary submodels of a homogeneous structure. J. Symbolic Logic, 66(3), 1286–1302. arXiv: math/9804157 DOI: 10.2307/2695107 MR: 1856742
type: article
abstract: We prove a main gap theorem for e-saturated submodels of a homogeneous structure. We also study the number of e-saturated models, which are not elementarily embeddable to each other
keywords: M: cla, (mod)
Sh:677
Shelah, S., & Spinas, O. (2000). On incomparability and related cardinal functions on ultraproducts of Boolean algebras. Math. Japon., 52(3), 345–358. arXiv: math/9903116 MR: 1796651
type: article
abstract: Let C denote any of the following cardinal characteristics of Boolean algebras: incomparability, spread, character, \pi-character, hereditary Lindelöf number, hereditary density. It is shown to be consistent that there exists a sequence \langle B_i:i<\kappa\rangle of Boolean algebras and an ultrafilter D on \kappa such that C(\prod_{i<\kappa}B_i/D)<|\prod_{i<\kappa}C(B_i)/D|. This answers a number of problems posed by Monk.
keywords: S: for, (set), (ba), (large), (up), (inv(ba))
Sh:678
Eklof, P. C., & Shelah, S. (1999). Absolutely rigid systems and absolutely indecomposable groups. In Abelian groups and modules (Dublin, 1998), Birkhäuser, Basel, pp. 257–268. arXiv: math/0010264 MR: 1735574
type: article
abstract: We give a new proof that there are arbitrarily large indecomposable abelian groups; moreover, the groups constructed are absolutely indecomposable, that is, they remain indecomposable in any generic extension. However, any absolutely rigid family of groups has cardinality less than the partition cardinal \kappa(\omega).
keywords: O: alg, (ab), (stal), (inf-log)
Sh:679
Shelah, S. (2002). A partition theorem. Sci. Math. Jpn., 56(2), 413–438. arXiv: math/0003163 MR: 1922806
type: article
abstract: We prove the following: there is a primitive recursive function f_{-}^*(-,-), in the three variables, such that: for every natural numbers t,n>0, and c, for any natural number k\geq f^*_t(n,c) the following holds. Assume \Lambda is an alphabet with n>0 letters, M is the family of non empty subsets of \{1,\ldots,k\} with \leq t members and V is the set of functions from M to \Lambda and lastly d is a c–colouring of V (i.e., a function with domain V and range with at most c members). Then there is a d–monochromatic V–line, which means that there are w \subseteq \{1,\ldots,k\}, with at least t members and function \rho from \{u\in M: u not a subset of w\} to \Lambda such that letting L=\{\eta\in V:\eta extend \rho and for each s=1,\ldots,t it is constant on \{u\in M:u\subseteq w has s members \}\}, we have d\restriction L is constant (for t=1 those are the Hales Jewett numbers).
keywords: S: ods, (pc), (fc)
Sh:680
Ciesielski, K. C., & Shelah, S. Uniformly antisymmetric functions with bounded range. Real Anal. Exchange, 24(2), 615–619. arXiv: math/9805151 MR: 1704738
type: article
abstract: The goal of this note is to construct a uniformly antisymmetric function f:R\to R with a bounded countable range. This answers Problem 1(b) of Ciesielski and Larson. (See also list of problems in Thomson and Problem 2(b) from Ciesielski’s survey.) A problem of existence of uniformly antisymmetric function f:R\to R with finite range remains open.
keywords: S: ico, S: str, (set)
Sh:681
Göbel, R., Shelah, S., & Strüngmann, L. H. (2004). Generalized E-rings. In Rings, modules, algebras, and abelian groups, Vol. 236, Dekker, New York, pp. 291–306. arXiv: math/0404271 MR: 2050718
type: article
abstract: A ring R is called an E-ring if the canonical homomorphism from R to the endomorphism ring End(R_{\mathbb Z}) of the additive group R_{\mathbb Z}, taking any r \in R to the endomorphism left multiplication by r turns out to be an isomorphism of rings. In this case R_{\mathbb Z} is called an E-group. Obvious examples of E-rings are subrings of {\mathbb Q}. However there is a proper class of examples constructed recently. E-rings come up naturally in various topics of algebra. This also led to a generalization: an abelian group G is an {\mathbb E}-group if there is an epimorphism from G onto the additive group of End(G). If G is torsion-free of finite rank, then G is an E-group if and only if it is an {\mathbb E}-group. The obvious question was raised a few years ago which we will answer by showing that the two notions do not coincide. We will apply combinatorial machinery to non-commutative rings to produce an abelian group G with (non-commutative) End(G) and the desired epimorphism with prescribed kernel H. Hence, if we let H=0, we obtain a non-commutative ring R such that End(R_{{\mathbb Z}}) \cong R but R is not an E-ring.
keywords: M: non, (ab), (large)
Sh:682
Göbel, R., & Shelah, S. (1999). Almost free splitters. Colloq. Math., 81(2), 193–221. arXiv: math/9910161 DOI: 10.4064/cm-81-2-193-221 MR: 1715347
See [Sh:E22]
type: article
abstract: Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is {\rm Ext}_R(G,G)=0 holds. For simplicity we will call such modules splitters. Our investigation continues [GbSh:647]. In [GbSh:647] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [GbSh:647] we concentrated on splitters which are larger then the continuum and such that countable submodules are not necessarily free. The ‘opposite’ case of \aleph_1-free splitters of cardinality less or equal to \aleph_1 was singled out because of basically different techniques. This is the target of the present paper. If the splitter is countable, then it must be free over some subring of the rationals by a result of Hausen. We can show that all \aleph_1-free splitters of cardinality \aleph_1 are free indeed.
keywords: O: alg, (ab), (af), (unif)
Sh:683
Kolman, O., & Shelah, S. (1999). Almost disjoint pure subgroups of the Baer-Specker group. In Abelian groups and modules (Dublin, 1998), Birkhäuser, Basel, pp. 225–230. arXiv: math/0102057 MR: 1735570
type: article
abstract: We prove in ZFC that the Baer-Specker group {\bf Z}^\omega has 2^{\aleph_1} non-free pure subgroups of cardinality \aleph_1 which are almost disjoint: there is no non-free subgroup embeddable in any pair.
keywords: S: ico, S: str, (ab)
Sh:684
Mildenberger, H., & Shelah, S. (2000). Changing cardinal characteristics without changing \omega-sequences or confinalities. Ann. Pure Appl. Logic, 106(1-3), 207–261. arXiv: math/9901096 DOI: 10.1016/S0168-0072(00)00026-9 MR: 1785760
type: article
abstract: We show: There are pairs of universes V_1\subseteq V_2 and there is a notion of forcing P\in V_1 such that the change mentioned in the title occurs when going from V_1[G] to V_2[G] for a P–generic filter G over V_2. We use forcing iterations with partial memories. Moreover, we implement highly transitive automorphism groups into the forcing orders.
keywords: S: str, (set), (inv), (pure(for))
Sh:685
Džamonja, M., & Shelah, S. (2000). On versions of \clubsuit on cardinals larger than \aleph_1. Math. Japon., 51(1), 53–61. arXiv: math/9911228 MR: 1739051
type: article
abstract: We give two results on guessing unbounded subsets of \lambda^+. The first is a positive result and applies to the situation of \lambda regular and at least equal to \aleph_3, while the second is a negative consistency result which applies to the situation of \lambda a singular strong limit with 2^\lambda>\lambda^+. The first result shows that in ZFC there is a guessing of unbounded subsets of S^{\lambda^+}_\lambda. The second result is a consistency result (assuming a supercompact cardinal exists) showing that a natural guessing fails. A result of Shelah in [Sh:667] shows that if 2^\lambda=\lambda^+ and \lambda is a strong limit singular, then the corresponding guessing holds. Both results are also connected to an earlier result of Džamonja-Shelah in which they showed that a certain version of \clubsuit holds at a successor of singular just in ZFC. The first result here shows that a result of [DjSh:545] can to a certain extent be extended to the successor of a regular. The negative result here gives limitations to the extent to which one can hope to extend the mentioned Džamonja-Shelah result.
keywords: S: for, (set), (iter)
Sh:686
Rosłanowski, A., & Shelah, S. (2001). The yellow cake. Proc. Amer. Math. Soc., 129(1), 279–291. arXiv: math/9810179 DOI: 10.1090/S0002-9939-00-05538-6 MR: 1694876
type: article
abstract: We consider the following property:

(*) For every function f:R\times R\longrightarrow R there are functions g^0_n,g^1_n:R\longrightarrow R (for n<\omega) such that (\forall x,y\in R)(f(x,y)=\sum_{n<\omega}g^0_n(x)g^1_n(y)).

We show that, despite some expectation suggested by [Sh:675], (*) does not imply MA(\sigma-centered). Next, we introduce cardinal characteristics of the continuum responsible for the failure of (*).
keywords: S: str, (set)
Sh:687
Laskowski, M. C., & Shelah, S. (2003). Karp complexity and classes with the independence property. Ann. Pure Appl. Logic, 120(1-3), 263–283. arXiv: math/0303345 DOI: 10.1016/S0168-0072(02)00080-5 MR: 1949710
type: article
abstract: A class {\bf K} of structures is controlled if for all cardinals \lambda, the relation of L_{\infty,\lambda}-equivalence partitions {\bf K} into a set of equivalence classes (as opposed to a proper class). We prove that no pseudo-elementary class with the independence property is controlled. By contrast, there is a pseudo-elementary class with the strict order property that is controlled.
keywords: M: non, (mod)
Sh:688
Goldstern, M., & Shelah, S. (1999). There are no infinite order polynomially complete lattices, after all. Algebra Universalis, 42(1-2), 49–57. arXiv: math/9810050 DOI: 10.1007/s000120050122 MR: 1736340
type: article
abstract: If L is a lattice with the interpolation property whose cardinality is a strong limit cardinal of uncountable cofinality, then some finite power L^n has an antichain of size \kappa. Hence there are no infinite opc lattices (i.e., lattices on which every n-ary monotone function is a polynomial).

However, the existence of strongly amorphous sets implies (in ZF) the existence of infinite opc lattices.
keywords: O: alg, (ua)
Sh:689
Cherlin, G. L., Shelah, S., & Shi, N. (1999). Universal graphs with forbidden subgraphs and algebraic closure. Adv. In Appl. Math., 22(4), 454–491. arXiv: math/9809202 DOI: 10.1006/aama.1998.0641 MR: 1683298
type: article
abstract: We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an associated "algebraic closure" operator. The main applications are new examples of universal graphs with forbidden subgraphs and simplified treatments of some previously known cases.
keywords: M: odm, (mod), (graph), (univ)
Sh:690
Eisworth, T., Nyikos, P. J., & Shelah, S. (2003). Gently killing S-spaces. Israel J. Math., 136, 189–220. arXiv: math/9812133 DOI: 10.1007/BF02807198 MR: 1998110
type: article
abstract: We produce a model of ZFC in which there are no locally compact first countable S–spaces, and in which 2^{\aleph_0}<2^{\aleph_1}. A consequence of this is that in this model there are no locally compact, separable, hereditarily normal spaces of size \aleph_1, answering a question of the second author
keywords: S: for, (iter), (gt)
Sh:691
Džamonja, M., & Shelah, S. (2003). Weak reflection at the successor of a singular cardinal. J. London Math. Soc. (2), 67(1), 1–15. arXiv: math/0003118 DOI: 10.1112/S0024610702003757 MR: 1942407
See [Sh:E20]
type: article
abstract: The notion of stationary reflection is one of the most important notions of combinatorial set theory. We investigate weak reflection, which is, as its name suggests, a weak version of stationary reflection. Our main result is that modulo a large cardinal assumption close to 2-hugeness, there can be a regular cardinal \kappa such that the first cardinal weakly reflecting at \kappa is the successor of a singular cardinal. This answers a question of Cummings, Džamonja and Shelah.
keywords: S: for, (set), (large), (ref)
Sh:692
Džamonja, M., & Shelah, S. (2004). On \vartriangleleft^*-maximality. Ann. Pure Appl. Logic, 125(1-3), 119–158. arXiv: math/0009087 DOI: 10.1016/j.apal.2003.11.001 MR: 2033421
type: article
abstract: This paper investigates a connection between the ordering \triangleleft^\ast among theories in model theory and the (N)SOP{}_n hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP{}_2 and SOP{}_1, and gives a strong connection between SOP{}_1 and the maximality in Keisler ordering. Together with the known results about the connection between the (N)SOP{}_n hierarchy and the existence of universal models in the absence of GCH, the paper provides a step toward the classification of unstable theories without the strict order property.
keywords: M: cla, S: ico, (mod)
Sh:693
Shelah, S., & Trlifaj, J. (2001). Spectra of the \Gamma-invariant of uniform modules. J. Pure Appl. Algebra, 162(2-3), 367–379. arXiv: math/0009060 DOI: 10.1016/S0022-4049(00)00118-3 MR: 1843814
type: article
abstract: For a ring R, denote by {\rm Spec}^R_\kappa (\Gamma) the \kappa-spectrum of the \Gamma-invariant of strongly uniform right R-modules. Recent realization techniques of Goodearl and Wehrung show that {\rm Spec}^R_{\aleph_1} (\Gamma) is full for suitable von Neumann regular algebras R, but the techniques do not extend to cardinals \kappa > \aleph_1. By a direct construction, we prove that for any field F and any regular uncountable cardinal \kappa there is an F-algebra R such that {\rm Spec}^R_\kappa (\Gamma) is full. We also derive some consequences for the complexity of Ziegler spectra of infinite dimensional algebras.
keywords: O: alg, (ab), (stal)
Sh:694
Jech, T. J., & Shelah, S. (2001). Simple complete Boolean algebras. Proc. Amer. Math. Soc., 129(2), 543–549. arXiv: math/0406438 DOI: 10.1090/S0002-9939-00-05566-0 MR: 1707521
type: article
abstract: For every regular cardinal \kappa there exists a simple complete Boolean algebra with \kappa generators.
keywords: S: ico, S: for, (set), (ba), (pure(for))
Sh:695
Ciesielski, K. C., & Shelah, S. (2000). Category analogue of sup-measurability problem. J. Appl. Anal., 6(2), 159–172. arXiv: math/9905147 DOI: 10.1515/JAA.2000.159 MR: 1805097
type: article
abstract: A function F\colon{\mathbb R}^2\to{\mathbb R} is sup-measurable if F_f\colon{\mathbb R}\to{\mathbb R} given by F_f(x)=F(x,f(x)), x\in{\mathbb R}, is measurable for each measurable function f\colon{\mathbb R}\to{\mathbb R}. It is known that under different set theoretical assumptions, including CH, there are sup-measurable non-measurable functions, as well as their category analog. In this paper we will show that the existence of category analog of sup-measurable non-measurable functions is independent of ZFC. A problem whether the similar is true for the original measurable case remains open.
keywords: S: str, (set), (leb)
Sh:696
Goldstern, M., & Shelah, S. (2002). Antichains in products of linear orders. Order, 19(3), 213–222. arXiv: math/9902054 DOI: 10.1023/A:1021289412771 MR: 1942184
type: article
abstract: We show that: For many cardinals \lambda, for all n\in \{2,3,4,\ldots\} There is a linear order L such that L^n has no (incomparability-)antichain of cardinality \lambda, while L^{n+1} has an antichain of cardinality \lambda. For any nondecreasing sequence (\lambda_n: n \in \{2,3,4,\ldots\}) of infinite cardinals it is consistent that there is a linear order L such that L^n has an antichain of cardinality \lambda_n, but not one of cardinality \lambda_n^+.
keywords: S: ico, (set)
Sh:697
Hajnal, A., Juhász, I., & Shelah, S. (2000). Strongly almost disjoint families, revisited. Fund. Math., 163(1), 13–23. arXiv: math/9812114 MR: 1750332
type: article
abstract: The relations M(\kappa,\lambda,\mu)\to B (resp. B(\sigma)) meaning that if {\mathcal A}\subset [\kappa]^\lambda with |{\mathcal A}|=\kappa is \mu-almost disjoint then {\mathcal A} has property B (resp. has a \sigma-transversal) had been introduced and studied under GCH by Erdos and Hajnal in 1961. Our two main results here say the following:

Assume GCH and \varrho be any regular cardinal with a supercompact [resp. 2-huge] cardinal above \varrho. Then there is a \varrho-closed forcing P such that, in V^P, we have both GCH and M(\varrho^{(+\varrho+1)},\varrho^+,\varrho)\nrightarrow B (resp. M(\varrho^{(+\varrho+1)},\lambda,\varrho) \nrightarrow B(\varrho^+) for all \lambda\le\varrho^{(+\varrho+1)}).
keywords: S: ico, S: for, (set), (large)
Sh:698
Shelah, S. (2002). On the existence of large subsets of [\lambda]^{<\kappa} which contain no unbounded non-stationary subsets. Arch. Math. Logic, 41(3), 207–213. arXiv: math/9908159 DOI: 10.1007/s001530000054 MR: 1901184
type: article
abstract: Here we deal with some problems of Mate, written as it developed. The first section deals with the existence of stationary subsets of [\lambda]^{<\kappa} with no unbounded subsets which are not stationary, where \kappa is regular uncountable \leq\lambda. In the section section we deal with the existence of such clubs.
keywords: S: ico, (set), (normal), (app(pcf))
Sh:699
Halbeisen, L. J., & Shelah, S. (2001). Relations between some cardinals in the absence of the axiom of choice. Bull. Symbolic Logic, 7(2), 237–261. arXiv: math/0010268 DOI: 10.2307/2687776 MR: 1839547
type: article
abstract: If we assume the axiom of choice, then every two cardinal numbers are comparable. In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using the axiom of choice.
keywords: S: ods, (AC)
Sh:700
Shelah, S. (2004). Two cardinal invariants of the continuum (\mathfrak d<\mathfrak a) and FS linearly ordered iterated forcing. Acta Math., 192(2), 187–223. Previous title “Are \mathfrak a and \mathfrak d your cup of tea?” arXiv: math/0012170 DOI: 10.1007/BF02392740 MR: 2096454
See [Sh:700a]
type: article
abstract: We show that consistently, every MAD family has cardinality strictly bigger than the dominating number, that is \mathfrak{a} > \mathfrak{d}, thus solving one of the oldest problems on cardinal invariants of the continuum. The method is a contribution to the theory of iterated forcing for making the continuum large.
keywords: S: for, S: str, (set), (iter), (inv)
Sh:701
Göbel, R., Rodrı́guez Blancas, J. L., & Shelah, S. (2002). Large localizations of finite simple groups. J. Reine Angew. Math., 550, 1–24. arXiv: math/9912191 DOI: 10.1515/crll.2002.072 MR: 1925906
type: article
abstract: We answer a question of Emanuel Farjoun on homotopy groups using cancellation theory
keywords: M: non, (stal)
Sh:702
Shelah, S. (2000). On what I do not understand (and have something to say), model theory. Math. Japon., 51(2), 329–377. arXiv: math/9910158 MR: 1747306
type: article
abstract: This is a non-standard paper, containing some problems, mainly in model theory, which I have, in various degrees, been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (“See..." means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall ’97 and reflect my knowledge then. The other half, concentrating on set theory, is in print [Sh:666], but the two halves are independent. We thank A. Blass, G. Cherlin and R. Grossberg for some corrections.
keywords: M: odm, (mod)
Sh:703
Shelah, S. (2003). On ultraproducts of Boolean algebras and irr. Arch. Math. Logic, 42(6), 569–581. arXiv: math/0012171 DOI: 10.1007/s00153-002-0167-6 MR: 2001060
type: article
abstract: We prove the consistency of {\rm irr}(\prod\limits_{i< \kappa} B_i/D)<\prod\limits_{i<\kappa}{\rm irr}(B_i)/D, where D is an ultrafilter on \kappa and each B_i is a Boolean Algebra. This solves the last problem of this form from the Monk’s list of problems, that is number 35. The solution applies to many other properties, e.g., Souslinity. Next, we get similar results with \kappa=\aleph_1 (easily we cannot have it for \kappa = \aleph_0) and Boolean Algebras B_i (i<\kappa) of cardinality <\beth_{\omega_1}.
keywords: S: for, (set), (ba), (large), (inv(ba))
Sh:704
Shelah, S. (2002). Superatomic Boolean algebras: maximal rigidity. In Set theory (Piscataway, NJ, 1999), Vol. 58, Amer. Math. Soc., Providence, RI, pp. 107–128. arXiv: math/0009075 DOI: 10.1090/dimacs/058/09 MR: 1903854
type: article
abstract: We prove that for any superatomic Boolean Algebra of cardinality >\beth_\omega there is an automorphism moving uncountably many atoms. Similarly for larger cardinals any of those results are essentially best possible.
keywords: O: alg, (set), (ba), (auto)
Sh:706
Shelah, S. (2012). Universality among graphs omitting a complete bipartite graph. Combinatorica, 32(3), 325–362. arXiv: math/0102058 DOI: 10.1007/s00493-012-2033-4 MR: 2965281
type: article
abstract: For cardinals \lambda,\kappa,\theta we consider the class of graphs of cardinality \lambda which has no subgraph which is (\kappa,\theta)-complete bipartite graph. The question is whether in such a class there is a universal one under (weak) embedding. We solve this problem completely under GCH. Under various assumptions mostly related to cardinal arithmetic we prove nonexistence of universals for this problem and some related ones.
keywords: S: ico, (set), (graph), (univ)
Sh:708
Gitik, M., & Shelah, S. (2001). On some configurations related to the Shelah weak hypothesis. Arch. Math. Logic, 40(8), 639–650. arXiv: math/9909087 DOI: 10.1007/s001530100076 MR: 1867686
type: article
abstract: We show that some cardinal arithmetic configurations related to the negation of the Shelah Weak Hypothesis and natural from the forcing point of view are impossible.
keywords: S: pcf, (set)
Sh:709
Kolman, O., & Shelah, S. (2000). Infinitary axiomatizability of slender and cotorsion-free groups. Bull. Belg. Math. Soc. Simon Stevin, 7(4), 623–629. arXiv: math/9910162 http://projecteuclid.org/euclid.bbms/1103055621 MR: 1806941
type: article
abstract: The classes of slender and cotorsion-free abelian groups are axiomatizable in the infinitary logics L_{\infty\omega_1} and L_{\infty\omega} respectively. The Baer-Specker group {\mathbb Z}^\omega is not L_{\infty\omega_1}-equivalent to a slender group.
keywords: O: alg, (ab), (stal)
Sh:710
Džamonja, M., & Shelah, S. (2006). On properties of theories which preclude the existence of universal models. Ann. Pure Appl. Logic, 139(1-3), 280–302. arXiv: math/0009078 DOI: 10.1016/j.apal.2005.06.001 MR: 2206258
type: article
abstract: In this paper we investigate some properties of first order theories which prevent them from having universal models under certain cardinal assumptions. Our results give a new syntactical condition, oak property, which is a sufficient condition for a theory not to have universal models in cardinality \lambda when certain cardinal arithmetic assumptions implying the failure of GCH (and close to the failure of SCH) hold.
keywords: M: cla, (mod), (univ)
Sh:711
Shelah, S. (2005). On nicely definable forcing notions. J. Appl. Anal., 11(1), 1–17. arXiv: math/0303293 DOI: 10.1515/JAA.2005.1 MR: 2151390
type: article
abstract: We prove that if \mathbb Q is a nw-nep forcing then it cannot add a dominating real. We also prove that Amoeba forcing cannot be {\mathcal P}(X)/I if I is an \aleph_1-complete ideal.
keywords: S: str, (set), (pure(for))
Sh:712
Fuchino, S., Geschke, S., Shelah, S., & Soukup, L. (2001). On the weak Freese-Nation property of complete Boolean algebras. Ann. Pure Appl. Logic, 110(1-3), 89–105. arXiv: math/9911230 DOI: 10.1016/S0168-0072(01)00023-9 MR: 1846760
type: article
abstract: The following results are proved: (a) In a Cohen model, there is always a ccc complete Boolean algebras without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a ccc complete Boolean algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of \neg0^\#, the weak Freese-Nation property of ({\mathcal P}(\omega),{\subseteq}) is equivalent to the weak Freese-Nation property of any of {\mathbb C}(\kappa) or {\mathbb R}(\kappa) for uncountable \kappa. (d) Modulo consistency of (\aleph_{\omega+1},\aleph_\omega) \longrightarrow(\aleph_1,\aleph_0), it is consistent with GCH that the assertion in (c) does not hold and also that adding \aleph_\omega Cohen reals destroys the weak Freese-Nation property of ({\mathcal P}(\omega),{\subseteq}).
keywords: S: ico, (ba)
Sh:713
Matet, P., Péan, C., & Shelah, S. (2016). Cofinality of normal ideals on [\lambda]^{<\kappa} I. Arch. Math. Logic, 55(5-6), 799–834. arXiv: math/0404318 DOI: 10.1007/s00153-016-0496-5 MR: 3523657
type: article
abstract: Given an ordinal \delta\leq\lambda and a cardinal \theta\leq\kappa, an ideal J on P_{\kappa}(\lambda) is said to be \lbrack\delta\rbrack^{<\theta}-normal if given B_e\in J for e\in P_\theta(\delta), the set of all a\in P_{\kappa}(\lambda) such that a\in B_e for some e\in P_{|a\cap\theta|}(a\cap\delta) lies in J. We give necessary and sufficient conditions for the existence of such ideals and we describe the least one and we compute its cofinality.
keywords: S: ico, S: pcf, (set), (normal)
Sh:714
Juhász, I., Shelah, S., Soukup, L., & Szentmiklóssy, Z. (2003). A tall space with a small bottom. Proc. Amer. Math. Soc., 131(6), 1907–1916. arXiv: math/0104198 DOI: 10.1090/S0002-9939-03-06662-0 MR: 1955280
type: article
abstract: We introduce a general method of constructing locally compact scattered spaces from certain families of sets and then, with the help of this method, we prove that if \kappa^{<\kappa} = \kappa then there is such a space of height \kappa^+ with only \kappa many isolated points. This implies that there is a locally compact scattered space of height {\omega}_2 with \omega_1 isolated points in ZFC, solving an old problem of the first author.
keywords: O: top, (ba), (gt)
Sh:715
Shelah, S. (2004). Classification theory for elementary classes with the dependence property—a modest beginning. Sci. Math. Jpn., 59(2), 265–316. arXiv: math/0009056 MR: 2062198
type: article
abstract: Our thesis is that for the family of classes of the form EC(T),T a complete first order theory with the dependence property (which is just the negation of the independence property) there is a substantial theory which means: a substantial body of basic results for all such classes and some complimentary results for the first order theories with the independence property, as for the family of stable (and the family of simple) first order theories. We examine some properties.
keywords: O: alg, (mod), (stal)
Sh:716
Göbel, R., & Shelah, S. (2001). Decompositions of reflexive modules. Arch. Math. (Basel), 76(3), 166–181. arXiv: math/0003165 DOI: 10.1007/s000130050557 MR: 1816987
type: article
abstract: We continue [GbSh:568], proving a stronger result under the special continuum hypothesis (CH). The original question of Eklof and Mekler related to dual abelian groups. We want to find a particular example of a dual group, which will provide a negative answer to the question. In order to derive a stronger and also more general result we will concentrate on reflexive modules over countable principal ideal domains R. Following H. Bass, an R-module G is reflexive if the evaluation map \sigma:G\longrightarrow G^{**} is an isomorphism. Here G^*={\rm Hom} (G,R) denotes the dual group of G. Guided by classical results the question about the existence of a reflexive R-module G of infinite rank with G\not\cong G\oplus R is natural. We will use a theory of bilinear forms on free R-modules which strengthens our algebraic results in [GbSh:568]. Moreover we want to apply a model theoretic combinatorial theorem from [Sh:e] which allows us to avoid the weak diamond principle. This has the great advantage that the used prediction principle is still similar to the diamond, but holds under CH.
keywords: O: alg, (ab), (stal)
Sh:717
Eklof, P. C., & Shelah, S. (2002). The structure of \mathrm{Ext}(A,\mathbb Z) and GCH: possible co-Moore spaces. Math. Z., 239(1), 143–157. arXiv: math/0303344 DOI: 10.1007/s002090100288 MR: 1879333
type: article
abstract: We consider what {\rm Ext}(A,{\mathbb{Z}}) can be when A is torsion-free and {\rm Hom}(A,{\mathbb{Z}})=0. We thereby give an answer to a question of Golasiński and Gonçalves which asks for the divisible Abelian groups which can be the type of a co-Moore space.
keywords: O: alg, (ab), (stal), (wh)
Sh:718
Shelah, S., & Väisänen, P. (2002). The number of L_{\infty\kappa}-equivalent nonisomorphic models for \kappa weakly compact. Fund. Math., 174(2), 97–126. arXiv: math/9911232 DOI: 10.4064/fm174-2-1 MR: 1927234
type: article
abstract: For a cardinal \kappa and a model {\mathcal M} of cardinality \kappa let {\rm No}({\mathcal M}) denote the number of non-isomorphic models of cardinality \kappa which are L_{\infty\kappa}–equivalent to {\mathcal M}. In [Sh:133] Shelah established that when \kappa is a weakly compact cardinal and \mu \leq \kappa is a nonzero cardinal, there exists a model {\mathcal M} of cardinality \kappa with {\rm No}({\mathcal M})=\mu. We prove here that if \kappa is a weakly compact cardinal, the question of the possible values of {\rm No}({\mathcal M}) for models {\mathcal M} of cardinality \kappa is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are \Sigma^1_1-definable over V_\kappa. In [ShVa:719] we prove that, consistent wise, the possible numbers of equivalence classes of \Sigma^1_1-equivalence relations can be completely controlled under the singular cardinal hypothesis. These results settle the problem of the possible values of {\rm No}({\mathcal M}) for models of weakly compact cardinality, provided that the singular cardinal hypothesis holds.
keywords: M: smt, M: non, (nni), (inf-log), (set-mod), (pure(large)), (ps-dst)
Sh:719
Shelah, S., & Väisänen, P. (2002). On equivalence relations second order definable over H(\kappa). Fund. Math., 174(1), 1–21. arXiv: math/9911231 DOI: 10.4064/fm174-1-1 MR: 1925484
type: article
abstract: Let \kappa be an uncountable regular cardinal. Call an equivalence relation on functions from \kappa into 2 \Sigma_1^1-definable over H(\kappa) if there is a first order sentence \phi and a parameter R\subseteq H(\kappa) such that functions f ,g \in {}^\kappa 2 are equivalent iff for some h\in {}^\kappa 2, the structure (H(\kappa),\in,R,f,g,h) satisfies \phi, where \in, R, f, g, and h are interpretations of the symbols appearing in \phi. All the values \mu, 1\leq\mu \leq\kappa^+ or \mu=2^\kappa, are possible numbers of equivalence classes for such a \Sigma_1^1-equivalence relation. Additionally, the possibilities are closed under unions of \leq\kappa-many cardinals and products of <\kappa-many cardinals. We prove that, consistent wise, these are the only restrictions under the singular cardinal hypothesis. The result is that the possible numbers of equivalence classes of \Sigma_1^1-equivalence relations might consistent wise be exactly those cardinals which are in a prearranged set, provided that the singular cardinal hypothesis holds and that some necessary conditions are fulfilled.
keywords: M: smt, (nni), (inf-log), (set-mod), (pure(large)), (ps-dst)
Sh:720
Kojman, M., & Shelah, S. (2001). Fallen cardinals. Ann. Pure Appl. Logic, 109(1-2), 117–129. arXiv: math/0009079 DOI: 10.1016/S0168-0072(01)00045-8 MR: 1835242
type: article
abstract: We prove that for every singular cardinal \mu of cofinality \omega, the complete Boolean algebra {\rm comp}{\mathcal P}_\mu(\mu) contains as a complete subalgebra an isomorphic copy of the collapse algebra {\rm Comp}\;{\rm Col}(\omega_1,\mu^{\aleph_0}). Consequently, adding a generic filter to the quotient algebra {\mathcal P}_\mu(\mu)={\mathcal P}(\mu)/[\mu]^{<\mu} collapses \mu^{\aleph_0} to \aleph_1. Another corollary is that the Baire number of the space U(\mu) of all uniform ultrafilters over \mu is equal to \omega_2. The corollaries affirm two conjectures by Balcar and Simon.

The proof uses pcf theory.
keywords: S: ico, S: pcf, (set), (ba), (pure(for))
Sh:721
Göbel, R., Shelah, S., & Wallutis, S. L. (2001). On the lattice of cotorsion theories. J. Algebra, 238(1), 292–313. arXiv: math/0103154 DOI: 10.1006/jabr.2000.8619 MR: 1822193
type: article
abstract: We discuss the lattice of cotorsion theories for abelian groups. First we show that the sublattice of the well–studied rational cotorsion theories can be identified with the well–known lattice of types. Using a recently developed method for making Ext vanish we also prove that any power set together with the ordinary set inclusion (and thus any poset) can be embedded into the lattice of all cotorsion theories.
keywords: O: alg, (ab), (stal)
Sh:722
Bartoszyński, T., & Shelah, S. (2001). Continuous images of sets of reals. Topology Appl., 116(2), 243–253. arXiv: math/0001051 DOI: 10.1016/S0166-8641(00)00079-1 MR: 1855966
type: article
abstract: We show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.
keywords: S: str, (set)
Sh:723
Shelah, S. (2001). Consistently there is no non trivial ccc forcing notion with the Sacks or Laver property. Combinatorica, 21(2), 309–319. arXiv: math/0003139 DOI: 10.1007/s004930100027 MR: 1832454
type: article
abstract: The result in the title answers a problem of Boban Velickovic. A definable version of it (that is for Souslin forcing notions) has been answered in [Sh 480], and our proof follows it. Independently Velickovic proved this consistency, following [Sh 480] and some of his works, proving it from PFA and from OCA. We prove that moreover, consistently there is no ccc forcing with the Laver property. Note that if cov(meagre)=continuum (which follows e.g. from PFA) then there is a (non principal) Ramsey ultrafilter on \omega hence a forcing notion with the Laver property. So the results are incomparable.
keywords: S: str, (set), (pure(for))
Sh:724
Shelah, S. (2004). On nice equivalence relations on ^\lambda 2. Arch. Math. Logic, 43(1), 31–64. arXiv: math/0009064 DOI: 10.1007/s00153-003-0183-1 MR: 2036248
type: article
abstract: The main question here is the possible generalization of the following theorem on “simple" equivalence relation on {}^\omega 2 to higher cardinals.

Theorem: (1) Assume that (a) E is a Borel 2-place relation on {}^\omega 2, (b) E is an equivalence relation, (c) if \eta,\nu\in{}^\omega 2 and (\exists!n)(\eta(n)\neq \nu(n)), then \eta,\nu are not E–equivalent. Then there is a perfect subset of {}^\omega 2 of pairwise non E-equivalent members.

(2) Instead of “E is Borel”, “E is analytic (or even a Borel combination of analytic relations)” is enough.

(3) If E is a \Pi^1_2 relation which is an equivalence relation satisfying clauses (b)+(c) in V^{\rm Cohen}, then the conclusion of (1) holds.
keywords: S: dst, S: ods, (set), (ps-dst)
Sh:725
Mildenberger, H., & Shelah, S. (2004). On needed reals. Israel J. Math., 141, 1–37. arXiv: math/0104276 DOI: 10.1007/BF02772209 MR: 2063023
type: article
abstract: Following Blass, we call a real a “needed” for a binary relation R on the reals if in every R-adequate set we find an element from which a is Turing computable. We show that every real needed for {\bf Cof}({\mathcal N}) is hyperarithmetic. Replacing “R-adequate” by “R-adequate with minimal cardinality” we get related notion of being “weakly needed”. We show that is is consistent that the two notions do not coincide for the reaping relation. (They coincide in many models.) We show that not all hyperarithmetical reals are needed for the reaping relation. This answers some questions asked by Blass at the Oberwolfach conference in December 1999.
keywords: S: for, S: str, (set)
Sh:726
Shelah, S., & Väänänen, J. A. (2005). A note on extensions of infinitary logic. Arch. Math. Logic, 44(1), 63–69. arXiv: math/0009080 DOI: 10.1007/s00153-004-0212-8 MR: 2116833
type: article
abstract: We show that a strong form of the so called Lindström’s Theorem fails to generalize to extensions of L_{\kappa\omega} and L_{\kappa\kappa}: For weakly compact \kappa there is no strongest extension of L_{\kappa\omega} with the (\kappa,\kappa)-compactness property and the Löwenheim-Skolem theorem down to \kappa. With an additional set-theoretic assumption, there is no strongest extension of L_{\kappa\kappa} with the (\kappa,\kappa)-compactness property and the Löwenheim-Skolem theorem down to <\kappa.
keywords: M: smt, (set), (inf-log), (pure(large))
Sh:727
Göbel, R., & Shelah, S. (2001). Reflexive subgroups of the Baer-Specker group and Martin’s axiom. In Abelian groups, rings and modules (Perth, 2000), Vol. 273, Amer. Math. Soc., Providence, RI, pp. 145–158. arXiv: math/0009062 DOI: 10.1090/conm/273/04431 MR: 1817159
type: article
abstract: In two recent papers we answered a question raised in the book by Eklof and Mekler (p. 455, Problem 12) under the set theoretical hypothesis of \diamondsuit_{\aleph_1} which holds in many models of set theory, respectively of the special continuum hypothesis (CH). The objects are reflexive modules over countable principal ideal domains R, which are not fields. Following H. Bass, an R-module G is reflexive if the evaluation map \sigma: G\longrightarrow G^{**} is an isomorphism. Here G^*={\rm Hom}(G, R) denotes the dual module of G. We proved the existence of reflexive R-modules G of infinite rank with G \not\cong G \oplus R, which provide (even essentially indecomposable) counter examples to the question mentioned above. Is CH a necessary condition to find ‘nasty’ reflexive modules? In the last part of this paper we will show (assuming the existence of supercompact cardinals) that large reflexive modules always have large summands. So at least being essentially indecomposable needs an additional set theoretic assumption. However the assumption need not be CH as shown in the first part of this paper. We will use Martin’s axiom to find reflexive modules with the above decomposition which are submodules of the Baer-Specker module R^\omega.
keywords: O: alg, (ab), (stal)
Sh:728
Kennedy, J. C., & Shelah, S. (2003). On embedding models of arithmetic of cardinality \aleph_1 into reduced powers. Fund. Math., 176(1), 17–24. arXiv: math/0105134 DOI: 10.4064/fm176-1-2 MR: 1971470
type: article
abstract: In the early 1970’s S.Tennenbaum proved that all countable models of PA^- + \forall_1 -Th({\mathbb N}) are embeddable into the reduced product {\mathbb N}^\omega/{\mathcal F}, where {\mathcal F} is the cofinite filter. In this paper we show that if M is a model of PA^- + \forall_1 -Th({\mathbb N}), and |M|=\aleph_1, then M is embeddable into {\mathbb N}^\omega/D, where D is any regular filter on \omega.
keywords: M: odm, (mod), (univ), (up)
Sh:729
Shelah, S., & Strüngmann, L. H. (2001). The failure of the uncountable non-commutative Specker phenomenon. J. Group Theory, 4(4), 417–426. arXiv: math/0009045 DOI: 10.1515/jgth.2001.031 MR: 1859179
type: article
abstract: Higman proved in 1952 that every free group is non-commutatively slender, this is to say that if G is a free group and h is a homomorphism from the countable complete free product \bigotimes_\omega{\mathbb Z} to G, then there exists a finite subset F\subseteq\omega and a homomorphism \bar{h}: *_{i\in F} {\mathbb Z}\longrightarrow G such that h=\bar{h}\rho_F, where \rho_F is the natural map from \bigotimes_{i\in\omega}{\mathbb Z} to *_{i\in F}{\mathbb Z}. Corresponding to the abelian case this phenomenon was called the non-commutative Specker Phenomenon. In this paper we show that Higman’s result fails if one passes from countable to uncountable. In particular, we show that for non-trivial groups G_\alpha (\alpha\in\lambda) and uncountable cardinal \lambda there are 2^{2^\lambda} homomorphisms from the complete free product of the G_\alpha’s to the ring of integers.
keywords: S: ico, O: alg, (stal), (grp)
Sh:730
Shelah, S. (2000). A space with only Borel subsets. Period. Math. Hungar., 40(2), 81–84. arXiv: math/0009047 DOI: 10.1023/A:1010364023601 MR: 1805307
type: article
abstract: Miklós Laczkovich asked if there exists a Haussdorff (or even normal) space in which every subset is Borel yet it is not meager. The motivation of the last condition is that under {\rm MA}_\kappa every subspace of the reals of cardinality \kappa has the property that all subsets are {\rm F}_\sigma however Martin’s axiom also implies that these subsets are meager. Here we answer Laczkovich’ question.
keywords: S: str, (gt)
Sh:731
Mildenberger, H., & Shelah, S. (2002). The relative consistency of \mathfrak g<\mathrm{cf}(\mathrm{Sym}(\omega)). J. Symbolic Logic, 67(1), 297–314. arXiv: math/0009077 DOI: 10.2178/jsl/1190150045 MR: 1889552
type: article
abstract: We prove the consistency result from the title. By forcing we construct a model of \mathfrak{g}=\aleph_1, \mathfrak{b}={\rm cf}({\rm Sym}(\omega))=\aleph_2.
keywords: S: for, S: str, (set), (iter), (inv)
Sh:732
Bartoszyński, T., & Shelah, S. (2002). Perfectly meager sets and universally null sets. Proc. Amer. Math. Soc., 130(12), 3701–3711. arXiv: math/0102011 DOI: 10.1090/S0002-9939-02-06465-1 MR: 1920051
type: article
abstract: For a set of reals X: (a) X is perfectly meager (PM) if for every perfect set P\subseteq{\mathbb R}, P\cap X is meager in P. (b) X is universal null (UN) if every Borel isomorphic image of X has Lebesgue measure zero.

We show that it is consistent with ZFC that PM is a subset of UN.
keywords: S: for, S: str, (set), (leb), (meag)
Sh:733
Rosłanowski, A., & Shelah, S. (2001). Historic forcing for depth. Colloq. Math., 89(1), 99–115. arXiv: math/0006219 DOI: 10.4064/cm89-1-7 MR: 1853418
type: article
abstract: We show that, consistently, for some regular cardinals \theta<\lambda, there exist a Boolean algebra B such that |B|= \lambda^+ and for every subalgebra B'\subseteq B of size \lambda^+ we have {\rm Depth}(B')=\theta.
keywords: S: for, (set), (ba), (inv(ba))
Sh:735
Shelah, S., & Steprāns, J. (2002). Martin’s axiom is consistent with the existence of nowhere trivial automorphisms. Proc. Amer. Math. Soc., 130(7), 2097–2106. arXiv: math/0011166 DOI: 10.1090/S0002-9939-01-06280-3 MR: 1896046
type: article
abstract: Martin’s Axiom does not imply that all automorphisms of {\mathcal P}({\mathbb N})/[{\mathbb N}]^{<\aleph_0} are somewhere trivial.
keywords: S: for, S: str, (set), (auto)
Sh:736
Rosłanowski, A., & Shelah, S. (2006). Measured creatures. Israel J. Math., 151, 61–110. arXiv: math/0010070 DOI: 10.1007/BF02777356 MR: 2214118
type: article
abstract: Using forcing with measured creatures we build a universe of set theory in which (a) every sup-measurable function f:{\mathbb R}^2\longrightarrow{\mathbb R} is measurable, and (b) every function f:{\mathbb R}\longrightarrow{\mathbb R} is continuous on a non-measurable set. This answers von Weizsäcker’s problem (see Fremlin’s list of problems) and a question of Balcerzak, Ciesielski and Kharazishvili.
keywords: S: for, S: str, (set), (creatures)
Sh:737
Goldstern, M., & Shelah, S. (2002). Clones on regular cardinals. Fund. Math., 173(1), 1–20. arXiv: math/0005273 DOI: 10.4064/fm173-1-1 MR: 1899044
type: article
abstract: We investigate the structure of the lattice of clones on an infinite set X. We first observe that ultrafilters naturally induce clones; this yields a simple proof of Rosenberg’s theorem: there are 2^{2^{\lambda}} many maximal (= “precomplete”) clones on a set of size \lambda. The clones we construct do not contain all unary functions. We then investigate clones that do contain all unary functions. Using a strong negative partition theorem we show that for many cardinals \lambda (in particular, for all successors of regulars) there are 2^{2^\lambda } many such clones on a set of size \lambda. Finally, we show that on a weakly compact cardinal there are exactly 2 maximal clones which contain all unary functions.
keywords: O: alg, (ua)
Sh:738
Göbel, R., & Shelah, S. (2003). Philip Hall’s problem on non-abelian splitters. Math. Proc. Cambridge Philos. Soc., 134(1), 23–31. arXiv: math/0009091 DOI: 10.1017/S0305004102006096 MR: 1937789
type: article
abstract: Philip Hall raised around 1965 the following question which is stated in the Kourovka Notebook: Is there a non-trivial group which is isomorphic with every proper extension of itself by itself? We will decompose the problem into two parts: We want to find non-commutative splitters, that are groups G\neq 1 with {\rm Ext}(G,G)=1. The class of splitters fortunately is quite large so that extra properties can be added to G. We can consider groups G with the following properties: There is a complete group L with cartesian product L^\omega\cong G, {\rm Hom}(L^\omega, S_\omega)=0 (S_\omega the infinite symmetric group acting on \omega) and {\rm End}(L,L)={\rm Inn} L\cup\{0\}. We will show that these properties ensure that G is a splitter and hence obviously a Hall-group in the above sense. Then we will apply a recent result from our joint paper [GbSh:739] which also shows that such groups exist, in fact there is a class of Hall-groups which is not a set.
keywords: O: alg, (stal), (grp)
Sh:739
Göbel, R., & Shelah, S. (2002). Constructing simple groups for localizations. Comm. Algebra, 30(2), 809–837. arXiv: math/0009089 DOI: 10.1081/AGB-120013184 MR: 1883027
type: article
abstract: A group homomorphism \eta: A\to H is called a localization of A if every homomorphism \varphi:A\to H can be ‘extended uniquely’ to a homomorphism \Phi:H\to H in the sense that \Phi\eta=\varphi. This categorical concepts, obviously not depending on the notion of groups, extends classical localizations as known for rings and modules. Moreover this setting has interesting applications in homotopy theory. For localizations \eta:A\to H of (almost) commutative structures A often H resembles properties of A, e.g. size or satisfying certain systems of equalities and non-equalities. Perhaps the best known example is that localizations of finite abelian groups are finite abelian groups. This is no longer the case if A is a finite (non-abelian) group. Libman showed that A_n\to SO_{n-1}({\mathbb R}) for a natural embedding of the alternating group A_n is a localization if n even and n\geq 10. Answering an immediate question by Dror Farjoun and assuming the generalized continuum hypothesis GCH we recently showed in [GRSh:701] that any non-abelian finite simple has arbitrarily large localizations. In this paper we want to remove GCH so that the result becomes valid in ordinary set theory. At the same time we want to generalize the statement for a larger class of A’s.
keywords: M: non, O: alg, (stal), (grp)
Sh:740
Göbel, R., Paras, A. T., & Shelah, S. (2002). Groups isomorphic to all their non-trivial normal subgroups. Israel J. Math., 129, 21–27. arXiv: math/0009088 DOI: 10.1007/BF02773151 MR: 1910930
type: article
abstract: In answer to a question of P. Hall, we supply another construction of a group which is isomorphic to each of its non–trivial normal subgroups.
keywords: O: alg, (stal), (grp)
Sh:741
Göbel, R., & Shelah, S. (2002). Radicals and Plotkin’s problem concerning geometrically equivalent groups. Proc. Amer. Math. Soc., 130(3), 673–674. arXiv: math/0010303 DOI: 10.1090/S0002-9939-01-06108-1 MR: 1866018
type: article
abstract: If G and X are groups and N is a normal subgroup of X, then the G-closure of N in X is the normal subgroup {\overline X}^G=\bigcap\{{\rm ker}\varphi|\varphi:X\rightarrow G, \text{ with } N \subseteq{\rm ker}\varphi\} of X. In particular, {\overline 1}^G = R_GX is the G-radical of X. Plotkin calls two groups G and H geometrically equivalent, written G\sim H, if for any free group F of finite rank and any normal subgroup N of F the G–closure and the H–closure of N in F are the same. Quasiidentities are formulas of the form (\bigwedge_{i\le n} w_i = 1 \rightarrow w =1) for any words w, w_i \ (i\le n) in a free group. Generally geometrically equivalent groups satisfy the same quasiidentiies. Plotkin showed that nilpotent groups G and H satisfy the same quasiidenties if and only if G and H are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample.
keywords: O: alg, (stal), (grp)
Sh:742
Göbel, R., Shelah, S., & Wallutis, S. L. (2003). On universal and epi-universal locally nilpotent groups. Illinois J. Math., 47(1-2), 223–236. arXiv: math/0112252 http://projecteuclid.org/euclid.ijm/1258488149 MR: 2031317
type: article
abstract: In this paper we mainly consider the class LN of all locally nilpotent groups. We first show that there is no universal group in LN_\lambda if \lambda is a cardinal such that \lambda=\lambda^{\aleph_0}; here we call a group G universal (in LN_\lambda) if any group H\in LN_\lambda can be embedded into G where LN_\lambda denotes the class of all locally nilpotent groups of cardinality at most \lambda. However, our main interest is the construction of torsion-free epi-universal groups in LN_\lambda, where G\in LN_\lambda is said to be epi-universal if any group H\in LN_\lambda is an epimorphic image of G. Thus we give an affirmative answer to a question by Plotkin. To prove the torsion-freeness of the constructed locally nilpotent group we adjust the well-known commutator collecting process due to P. Hall to our situation. Finally, we briefly discuss how to use the same methods as for the class LN for other canonical classes of groups to construct epi-universal objects.
keywords: O: alg, (stal), (univ), (grp)
Sh:743
Droste, M., & Shelah, S. (2002). Outer automorphism groups of ordered permutation groups. Forum Math., 14(4), 605–621. arXiv: math/0010304 DOI: 10.1515/form.2002.026 MR: 1900174
type: article
abstract: An infinite linearly ordered set (S,\leq) is called doubly homogeneous if its automorphism group A(S) acts 2-transitively on it. We show that any group G arises as outer automorphism group G\cong{\rm Out}(A(S)) of the automorphism group A(S), for some doubly homogeneous chain (S,\leq).
keywords: O: alg, (stal), (auto)
Sh:744
Shelah, S. (2003). A countable structure does not have a free uncountable automorphism group. Bull. London Math. Soc., 35(1), 1–7. arXiv: math/0010305 DOI: 10.1112/S0024609302001534 MR: 1934424
type: article
abstract: Solecki proved that the group of automorphisms of a countable structure cannot be an uncountable free abelian group. See more in Just, Shelah and Thomas [JShT:654] where as a by product we can say something on on uncountable structures. We prove here the following Theorem: If {\mathbb A} is a countable model, then {\rm Aut}(M) cannot be a free uncountable group.
keywords: O: alg, (stal), (graph)
Sh:745
Nešetřil, J., & Shelah, S. (2003). On the order of countable graphs. European J. Combin., 24(6), 649–663. arXiv: math/0404319 DOI: 10.1016/S0195-6698(03)00064-7 MR: 1995579
type: article
abstract: A set of graphs is said to be independent if there is no homomorphism between distinct graphs from the set. We consider the existence problems related to the independent sets of countable graphs. While the maximal size of an independent set of countable graphs is 2^\omega the On Line problem of extending an independent set to a larger independent set is much harder. We prove here that singletons can be extended (“partnership theorem”). While this is the best possible in general, we give structural conditions which guarantee independent extensions of larger independent sets. This is related to universal graphs, rigid graphs and to the density problem for countable graphs.
keywords: S: ico
Sh:746
Larson, P. B., & Shelah, S. (2003). Bounding by canonical functions, with CH. J. Math. Log., 3(2), 193–215. arXiv: math/0011187 DOI: 10.1142/S021906130300025X MR: 2030084
type: article
abstract: We show that that a certain class of semi-proper iterations does not add \omega-sequences. As a result, starting from suitable large cardinals one can obtain a model in which the Continuum Hypothesis holds and every function from \omega_{1} to \omega_{1} is bounded by a canonical function on a club, and so \omega_{1} is the \omega_{2}-nd canonical function.
keywords: S: for, S: str, (set), (iter), (normal), (large)
Sh:747
Goldstern, M., & Shelah, S. (2009). Large intervals in the clone lattice. Algebra Universalis, 62(4), 367–374. arXiv: math/0208066 DOI: 10.1007/s00012-010-0047-6 MR: 2670171
type: article
abstract: We give three examples of large intervals in the lattice of (local) clones on an infinite set X, by exhibiting clones {\mathcal C}_1, {\mathcal C}_2, {\mathcal C}_3 such that:

(1) the interval [{\mathcal C}_1,{\mathcal O}] in the lattice of local clones is (as a lattice) isomorphic to \{0,1,2,\ldots\} under the divisibility relation,

(2) the interval [{\mathcal C}_2, {\mathcal O}] in the lattice of local clones is isomorphic to the congruence lattice of an arbitrary semilattice,

(3) the interval [{\mathcal C}_3,{\mathcal O}] in the lattice of all clones is isomorphic to the lattice of all filters on X.

These examples explain the difficulty of obtaining a satisfactory analysis of the clone lattice on infinite sets. In particular, (1) shows that the lattice of local clones is not dually atomic.
keywords: S: ico, O: alg, (stal), (ua)
Sh:748
Kikyo, H., & Shelah, S. (2002). The strict order property and generic automorphisms. J. Symbolic Logic, 67(1), 214–216. arXiv: math/0010306 DOI: 10.2178/jsl/1190150038 MR: 1889545
type: article
abstract: If T is an model complete theory with the strict order property, then the theory of the models of T with an automorphism has no model companion.
keywords: M: cla, (mod), (auto)
Sh:749
Eklof, P. C., & Shelah, S. (2003). On the existence of precovers. Illinois J. Math., 47(1-2), 173–188. arXiv: math/0011228 http://projecteuclid.org/euclid.ijm/1258488146 MR: 2031314
type: article
abstract: It is proved undecidable in ZFC + GCH whether every {\mathbb Z}-module has a ^{\perp}\{{\mathbb Z}\}-precover.
keywords: O: alg, (ab), (stal), (wh)
Sh:750
Shelah, S. (2011). On \lambda strong homogeneity existence for cofinality logic. Cubo, 13(2), 59–72. arXiv: 0902.0439 DOI: 10.4067/s0719-06462011000200003 MR: 2908010
type: article
abstract: (none)
keywords: M: smt, (mod)
Sh:751
Eda, K., & Shelah, S. (2002). The non-commutative Specker phenomenon in the uncountable case. J. Algebra, 252(1), 22–26. arXiv: math/0011231 DOI: 10.1016/S0021-8693(02)00045-5 MR: 1922382
type: article
abstract: An infinitary version of the notion of free products has been introduced and investigated by G.Higman. Let G_i (for i\in I) be groups and \ast_{i\in X} G_i the free product of G_i (i\in X) for X \Subset I and p _{XY}:\ast_{i\in Y} G_{i}\rightarrow \ast_{i\in X} G_{i} the canonical homomorphism for X\subseteq Y \Subset I. ( X\Subset I denotes that X is a finite subset of I.) Then, the unrestricted free product is the inverse limit \lim (\ast_{i\in X} G_i, p_{XY}: X\subseteq Y\Subset I).

We remark \ast_{i\in\emptyset} G_i=\{e\}. We prove:

Theorem: Let F be a free group. Then, for each homomorphism h: \lim \ast G_i \to F there exist countably complete ultrafilters u_0,\cdots,u_m on I such that h = h\cdot p_{U_0\cup \cdots\cup U_m} for every U_0\in u_0,\cdots ,U_m\in u_m.

If the cardinality of the index set I is less than the least measurable cardinal, then there exists a finite subset X_0 of I and a homomorphism \overline{h}:\ast _{i\in X_0}G_i\to F such that h=\overline{h}\cdot p_{X_0}, where p_{X_0}:\lim\ast G_i\to \ast_{i\in X_0}G_i is the canonical projection.
keywords: O: alg, (stal), (grp)
Sh:752
Eklof, P. C., & Shelah, S. (2002). Whitehead modules over large principal ideal domains. Forum Math., 14(3), 477–482. arXiv: math/0011230 DOI: 10.1515/form.2002.021 MR: 1899295
type: article
abstract: We consider the Whitehead problem for principal ideal domains of large size. It is proved, in ZFC, that some p.i.d.’s of size \geq\aleph_{2} have non-free Whitehead modules even though they are not complete discrete valuation rings.
keywords: O: alg, (ab), (stal), (wh)
Sh:753
Mildenberger, H., & Shelah, S. (2002). The splitting number can be smaller than the matrix chaos number. Fund. Math., 171(2), 167–176. arXiv: math/0011188 DOI: 10.4064/fm171-2-4 MR: 1880382
type: article
abstract: Let \chi be the minimum cardinal of a subset of 2^\omega that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of creature forcing we show that {\mathfrak s}<\chi is consistent. We thus answer a question by Vojtáš. We give two kinds of models for the strict inequality. The first is the combination of an \aleph_2-iteration of some proper forcing with adding \aleph_1 random reals. The second kind of models is got by adding \delta random reals to a model of {\rm MA}_{<\kappa} for some \delta\in [\aleph_1,\kappa). It was a conjecture of Blass that {\mathfrak s}=\aleph_1<\chi= \kappa holds in such a model. For the analysis of the second model we again use the creature forcing from the first model.
keywords: S: for, S: str, (set), (iter), (inv)
Sh:754
Shelah, S., & Strüngmann, L. H. (2003). It is consistent with ZFC that B_1-groups are not B_2. Forum Math., 15(4), 507–524. arXiv: math/0012172 DOI: 10.1515/form.2003.028 MR: 1978332
type: article
abstract: A torsion-free abelian group B of arbitrary rank is called a B_1-group if {\rm Bext}^1(B,T)=0 for every torsion abelian group T, where {\rm Bext}^1 denotes the group of equivalence classes of all balanced exact extensions of T by B. It is a long-standing problem whether or not the class of B_1-groups coincides with the class of B_2-groups. A torsion-free abelian group B is called a B_2-group if there exists a continuous well-ordered ascending chain of pure subgroups, 0=B_0 \subset B_1 \subset\cdots\subset B_\alpha\subset\cdots\subset B_\lambda=B= \bigcup\limits_{\alpha\in\lambda} B_\alpha such that B_{\alpha+1} =B_\alpha+G_\alpha for every \alpha\in\lambda for some finite rank Butler group G_\alpha. Both, B_1-groups and B_2-groups are natural generalizations of finite rank Butler groups to the infinite rank case and it is known that every B_2-group is a B_1-group. Moreover, assuming V=L it was proven that the two classes coincide. Here we demonstrate that it is undecidable in ZFC whether or not all B_1-groups are B_2-groups. Using Cohen forcing we prove that there is a model of ZFC in which there exists a B_1-group that is not a B_2-group.
keywords: S: for, (ab)
Sh:755
Shelah, S. (2002). Weak diamond. Sci. Math. Jpn., 55(3), 531–538. arXiv: math/0107207 MR: 1901038
type: article
abstract: Under some cardinal arithmetic assumptions, we prove that every stationary of \lambda of a right cofinality has the weak diamond. This is a strong negation of uniformization. We then deal with a weaker version of the weak diamond- colouring restrictions. We then deal with semi- saturated (normal) filters.
keywords: S: ico, (set), (normal), (wd)
Sh:756
Hyttinen, T., & Shelah, S. (2002). Forcing a Boolean algebra with predesigned automorphism group. Proc. Amer. Math. Soc., 130(10), 2837–2843. arXiv: math/0102044 DOI: 10.1090/S0002-9939-02-06399-2 MR: 1908905
type: article
abstract: For suitable groups G we will show that one can add a Boolean algebra B by forcing in such a way that Aut(B) is almost isomorphic to G. In particular, we will give a positive answer to the following question due to J. Roitman: Is \aleph_{\omega} a possible number of automorphisms of a rich Boolean algebra?
keywords: S: for, (set), (ba), (auto)
Sh:757
Shelah, S. (2004). Quite complete real closed fields. Israel J. Math., 142, 261–272. arXiv: math/0112212 DOI: 10.1007/BF02771536 MR: 2085719
type: article
abstract: We prove that any ordered field can be extended to one for which every decreasing sequence, of closed intervals has a non empty intersection.
keywords: O: alg, (stal)
Sh:758
Matsubara, Y., & Shelah, S. (2002). Nowhere precipitousness of the non-stationary ideal over \mathcal P_{\kappa}\lambda. J. Math. Log., 2(1), 81–89. arXiv: math/0102045 DOI: 10.1142/S021906130200014X MR: 1900548
type: article
abstract: We prove that if \lambda is a strong limit singular cardinal and \kappa a regular uncountable cardinal <\lambda, then NS_{\kappa\lambda}, the non-stationary ideal over {\mathcal P}_{\kappa}\lambda, is nowhere precipitous. We also show that under the same hypothesis every stationary subset of {\mathcal P}_{\kappa}\lambda can be partitioned into \lambda^{< \kappa} disjoint stationary sets.
keywords: S: for, (set), (normal), (app(pcf))
Sh:759
Baldwin, J. T., & Shelah, S. (2001). Model companions of T_\mathrm{Aut} for stable T. Notre Dame J. Formal Logic, 42(3), 129–142 (2003). arXiv: math/0105136 DOI: 10.1305/ndjfl/1063372196 MR: 2010177
type: article
abstract: Let T be a complete first order theory in a countable relational language L. We assume relation symbols have been added to make each formula equivalent to a predicate. Adjoin a new unary function symbol \sigma to obtain the language L_\sigma; T_\sigma is obtained by adding axioms asserting that \sigma is an L-automorphism. We provide necessary and sufficient conditions for T_{\rm Aut} to have a model companion when T is stable. Namely, we introduce a new condition: T admits obstructions, and show that T_{\rm Aut} has a model companion iff and only if T does not admit obstructions. This condition is weakening of the finite cover property: if a stable theory T has the finite cover property then T admits obstructions.
keywords: M: cla, O: alg, (mod), (auto), (sta)
Sh:760
Blass, A. R., Gurevich, Y., & Shelah, S. (2002). On polynomial time computation over unordered structures. J. Symbolic Logic, 67(3), 1093–1125. arXiv: math/0102059 DOI: 10.2178/jsl/1190150152 MR: 1926601
type: article
abstract: This paper is motivated by the question whether there exists a logic capturing polynomial time computation over unordered structures. We consider several algorithmic problems near the border of the known, logically defined complexity classes contained in polynomial time. We show that fixpoint logic plus counting is stronger than might be expected, in that it can express the existence of a complete matching in a bipartite graph. We revisit the known examples that separate polynomial time from fixpoint plus counting. We show that the examples in a paper of Cai, Fürer, and Immerman, when suitably padded, are in choiceless polynomial time yet not in fixpoint plus counting. Without padding, they remain in polynomial time but appear not to be in choiceless polynomial time plus counting. Similar results hold for the multipede examples of Gurevich and Shelah, except that their final version of multipedes is, in a sense, already suitably padded. Finally, we describe another plausible candidate, involving determinants, for the task of separating polynomial time from choiceless polynomial time plus counting.
keywords: O: fin, (fmt)
Sh:761
Shelah, S. (2003). A partition relation using strongly compact cardinals. Proc. Amer. Math. Soc., 131(8), 2585–2592. arXiv: math/0103155 DOI: 10.1090/S0002-9939-02-06789-8 MR: 1974659
type: article
abstract: If \kappa is strongly compact, \lambda>\kappa is regular, then (2^{<\lambda})^+\to (\lambda+\eta)^2_\theta holds for \eta, \theta<\kappa.
keywords: S: ico, (set), (pc), (pure(large))
Sh:762
Brendle, J., & Shelah, S. (2003). Evasion and prediction. IV. Strong forms of constant prediction. Arch. Math. Logic, 42(4), 349–360. arXiv: math/0103153 DOI: 10.1007/s001530200143 MR: 2018086
type: article
abstract: Say that a function \pi:n^{<\omega}\to n (henceforth called a predictor) k–constantly predicts a real x\in n^\omega if for almost all intervals I of length k, there is i\in I such that x(i)=\pi(x\restriction i). We study the k–constant prediction number {\mathfrak v}_n^{\rm const}(k), that is, the size of the least family of predictors needed to k–constantly predict all reals, for different values of n and k, and investigate their relationship.
keywords: S: for, S: str, (set), (inv)
Sh:763
Fuchino, S., Greenberg, N., & Shelah, S. (2006). Models of real-valued measurability. Ann. Pure Appl. Logic, 142(1-3), 380–397. arXiv: math/0601087 DOI: 10.1016/j.apal.2006.04.003 MR: 2250550
See [Sh:E85]
type: article
abstract: Solovay’s random-real forcing is the standard way of producing real-valued measurable cardinals. Following questions of Fremlin, by giving a new construction, we show that there are combinatorial, measure-theoretic properties of Solovay’s model that do not follow from the existence of real-valued measurability.
keywords: S: str, (set), (leb)
Sh:764
Shelah, S., & Shioya, M. (2006). Nonreflecting stationary sets in \mathcal P_\kappa\lambda. Adv. Math., 199(1), 185–191. arXiv: math/0405013 DOI: 10.1016/j.aim.2005.01.012 MR: 2187403
See [Sh:E86]
type: article
abstract: Let \kappa be a regular uncountable cardinal and \lambda\geq \kappa^+. The principle of stationary reflection for {\mathcal P}_\kappa\lambda has been successful in settling problems of infinite combinatorics in the case \kappa=\omega_1. For a greater \kappa the principle is known to fail at some \lambda. This note shows that it fails at every \lambda if \kappa is the successor of a regular uncountable cardinal or \kappa is countably closed.
keywords: S: ico, (set), (normal), (ref)
Sh:765
Juhász, I., Shelah, S., Soukup, L., & Szentmiklóssy, Z. (2004). Cardinal sequences and Cohen real extensions. Fund. Math., 181(1), 75–88. arXiv: math/0404322 DOI: 10.4064/fm181-1-3 MR: 2071695
type: article
abstract: We show that if we add any number of Cohen reals to the ground model then, in the generic extension, a locally compact scattered space has at most (2^{\aleph_0})^V many levels of size \omega. We also give a complete ZFC characterization of the cardinal sequences of regular scattered spaces. Although the classes of the regular and of the 0-dimensional scattered spaces are different, we prove that they have the same cardinal sequences.
keywords: O: top, (set), (ba), (gt)
Sh:766
Fuchs, L., & Shelah, S. (2003). On a non-vanishing Ext. Rend. Sem. Mat. Univ. Padova, 109, 235–239. arXiv: math/0405015 MR: 1997989
type: article
abstract: The existence of valuation domains admitting non-standard uniserial modules for which certain Exts do not vanish was proved under Jensen’s Diamond Principle. In this note, the same is verified using the ZFC axioms alone.
keywords: O: alg, (ab), (wh)
Sh:767
Shelah, S., & Tsuboi, A. (2002). Definability of initial segments. Notre Dame J. Formal Logic, 43(2), 65–73 (2003). arXiv: math/0104277 DOI: 10.1305/ndjfl/1071509428 MR: 2033316
type: article
abstract: We consider implicit definability of the standard part \{0,1,...\} in nonstandard models of Peano arithmetic (PA), and we ask whether there is a model of PA in which the standard part is implicitly definable. In §1, we define a certain class of formulas, and show that in any model of PA the standard part is not implicitly defined by using such formulas. In §2 we construct a model of PA in which the standard part is implicitly defined. To construct such a model, first we assume a set theoretic hypothesis \diamondsuit_{S_\lambda^{\lambda^+}}, which is an assertion of the existence of a very general set. Then we shall eliminate the hypothesis using absoluteness for the existence of a model having a tree structure with a certain property.
keywords: M: non, (mod)
Sh:768
Shelah, S., & Tsaban, B. (2003). Critical cardinalities and additivity properties of combinatorial notions of smallness. J. Appl. Anal., 9(2), 149–162. arXiv: math/0304019 DOI: 10.1515/JAA.2003.149 MR: 2021285
type: article
abstract: Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (\tau-covers). We deal with two types of combinatorial questions which arise from this study.

(a) Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum.

(b) We study the additivity numbers of the combinatorial notions corresponding to the topological diagonalization notions.

This gives new insights on the structure of the eventual dominance ordering on the Baire space, the almost inclusion ordering on the Rothberger space, and the interactions between them.
keywords: S: ico, (inv)
Sh:769
Kennedy, J. C., & Shelah, S. (2002). On regular reduced products. J. Symbolic Logic, 67(3), 1169–1177. arXiv: math/0105135 DOI: 10.2178/jsl/1190150156 MR: 1926605
type: article
abstract: Assume \langle\aleph_0,\aleph_1\rangle\rightarrow\langle \lambda,\lambda^+\rangle. Assume M is a model of a first order theory T of cardinality at most \lambda^+ in a vocabulary {\mathcal L}(T) of cardinality \leq\lambda. Let N be a model with the same vocabulary. Let \Delta be a set of first order formulas in {\mathcal L}(T) and let D be a regular filter on \lambda. Then M is \Delta-embeddable into the reduced power N^\lambda/D, provided that every \Delta-existential formula true in M is true also in N. We obtain the following corollary: for M as above and D a regular ultrafilter over \lambda, M^\lambda/ D is \lambda^{++}-universal. Our second result is as follows: For i<\mu let M_i and N_i be elementarily equivalent models of a vocabulary which has has cardinality \le\lambda. Suppose D is a regular filter on \mu and \langle \aleph_0,\aleph_1\rangle\rightarrow\langle\lambda,\lambda^+ \rangle holds. We show that then the second player has a winning strategy in the Ehrenfeucht-Fraisse game of length \lambda^+ on \prod_i M_i/D and \prod_i N_i/D. This yields the following corollary: Assume GCH and \lambda regular (or just \langle\aleph_0,\aleph_1 \rangle \rightarrow \langle \lambda,\lambda^+ \rangle and 2^\lambda=\lambda^+). For L, M_i and N_i as above, if D is a regular filter on \lambda, then \prod_i M_i/D\cong\prod_i N_i/D.
keywords: S: ico, (mod), (up)
Sh:770
Hellsten, A., Hyttinen, T., & Shelah, S. (2002). Potential isomorphism and semi-proper trees. Fund. Math., 175(2), 127–142. arXiv: math/0112288 DOI: 10.4064/fm175-2-3 MR: 1969631
type: article
abstract: We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notions of semi-proper and weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. We prove the existence of semi-proper trees under certain cardinal arithmetic assumptions. We also show the consistency of the non-existence of weakly semi-proper trees assuming the consistency of some large cardinals.
keywords: M: odm, (set-mod)
Sh:771
Shelah, S. (2011). Polish algebras, shy from freedom. Israel J. Math., 181, 477–507. arXiv: math/0212250 DOI: 10.1007/s11856-011-0020-x MR: 2773054
type: article
abstract: Every Polish group is not free whereas some F_\sigma group may be free. Also every automorphism group of a structure of cardinality, e.g. \beth_\omega is not free.
keywords: S: dst, (mod), (stal), (grp), (ps-dst)
Sh:772
Shelah, S., & Strüngmann, L. H. (2003). Kulikov’s problem on universal torsion-free abelian groups. J. London Math. Soc. (2), 67(3), 626–642. arXiv: math/0112253 DOI: 10.1112/S0024610703004216 MR: 1967696
type: article
abstract: Let T be an abelian group and \lambda an uncountable regular cardinal. We consider the question of whether there is a \lambda-universal group G^* among all torsion-free abelian groups G of cardinality less than or equal to \lambda satisfying {\rm Ext}(G,T)=0. Here G^* is said to be \lambda-universal for T if, whenever a torsion-free abelian group G of cardinality less than or equal to \lambda satisfies {\rm Ext}(G,T)=0, then there is an embedding of G into G^*. For large classes of abelian groups T and cardinals \lambda it is shown that the answer is consistently no. In particular, for T torsion, this solves a problem of Kulikov.
keywords: O: alg, (ab), (wh)
Sh:773
Shelah, S., & Strüngmann, L. H. (2002). Cotorsion theories cogenerated by \aleph_1-free abelian groups. Rocky Mountain J. Math., 32(4), 1617–1626. arXiv: math/0107208 DOI: 10.1216/rmjm/1181070044 MR: 1987629
type: article
abstract: Given an \aleph_1-free abelian group G we characterize the class {\mathfrak C_G} of all torsion abelian groups T satisfying {\rm Ext}(G,T)=0 assuming the continuum hypothesis CH. Moreover, in Gödel’s constructable universe we prove that this characterizes {\mathfrak C}_G for arbitrary torsion-free abelian G. It follows that there exist some ugly \aleph_1-free abelian groups.
keywords: S: ico, (ab), (cont)
Sh:774
Bartoszyński, T., Shelah, S., & Tsaban, B. (2003). Additivity properties of topological diagonalizations. J. Symbolic Logic, 68(4), 1254–1260. arXiv: math/0112262 DOI: 10.2178/jsl/1067620185 MR: 2017353
type: article
abstract: In a work of Just, Miller, Scheepers and Szeptycki it was asked whether certain diagonalization properties for sequences of open covers are provably closed under taking finite or countable unions. In a recent work, Scheepers proved that one of the classes in question is closed under taking countable unions. In this paper we show that none of the remaining classes is provably closed under taking finite unions, and thus settle the problem. We also show that one of these properties is consistently (but not provably) closed under taking unions of size less than the continuum, by relating a combinatorial version of this problem to the Near Coherence of Filters (NCF) axiom, which asserts that the Rudin-Keisler ordering is downward directed.
keywords: S: str, (set), (gt)
Sh:775
Shelah, S. (2005). Super black box (ex. Middle diamond). Arch. Math. Logic, 44(5), 527–560. arXiv: math/0212249 DOI: 10.1007/s00153-004-0239-x MR: 2210145
type: article
abstract: This is a slightly corrected version of an old work.

Under certain cardinal arithmetic assumptions, we prove that for every large enough regular \lambda cardinal, for many regular \kappa < \lambda, many stationary subsets of \lambda concentrating on cofinality \kappa have super BB. In particular, we have the super BB on \{\delta < \lambda \colon cf(\delta) = \kappa\}. This is a strong negation of uniformization.

We have added some details. Works continuing it are [Sh:898] and [Sh:1028]. We thank Ari Brodski and Adi Jarden for their helpful comments.

In this paper we had earlier used the notion “middle diamond" which is now replaced by “super BB”, that is, “super black box”, in order to be consistent with other papers (see [Sh:898]).
keywords: S: ico, (set), (BB), (wd)
Sh:776
Hyttinen, T., Shelah, S., & Väänänen, J. A. (2002). More on the Ehrenfeucht-Fraïssé game of length \omega_1. Fund. Math., 175(1), 79–96. arXiv: math/0212234 DOI: 10.4064/fm175-1-5 MR: 1971240
type: article
abstract: Let A and B be two first order structures of the same relational vocabulary L. The Ehrenfeucht-Fraı̈ssé-game of length \gamma of A and B denoted by EFG_\gamma(A,B) is defined as follows: There are two players called \forall and \exists. First \forall plays x_0 and then \exists plays y_0. After this \forall plays x_1, and \exists plays y_1, and so on. Eventually a sequence \langle(x_\beta,y_\beta):\beta< \gamma\rangle has been played. The rules of the game say that both players have to play elements of A\cup B. Moreover, if \forall plays his x_\beta in A (B), then \exists has to play his y_\beta in B (A). Thus the sequence \langle(x_\beta,y_\beta): \beta<\gamma\rangle determines a relation \pi\subseteq A\times B. Player \exists wins this round of the game if \pi is a partial isomorphism. Otherwise \forall wins. The game EFG_\gamma^\delta(A,B) is defined similarly except that the players play sequences of length <\delta at a time. Theorem 1: The following statements are equiconsistent relative to ZFC: (A) There is a weakly compact cardinal. (B) CH and EF_{\omega_1}(A,B) is determined for all models A,B of cardinality \aleph_2. Theorem 2: Assume that 2^\omega<2^{\omega_3} and T is a countable complete first order theory. Suppose that one of (i)-(iii) below holds. Then there are A ,B\models T of power \omega_3 such that for all cardinals 1<\theta\leq\omega_3, EF^\theta_{\omega_1}(A,B) is non-determined. [(i)] T is unstable. [(ii)] T is superstable with DOP or OTOP. [(iii)] T is stable and unsuperstable and 2^\omega\leq\omega_3.
keywords: M: smt, (inf-log), (set-mod)
Sh:777
Rosłanowski, A., & Shelah, S. (2007). Sheva-Sheva-Sheva: large creatures. Israel J. Math., 159, 109–174. arXiv: math/0210205 DOI: 10.1007/s11856-007-0040-8 MR: 2342475
type: article
abstract: We develop the theory of the forcing with trees and creatures for an inaccessible \lambda continuing Rosłanowski and Shelah [RoSh:470], [RoSh:672]. To make a real use of these forcing notions (that is to iterate them without collapsing cardinals) we need suitable iteration theorems, and those are proved as well. (In this aspect we continue Rosłanowski and Shelah [RoSh:655].)
keywords: S: for, (set), (creatures)
Sh:778
Mildenberger, H., & Shelah, S. (2003). Specialising Aronszajn trees by countable approximations. Arch. Math. Logic, 42(7), 627–647. arXiv: math/0112287 DOI: 10.1007/s00153-002-0168-5 MR: 2015092
type: article
abstract: We show that there are proper forcings based upon countable trees of creatures that specialize a given Aronszajn tree.
keywords: S: ico, S: for, (creatures)
Sh:779
Larson, P. B., & Shelah, S. (2017). Consistency of a strong uniformization principle. Colloq. Math., 146(1), 1–13. DOI: 10.4064/cm6542-3-2016 MR: 3570198
type: article
abstract: We prove the consistency of a strong uniformization for some \aleph_1 branches {}^{\omega>}\omega. As a consequence we get the consistency of a relative speaking on groups.
keywords: S: str
Sh:780
Göbel, R., & Shelah, S. (2003). Characterizing automorphism groups of ordered abelian groups. Bull. London Math. Soc., 35(3), 289–292. arXiv: math/0112264 DOI: 10.1112/S0024609302001881 MR: 1960938
type: article
abstract: We want to characterize the groups isomorphic to full automorphism groups of ordered abelian groups. The result will follow from classical theorems on ordered groups adding an argument from proofs used to realize rings as endomorphism rings of abelian groups.
keywords: O: alg, (ab)
Sh:781
Kojman, M., & Shelah, S. (2003). van der Waerden spaces and Hindman spaces are not the same. Proc. Amer. Math. Soc., 131(5), 1619–1622. arXiv: math/0112265 DOI: 10.1090/S0002-9939-02-06916-2 MR: 1950294
type: article
abstract: A Hausdorff topological space X is van der Waerden if for every sequence (x_n)_n in X there is a converging subsequence (x_n)_{n\in A} where A\subseteq\omega contains arithmetic progressions of all finite lengths. A Hausdorff topological space X is Hindman if for every sequence (x_n)_n in X there is an IP-converging subsequence (x_n)_{n\in FS(B)} for some infinite B\subseteq\omega. We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.
keywords: S: ico, S: str
Sh:782
Shelah, S. (2005). On the Arrow property. Adv. In Appl. Math., 34(2), 217–251. arXiv: math/0112213 DOI: 10.1016/j.aam.2002.03.001 MR: 2110551
type: article
abstract: Let X be a finite set of alternatives. A choice function c is a mapping which assigns to nonempty subsets S of X an element c(S) of S. A rational choice function is one for which there is a linear ordering on the alternatives such that c(S) is the maximal element of S according to that ordering. Arrow’s impossibility theorem asserts that under certain natural conditions, if there are at least three alternatives then every non-dictatorial social choice gives rise to a non-rational choice function. Gil Kalai asked if Arrow’s theorem can be extended to the case when the individual choices are not rational but rather belong to an arbitrary non-trivial symmetric class of choice functions. The main theorem of this paper gives an affirmative answer in a very general setting.
keywords: O: fin
Sh:783
Shelah, S. (2009). Dependent first order theories, continued. Israel J. Math., 173, 1–60. arXiv: math/0406440 DOI: 10.1007/s11856-009-0082-1 MR: 2570659
type: article
abstract: A dependent theory is a (first order complete theory) T which does not have the independence property. A major result here is: if we expand a model of T by the traces on it of sets definable in a bigger model then we preserve its being dependent. Another one justifies the cofinality restriction in the theorem (from a previous work) saying that pairwise perpendicular indiscernible sequences, can have arbitrary dual-cofinalities in some models containing them. We introduce "strongly dependent" and look at definable groups; and also at dividing, forking and relatives.
keywords: M: cla
Sh:784
Shelah, S. (2004). Forcing axiom failure for any \lambda>\aleph_1. Arch. Math. Logic, 43(3), 285–295. arXiv: math/0112286 DOI: 10.1007/s00153-003-0208-9 MR: 2052883
type: article
abstract: David Aspero asks on the possibility of having Forcing axiom FA_{{\aleph_2}}(\mathfrak{K}), where \mathfrak{K} is the class of forcing notions preserving stationarily of subsets of \aleph_1 and of \aleph_2. We answer negatively, in fact we show the negative result for any regular \lambda>\aleph_1 even demanding adding no new sequence of ordinals of length <\lambda.
keywords: S: ico, S: for, (set)
Sh:785
Göbel, R., Shelah, S., & Strüngmann, L. H. (2003). Almost-free E-rings of cardinality \aleph_1. Canad. J. Math., 55(4), 750–765. arXiv: math/0112214 DOI: 10.4153/CJM-2003-032-8 MR: 1994072
type: article
abstract: An E-ring is a unital ring R such that every endomorphism of the underlying abelian group R^+ is multiplication by some ring-element. The existence of almost-free E-rings of cardinality greater than 2^{\aleph_0} is undecidable in ZFC. While they exist in Goedel’s universe, they do not exist in other models of set theory. For a regular cardinal \aleph_1\leq\lambda\leq 2^{\aleph_0} we construct E-rings of cardinality \lambda in ZFC which have \aleph_1-free additive structure. For \lambda= \aleph_1 we therefore obtain the existence of almost-free E-rings of cardinality \aleph_1 in ZFC.
keywords: O: alg, (ab)
Sh:786
Shelah, S., & Steprāns, J. (2007). Possible cardinalities of maximal abelian subgroups of quotients of permutation groups of the integers. Fund. Math., 196(3), 197–235. arXiv: math/0212233 DOI: 10.4064/fm196-3-1 MR: 2353856
type: article
abstract: The maximality of Abelian subgroups play a role in various parts of group theory. For example, Mycielski has extended a classical result of Lie groups and shown that a maximal Abelian subgroup of a compact connected group is connected and, furthermore, all the maximal Abelian subgroups are conjugate. For finite symmetric groups the question of the size of maximal Abelian subgroups has been examined by Burns and Goldsmith in 1989 and Winkler in 1993. We show that there is not much interest in generalizing this study to infinite symmetric groups; the cardinality of any maximal Abelian subgroup of the symmetric group of the integers is 2^{\aleph_0}. Our purpose is also to examine the size of maximal Abelian subgroups for a class of groups closely related to the the symmetric group of the integers; these arise by taking an ideal on the integers, considering the subgroup of all permutations which respect the ideal and then taking the quotient by the normal subgroup of permutations which fix all integers except a set in the ideal. We prove that the maximal size of Abelian subgroups in such groups is sensitive to the nature of the ideal as well as various set theoretic hypotheses.
keywords: S: ico, S: for, (set), (ab)
Sh:787
Shelah, S., & Väisänen, P. (2002). Almost free groups and Ehrenfeucht-Fraïssé games for successors of singular cardinals. Ann. Pure Appl. Logic, 118(1-2), 147–173. arXiv: math/0212063 DOI: 10.1016/S0168-0072(02)00037-4 MR: 1934121
type: article
abstract: We strengthen non-structure theorems for almost free Abelian groups by studying long Ehrenfeucht-Fraı̈ssé games between a fixed group of cardinality \lambda and a free Abelian group. A group is called \epsilon-game-free if the isomorphism player has a winning strategy in the game (of the described form) of length \epsilon \in \lambda. We prove for a large set of successor cardinals \lambda = \mu^+ existence of nonfree (\mu \cdot \omega_1)-game-free groups of cardinality \lambda. We concentrate on successors of singular cardinals.
keywords: O: alg, (ab)
Sh:788
Komjáth, P., & Shelah, S. (2005). Finite subgraphs of uncountably chromatic graphs. J. Graph Theory, 49(1), 28–38. arXiv: math/0212064 DOI: 10.1002/jgt.20060 MR: 2130468
type: article
abstract: It is consistent that for every monotonically increasing function f:\omega\to\omega there is a graph with size and chromatic number \aleph_1 in which every n-chromatic subgraph has at least f(n) elements (n\geq 3). This solves a 250 problem of Erdős. It is also consistent that there is a graph X with {\rm Chr}(X)=|X|=\aleph_1 such that if Y is a graph all whose finite subgraphs occur in X then {\rm Chr}(Y)\leq \aleph_2.
keywords: S: ico, S: for
Sh:789
Shelah, S., & Usvyatsov, A. (2006). Banach spaces and groups—order properties and universal models. Israel J. Math., 152, 245–270. arXiv: math/0303325 DOI: 10.1007/BF02771986 MR: 2214463
type: article
abstract: We deal with two natural examples of almost-elementary classes: the class of all Banach spaces (over {\mathbb R} or {\mathbb C}) and the class of all groups. We show both of these classes do not have the strict order property, and find the exact place of each one of them in Shelah’s SOP_n (strong order property of order n) hierarchy. Remembering the connection between this hierarchy and the existence of universal models, we conclude, for example, that there are “few” universal Banach spaces (under isometry) of regular cardinalities.
keywords: M: cla, (mod), (grp)
Sh:790
Shelah, S., & Väänänen, J. A. (2006). Recursive logic frames. MLQ Math. Log. Q., 52(2), 151–164. arXiv: math/0405016 DOI: 10.1002/malq.200410058 MR: 2214627
type: article
abstract: We define the concept of a logic frame, which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive logic frame is called recursively (countably) compact, if every recursive (respectively, countable) finitely consistent theory has a model. We show that for logic frames built from the cardinality quantifiers "there exists at least \lambda" recursive compactness always implies countable compactness. On the other hand we show that a recursively compact extension need not be countably compact.
keywords: M: smt, (mod)
Sh:791
Shelah, S., & Zapletal, J. (2002). Duality and the PCF theory. Math. Res. Lett., 9(5-6), 585–595. arXiv: math/0212041 DOI: 10.4310/MRL.2002.v9.n5.a2 MR: 1906062
type: article
abstract: We consider natural cardinal invariants {\mathfrak hm}_n and prove several duality theorems, saying roughly: if I is a suitably definable ideal and provably {\rm cov}(I)\geq{\mathfrak hm}_n, then {\rm non}(I) is provably small. The proofs integrate the determinacy theory, forcing and pcf theory.
keywords: S: for, S: pcf, S: str
Sh:792
Shelah, S., & Zapletal, J. (2003). Games with creatures. Comment. Math. Univ. Carolin., 44(1), 9–21. arXiv: math/0212042 MR: 2045842
type: article
abstract: Many forcing notions obtained using the creature technology are naturally connected with certain integer games
keywords: S: for, (set), (creatures)
Sh:793
Kojman, M., Kubiś, W., & Shelah, S. (2004). On two problems of Erdős and Hechler: new methods in singular madness. Proc. Amer. Math. Soc., 132(11), 3357–3365. arXiv: math/0406441 DOI: 10.1090/S0002-9939-04-07580-X MR: 2073313
type: article
abstract: For an infinite cardinal \mu, {\rm MAD}(\mu) denotes the set of all cardinalities of nontrivial maximal almost disjoint families over \mu. Erdős and Hechler proved the consistency of \mu\in {\rm MAD}(\mu) for a singular cardinal \mu and asked if it was ever possible for a singular \mu that \mu\notin {\rm MAD}(\mu), and also whether 2^{{\rm cf}\mu}<\mu\Longrightarrow\mu\in{\rm MAD} (\mu) for every singular cardinal \mu.

We introduce a new method for controlling {\rm MAD}(\mu) for a singular \mu and, among other new results about the structure of {\rm MAD}(\mu) for singular \mu, settle both problems affirmatively.
keywords: S: ico, (set)
Sh:794
Shelah, S. (2008). Reflection implies the SCH. Fund. Math., 198(2), 95–111. arXiv: math/0404323 DOI: 10.4064/fm198-2-1 MR: 2369124
type: article
abstract: We prove that, e.g., if \mu>{\rm cf}(\mu)=\aleph_0 and \mu> 2^{\aleph_0} and every stationary family of countable subsets of \mu^+ reflect in some subset of \mu^+ of cardinality \aleph_1, then the SCH for \mu^+ holds. (Moreover, for \mu^+, any scale for \mu^+ has a bad stationary set of cofinality \aleph_1.) This answers a question of Foreman and Todorčević, who got such conclusion from the simultaneous reflection of four stationary sets.
keywords: S: pcf, (set)
Sh:795
Juhász, I., & Shelah, S. (2003). Generic left-separated spaces and calibers. Topology Appl., 132(2), 103–108. arXiv: math/0212027 DOI: 10.1016/S0166-8641(02)00367-X MR: 1991801
type: article
abstract: We use a natural forcing to construct a left-separated topology on an arbitrary cardinal \kappa. The resulting left-separated space X_\kappa is also 0-dimensional T_2, hereditarily Lindelöf, and countably tight. Moreover if \kappa is regular then d(X_\kappa)=\kappa, hence \kappa is not a caliber of X_\kappa, while all other uncountable regular cardinals are.

We also prove it consistent that for every countable set A of uncountable regular cardinals there is a hereditarily Lindelöf T_3 space X such that \varrho=cf(\varrho) >\omega is a caliber of X exactly if \varrho\not\in A.
keywords: S: for
Sh:796
Komjáth, P., & Shelah, S. (2003). A partition theorem for scattered order types. Combin. Probab. Comput., 12(5-6), 621–626. arXiv: math/0212022 DOI: 10.1017/S0963548303005686 MR: 2037074
type: article
abstract: If \phi is a scattered order type, \mu a cardinal, then there exists a scattered order type \psi such that \psi \to [\phi]^{1}_{\mu,\aleph_0} holds.
keywords: S: ico
Sh:797
Shelah, S. (2012). Nice infinitary logics. J. Amer. Math. Soc., 25(2), 395–427. arXiv: 1005.2806 DOI: 10.1090/S0894-0347-2011-00712-1 MR: 2869022
type: article
abstract: We deal with soft model theory of infinitary logics. We find a logic between {mathbb L}_{\infty,\aleph_0} and {\mathbb L}_{\infty, \infty} which has some striking properties. First, it has interpolations (it was known that each of those logics fail interpolation though the pair has). Second, well ordering is not characterized in a strong way. Third, it can be characterized as the maximal such nice logic (in fact, is the maximal logic stronger than {\mathbb L}_{\infty, \aleph_0} and which satisfies “well ordering is not characterized in a strong way").
keywords: M: smt, (inf-log)
Sh:799
Matet, P., Rosłanowski, A., & Shelah, S. (2005). Cofinality of the nonstationary ideal. Trans. Amer. Math. Soc., 357(12), 4813–4837. arXiv: math/0210087 DOI: 10.1090/S0002-9947-05-04007-9 MR: 2165389
type: article
abstract: We show that the reduced cofinality of the nonstationary ideal NS_\kappa on a regular uncountable cardinal \kappa may be less than its cofinality, where the reduced cofinality of NS_\kappa is the least cardinality of any family F of nonstationary subsets of \kappa such that every nonstationary subset of \kappa can be covered by less than \kappa many members of F.
keywords: S: ico, S: pcf, (set), (normal)
Sh:801
Doron, M., & Shelah, S. (2005). A dichotomy in classifying quantifiers for finite models. J. Symbolic Logic, 70(4), 1297–1324. arXiv: math/0405091 DOI: 10.2178/jsl/1129642126 MR: 2194248
type: article
abstract: We consider a family \mathfrak{U} of finite universes. The second order quantifier Q_{\mathfrak{R}}, means for each U\in {\mathfrak{U}} quantifying over a set of n({\mathfrak{R}})-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called interpretability. We show that for every Q_{\mathfrak {R}}, ever Q_{\mathfrak {R}} is interpretable by quantifying over subsets of U and one to one functions on U both of bounded order, or the logic L(Q_{\mathfrak{R}}) (first order logic plus the quantifier Q_{\mathfrak{R}}) is undecidable.
keywords: M: smt, M: non, (fmt)
Sh:802
Kubiś, W., & Shelah, S. (2003). Analytic colorings. Ann. Pure Appl. Logic, 121(2-3), 145–161. arXiv: math/0212026 DOI: 10.1016/S0168-0072(02)00110-0 MR: 1982945
type: article
abstract: We investigate the existence of perfect homogeneous sets for analytic colorings. An analytic coloring of X is an analytic subset of [X]^N, where N>1 is a natural number. We define an absolute rank function on trees representing analytic colorings, which gives an upper bound for possible cardinalities of homogeneous sets and which decides whether there exists a perfect homogeneous set. We construct universal \sigma-compact colorings of any prescribed rank \gamma<\omega_1. These colorings consistently contain homogeneous sets of cardinality \aleph_\gamma but they do not contain perfect homogeneous sets. As an application, we discuss the so-called defectedness coloring of subsets of Polish linear spaces.
keywords: S: str
Sh:803
Shelah, S., & Strüngmann, L. H. (2009). Large indecomposable minimal groups. Q. J. Math., 60(3), 353–365. DOI: 10.1093/qmath/han012 MR: 2533663
type: article
abstract: Assuming V=L we prove that there exist indecomposable almost-free minimal groups of size \lambda for every regular cardinal \lambda below the first weakly compact cardinal. This is to say that there are indecomposable almost-free torsion-free abelian groups of cardinality \lambda which are isomorphic to all of their finite index subgroups.
keywords: O: alg, (ab)
Sh:805
Gitik, M., Schindler, R.-D., & Shelah, S. (2006). PCF theory and Woodin cardinals. In Logic Colloquium ’02, Vol. 27, Assoc. Symbol. Logic, La Jolla, CA, pp. 172–205. arXiv: math/0211433 MR: 2258707
type: article
abstract: We prove the following two results. Theorem A: Let \alpha be a limit ordinal. Suppose that 2^{|\alpha|}<\aleph_\alpha and 2^{|\alpha|^+}<\aleph_{| \alpha|^+}, whereas \aleph_\alpha^{|\alpha|}>\aleph_{| \alpha|^+}. Then for all n<\omega and for all bounded X\subset \aleph_{|\alpha|^+}, M_n^\#(X) exists.

Theorem B: Let \kappa be a singular cardinal of uncountable cofinality. If \{\alpha<\kappa\ | \ 2^\alpha=\alpha^+\} is stationary as well as co-stationary then for all n<\omega and for all bounded X\subset\kappa, M_n^\#(X) exists.

Theorem A answers a question of Gitik and Mitchell, and Theorem B yields a lower bound for an assertion discussed in Gitik, M., Introduction to Prikry type forcing notions, in: Handbook of set theory, Foreman, Kanamori, Magidor (see Problem 4 there).

The proofs of these theorems combine pcf theory with core model theory. Along the way we establish some ZFC results in cardinal arithmetic, motivated by Silver’s theorem and we obtain results of core model theory, motivated by the task of building a “stable core model.” Both sets of results are of independent interest.
keywords: S: pcf, (set)
Sh:806
Shelah, S. Martin’s axiom and maximal orthogonal families. Real Anal. Exchange, 28(2), 477–480. arXiv: math/0211438 DOI: 10.14321/realanalexch.28.2.0477 MR: 2010330
type: article
abstract: It is shown that Martin’s Axiom for \sigma-centred partial orders implies that every maximal orthogonal family in {\mathbb R}^{\mathbb N} is of size 2^{\aleph_0}.
keywords: S: str, (set)
Sh:807
Bartoszyński, T., & Shelah, S. (2003). Strongly meager sets of size continuum. Arch. Math. Logic, 42(8), 769–779. arXiv: math/0211023 DOI: 10.1007/s00153-003-0184-0 MR: 2020043
type: article
abstract: We will construct several models where there are no strongly meager sets of size 2^{\aleph_0}.
keywords: S: str, (set)
Sh:808
Goldstern, M., & Shelah, S. (2005). Clones from creatures. Trans. Amer. Math. Soc., 357(9), 3525–3551. arXiv: math/0212379 DOI: 10.1090/S0002-9947-04-03593-7 MR: 2146637
type: article
abstract: We show that (consistently) there is a clone C on a countable set such that the interval of clones above C is linearly ordered and has no coatoms.
keywords: S: for, (ua)
Sh:809
Shelah, S., & Steprāns, J. (2005). Comparing the uniformity invariants of null sets for different measures. Adv. Math., 192(2), 403–426. arXiv: math/0405092 DOI: 10.1016/j.aim.2004.04.010 MR: 2128705
type: article
abstract: It is shown to be consistent with set theory that the uniformity invariant for Lebesgue measure is strictly greater than the corresponding invariant for Hausdorff r-dimensional measure where 0<r<1.
keywords: S: for, S: str, (set)
Sh:811
Geschke, S., & Shelah, S. (2003). Some notes concerning the homogeneity of Boolean algebras and Boolean spaces. Topology Appl., 133(3), 241–253. arXiv: math/0211399 DOI: 10.1016/S0166-8641(03)00103-2 MR: 2000501
type: article
abstract: We consider homogeneity properties of Boolean algebras that have nonprincipal ultrafilters which are countably generated. It is shown that a Boolean algebra B is homogeneous if it is the union of countably generated nonprincipal ultrafilters and has a dense subset D such that for every a\in D the relative algebra B\restriction a:=\{b\in B:b\leq a\} is isomorphic to B. In particular, the free product of countably many copies of an atomic Boolean algebra is homogeneous. Moreover, a Boolean algebra B is homogeneous if it satisfies the following conditions: (i) B has a countably generated ultrafilter, (ii) B is not c.c.c., and (iii) for every a\in B\setminus\{0\} there are finitely many automorphisms h_1,\dots,h_n of B such that 1=h_1(a)\cup\dots\cup h_n(a).
keywords: S: ico, (set)
Sh:812
Shelah, S., Väisänen, P., & Väänänen, J. A. (2005). On ordinals accessible by infinitary languages. Fund. Math., 186(3), 193–214. DOI: 10.4064/fm186-3-1 MR: 2191236
type: article
abstract: Let \lambda be an infinite cardinal number. The ordinal number \delta(\lambda) is the least ordinal \gamma such that if \phi is any sentence of L_{\lambda^+\omega}, with a unary predicate D and a binary predicate \prec, and \phi has a model M with \langle D^M,\prec^M\rangle a well-ordering of type \ge\gamma, then \phi has a model M' where \langle D^{M'}, \prec^{M'}\rangle is non-well-ordered. One of the interesting properties of this number is that the Hanf number of L_{\lambda^+\omega} is exactly \beth_{\delta(\lambda)}. We show the following theorem.

Theorem Suppose \aleph_0<\lambda<\theta\leq\kappa are cardinal numbers such that \lambda^{<\lambda}=\lambda, {\rm cf}(\theta)\geq \lambda^+ and \mu^\lambda<\theta whenever \mu<\theta, and \kappa^\lambda = \kappa. Then there is a forcing extension preserving all cofinalities, adding no new sets of cardinality < \lambda such that in the extension 2^\lambda = \kappa and \delta(\lambda)= \theta.
keywords: M: smt, (mod)
Sh:813
Matet, P., Péan, C., & Shelah, S. (2005). Cofinality of normal ideals on P_\kappa(\lambda). II. Israel J. Math., 150, 253–283. DOI: 10.1007/BF02762383 MR: 2255811
type: article
abstract: We investigate sufficient conditions for the existence of members of {\mathcal E}, a normal filter on [\lambda]^{<\kappa}, which contains no unbounded, nonstationary subsets continuing [Sh:698].
keywords: S: ico, (set), (normal)
Sh:814
Eklof, P. C., Shelah, S., & Trlifaj, J. (2004). On the cogeneration of cotorsion pairs. J. Algebra, 277(2), 572–578. arXiv: math/0405117 DOI: 10.1016/j.jalgebra.2003.09.018 MR: 2067620
type: article
abstract: Let R be a Dedekind domain. Enochs’ solution of the Flat Cover Conjecture was extended as follows: (*) If \mathfrak C is a cotorsion pair generated by a class of cotorsion modules, then \mathfrak C is cogenerated by a set. We show that (*) is the best result provable in ZFC in case R has a countable spectrum: the Uniformization Principle UP^{+} implies that \mathfrak C is not cogenerated by a set whenever \mathfrak C is a cotorsion pair generated by a set which contains a non-cotorsion module.
keywords: O: alg, (ab)
Sh:815
Baizhanov, B. S., Baldwin, J. T., & Shelah, S. (2005). Subsets of superstable structures are weakly benign. J. Symbolic Logic, 70(1), 142–150. arXiv: math/0303324 DOI: 10.2178/jsl/1107298514 MR: 2119127
type: article
abstract: Baizhanov and Baldwin introduced the notion of benign and weakly benign sets to investigate the preservation of stability by naming arbitrary subsets of a stable structure. They connected the notion with works of Baldwin, Benedikt, Bouscaren, Casanovas, Poizat, and Ziegler. Stimulated by those results, we investigate here the existence of benign or weakly benign sets.
keywords: M: cla, (mod)
Sh:816
Shelah, S. (2009). What majority decisions are possible. Discrete Math., 309(8), 2349–2364. arXiv: math/0405119 DOI: 10.1016/j.disc.2008.05.010 MR: 2510361
type: article
abstract: Suppose we are given a family of choice functions on pairs from a given finite set (with at least three elements) closed under permutations of the given set. The set is considered the set of alternatives (say candidates for an office). The question is, what are the choice functions \mathbf c on pairs of this set of the following form: for some (finite) family of “voters", each having a preference, i.e., a choice from each pair from the given family, \mathbf c\{x,y\} is chosen by the preference of the majority of voters. We give full characterization.
keywords: O: fin
Sh:817
Shelah, S. (2004). Spectra of monadic second order sentences. Sci. Math. Jpn., 59(2), 351–355. arXiv: math/0405158 MR: 2062201
type: article
abstract: For a monadic sentence \psi in the finite vocabulary we show that the spectra, the set of cardinalities of models of \psi is almost periodic under reasonable conditions. The first is that every model is so called “weakly k-decomposable". The second is that we restrict ourselves to a nice class of models constructed by some recursion.
keywords: M: smt, O: fin, (fmt)
Sh:818
Kramer, L., Shelah, S., Tent, K., & Thomas, S. (2005). Asymptotic cones of finitely presented groups. Adv. Math., 193(1), 142–173. arXiv: math/0306420 DOI: 10.1016/j.aim.2004.04.012 MR: 2132762
type: article
abstract: Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that {\mathbb R}\text{-rank}(S) \geq 2 and let \Gamma be a uniform lattice in G.

(a) If CH holds, then \Gamma has a unique asymptotic cone up to homeomorphism.

(b) If CH fails, then \Gamma has 2^{2^{\omega}} asymptotic cones up to homeomorphism.
keywords: S: str, (grp)
Sh:819
Eisworth, T., & Shelah, S. (2009). Successors of singular cardinals and coloring theorems. II. J. Symbolic Logic, 74(4), 1287–1309. arXiv: 0806.0031 DOI: 10.2178/jsl/1254748692 MR: 2583821
type: article
abstract: In this paper, we investigate the extent to which techniques used in [10], [2], and [3] – developed to prove coloring theorems at successors of singular cardinals of uncountable cofinality – can be extended to cover the countable cofinality case.
keywords: S: ico, (set)
Sh:820
Shelah, S. (2017). Universal structures. Notre Dame J. Form. Log., 58(2), 159–177. arXiv: math/0405159 DOI: 10.1215/00294527-3800985 MR: 3634974
type: article
abstract: We deal with the existence of universal members in a given cardinality for several classes. First we deal with classes of Abelian groups, specifically with the existence of universal members in cardinalities which are strong limit singular of countable cofinality. Second, we then deal with (variants of) the oak property (from a work of Dzamonja and the author), a property of complete first order theories, sufficient for the non-existence of universal models under suitable cardinal assumptions. Third, we prove that the oak property holds for the class of groups (naturally interpreted, so for quantifier free formulas) and deal more with the existence of universals.
keywords: M: cla, (mod), (grp)
Sh:821
Hyttinen, T., Lessmann, O., & Shelah, S. (2005). Interpreting groups and fields in some nonelementary classes. J. Math. Log., 5(1), 1–47. arXiv: math/0406481 DOI: 10.1142/S0219061305000390 MR: 2151582
type: article
abstract: This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem:

Theorem: Let \mathfrak{C} be a large homogeneous model of a stable diagram D. Let p, q \in S_D(A), where p is quasiminimal and q unbounded. Let P = p(\mathfrak{C}) and Q = q(\mathfrak{C}). Suppose that there exists an integer n < \omega such that dim(a_1\dots a_{n}/A \cup C)=n, for any independent a_1,\dots, a_{n} \in P and finite subset C \subseteq Q, but dim(a_1\dots a_n a_{n+1}/A\cup C)\leq n, for some independent a_1,\dots,a_n,a_{n+1}\in P and some finite subset C\subseteq Q. Then \mathfrak{C} interprets a group G which acts on the geometry P' obtained from P. Furthermore, either \mathfrak{C} interprets a non-classical group, or n = 1,2,3 and

If n = 1 then G is abelian and acts regularly on P'.

If n = 2 the action of G on P' is isomorphic to the affine action of K \rtimes K^* on the algebraically closed field K.

If n = 3 the action of G on P' is isomorphic to the action of PGL_2(K) on the projective line \mathbb{P}^1(K) of the algebraically closed field K.
keywords: M: cla, (mod), (grp)
Sh:822
Börner, F., Goldstern, M., & Shelah, S. (2023). Automorphisms and strongly invariant relations. Algebra Universalis, 84(4), Paper No. 27, 23. arXiv: math/0309165 DOI: 10.1007/s00012-023-00818-4 MR: 4629461
type: article
abstract: We investigate characterizations of the Galois connection {\rm sInv}–{\rm Aut} between sets of finitary relations on a base set A and their automorphisms. In particular, for A=\omega_1, we construct a countable set R of relations that is closed under all invariant operations on relations and under arbitrary intersections, but is not closed under {\rm sInv}{\rm Aut}.

Our structure (A,R) has an \omega-categorical first order theory. A higher order definable well-order makes it rigid, but any reduct to a finite language is homogeneous.
keywords: M: odm, (mod), (auto), (ua)
Sh:823
Bergman, G. M., & Shelah, S. (2006). Closed subgroups of the infinite symmetric group. Algebra Universalis, 55(2-3), 137–173. arXiv: math/0401305 DOI: 10.1007/s00012-006-1959-z MR: 2280223
type: article
abstract: Let S={\rm Sym}(\omega) be the group of all permutations of the natural numbers, and for subgroups G_1,G_2\leq S let us write G_1\approx G_2 if there exists a finite set U\subseteq S such that \langle G_1\cup U\rangle=\langle G_2\cup U\rangle. It is shown that the subgroups closed in the function topology on S lie in precisely four equivalence classes under this relation. Given an arbitrary subgroup G\leq S, which of these classes the closure of G belongs to depends on which of the following statements about pointwise stabilizer subgroups G_{(\Gamma)} of finite subsets \Gamma\subseteq\omega holds:

(i) For every finite set \Gamma, the subgroup G_{(\Gamma)} has at least one infinite orbit in \omega.

(ii) There exist finite sets \Gamma such that all orbits of G_{(\Gamma)} are finite, but none such that the cardinalities of these orbits have a common finite bound.

(iii) There exist finite sets \Gamma such that the cardinalities of the orbits of G_{(\Gamma)} have a common finite bound, but none such that G_{(\Gamma)}=\{1\}.

(iv) There exist finite sets \Gamma such that G_{(\Gamma)}=\{1\}.
keywords: S: str, O: alg, (grp)
Sh:824
Shelah, S. (2005). Two cardinals models with gap one revisited. MLQ Math. Log. Q., 51(5), 437–447. arXiv: math/0404149 DOI: 10.1002/malq.200410036 MR: 2163755
type: article
abstract: We succeed to say something on the identities of (\mu^+,\mu) when \mu>\theta>{\rm cf}(\mu), \mu strong limit \theta–compact. This hopefully will help to prove the consistency of “some pair (\mu^+,\mu) is not compact", however, this has not been proved.
keywords: M: odm, S: ico, (set), (mod)
Sh:825
Kanovei, V., & Shelah, S. (2004). A definable nonstandard model of the reals. J. Symbolic Logic, 69(1), 159–164. arXiv: math/0311165 DOI: 10.2178/jsl/1080938834 MR: 2039354
type: article
abstract: We prove, in ZFC, the existence of a definable, countably saturated elementary extension of the reals.
keywords: M: odm, (mod), (cont), (up)
Sh:826
Bartoszyński, T., & Shelah, S. (2008). On the density of Hausdorff ultrafilters. In Logic Colloquium 2004, Vol. 29, Assoc. Symbol. Logic, Chicago, IL, pp. 18–32. arXiv: math/0311064 MR: 2401857
type: article
abstract: An ultrafilter U is Hausdorff if for any two functions f,g \in \omega^\omega, f(U)=g(U) iff f \restriction X=g\restriction X for some X \in U. We will show that the statement that Hausdorff ultrafilters are dense in the Rudin-Keisler order is independent of ZFC
keywords: S: for, S: str, (set), (iter), (ultraf)
Sh:827
Kojman, M., & Shelah, S. (2006). Almost isometric embedding between metric spaces. Israel J. Math., 155, 309–334. arXiv: math/0406530 DOI: 10.1007/BF02773958 MR: 2269433
type: article
abstract: We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations:

(1) There is a countable universal metric space.

(2) There may exist fewer than continuum separable metric spaces on \aleph_1 so that every separable metric space is almost isometrically embedded into one of them when the continuum hypothesis fails.

(3) There is no collection of fewer than continuum metric spaces of cardinality \aleph_2 so that every ultra-metric space of cardinality \aleph_2 is almost isometrically embedded into one of them if \aleph_2<2^{\aleph_0}.

We also prove that various spaces X satisfy that if a space X is almost isometric to X than Y is isometric to X.
keywords: S: str, O: top, (nni), (univ)
Sh:828
Kellner, J., & Shelah, S. (2005). Preserving preservation. J. Symbolic Logic, 70(3), 914–945. arXiv: math/0405081 DOI: 10.2178/jsl/1122038920 MR: 2155272
type: article
abstract: We prove that the property “P doesn’t make the old reals Lebesgue null” is preserved under countable support iterations of proper forcings, under the additional assumption that the forcings are nep (a generalization of Suslin proper) in an absolute way. We also give some results for general Suslin ccc ideals.
keywords: S: for, S: str, (set), (pure(for))
Sh:829
Shelah, S. (2006). More on the revised GCH and the black box. Ann. Pure Appl. Logic, 140(1-3), 133–160. arXiv: math/0406482 DOI: 10.1016/j.apal.2005.09.013 MR: 2224056
type: article
abstract: We strengthen the revised GCH theorem by showing, e.g., that for \lambda={\rm cf}(\lambda)>\beth_\omega, for all but finitely many regular \kappa<\beth_\omega, \lambda is accessible on cofinality \kappa in a weak version of it holds. In particular, \lambda=2^\mu=\mu^+>\beth_\omega implies the diamond on \lambda is restricted to cofinality \kappa for all but finitely many \kappa\in{\rm Reg}\cap \beth_\omega and we strengthen the results on the middle diamond. Moreover, we get stronger results on the middle diamond.
keywords: S: ico, S: pcf, (set)
Sh:830
Shelah, S. (2006). The combinatorics of reasonable ultrafilters. Fund. Math., 192(1), 1–23. arXiv: math/0407498 DOI: 10.4064/fm192-1-1 MR: 2283626
type: article
abstract: We are interested in generalizing part of the theory of ultrafilters on \omega to larger cardinals. Here we set the scene for further investigations introducing properties of ultrafilters in strong sense dual to being normal.
keywords: S: ico, (set), (ultraf)
Sh:831
Göbel, R., & Shelah, S. (2005). How rigid are reduced products? J. Pure Appl. Algebra, 202(1-3), 230–258. DOI: 10.1016/j.jpaa.2005.02.002 MR: 2163410
type: article
abstract: For any cardinal \mu let {\mathbb Z}^\mu be the additive group of all integer-valued functions f:\mu\to {\mathbb Z}. The support of f is [f]=\{i\in\mu: f(i)=f_i\ne 0\}. Also let {\mathbb Z}_\mu= {\mathbb Z}^\mu/{\mathbb Z}^{<\mu} with {\mathbb Z}^{<\mu}= \{f\in {\mathbb Z}^\mu: \left|[f]\right|<\mu\}. If \mu\le \chi are regular cardinals we analyze the question when Hom({\mathbb Z}_\mu,{\mathbb Z}_\chi) = 0 and obtain a complete answer under GCH and independence results in Section 8. These results and some extensions are applied to a problem on groups: Let the norm \|G\| of a group G be the smallest cardinal \mu with Hom({\mathbb Z}_\mu,G) \ne 0 - this is an infinite, regular cardinal (or \infty). As a consequence we characterize those cardinals which appear as norms of groups. This allows us to analyze another problem on radicals: The norm \|R\| of a radical R is the smallest cardinal \mu for which there is a family \{ G_i: i\in \mu\} of groups such that R does not commute with the product \prod_{i\in\mu}G_i. Again these norms are infinite, regular cardinals and we show which cardinals appear as norms of radicals. The results extend earlier work (Arch. Math. 71 (1998) 341–348; Pacific J. Math. 118(1985) 79–104; Colloq. Math. Soc. János Bolyai 61 (1992) 77–107) and a seminal result by Łoś on slender groups (His elegant proof appears here in new light; Proposition 4.5.), see Fuchs [Vol. 2] (Infinite Abelian Groups, vols. I and II, Academic Pess, New York, 1970 and 1973). An interesting connection to earlier (unpublished) work on model theory by (unpublished, circulated notes, 1973) is elaborated in Section 3.
keywords: O: alg, (ab), (stal)
Sh:833
Göbel, R., & Shelah, S. (2005). On Crawley modules. Comm. Algebra, 33(11), 4211–4218. arXiv: math/0504198 DOI: 10.1080/00927870500261520 MR: 2183994
type: article
abstract: This continues recent work in a paper by Corner, Göbel and Goldsmith. A particular question was left open: Is it possible to carry over the results concerning the undecidability of torsion–free Crawley groups to modules over the ring of p-adic integers? We will confirm this and also strengthen one of the older results in by replacing the hypothesis of \diamondsuit by CH.
keywords: O: alg, (set), (ab)
Sh:834
Göbel, R., & Shelah, S. (2006). Torsionless linearly compact modules. In Abelian groups, rings, modules, and homological algebra, Vol. 249, Chapman & Hall/CRC, Boca Raton, FL, pp. 153–158. DOI: 10.1201/9781420010763.ch14 MR: 2229109
type: article
abstract: (none)
keywords: O: alg, (ab), (sta)
Sh:836
Shelah, S. (2006). On long EF-equivalence in non-isomorphic models. In Logic Colloquium ’03, Vol. 24, Assoc. Symbol. Logic, La Jolla, CA, pp. 315–325. arXiv: math/0404222 MR: 2207360
type: article
abstract: There has been much interest on constructing models which are not isomorphic of cardinality \lambda but are equivalent under the Ehrenfeucht–Fraissé game of length \alpha even for every \alpha< \lambda. So under G.C.H. we know much. We deal here with constructions of such pairs of models proven in ZFC and get the existence under mild conditions.
keywords: M: non, M: odm, (mod), (inf-log)
Sh:837
Shelah, S., & Usvyatsov, A. (2011). Model theoretic stability and categoricity for complete metric spaces. Israel J. Math., 182, 157–198. arXiv: math/0612350 DOI: 10.1007/s11856-011-0028-2 MR: 2783970
type: article
abstract: We deal with the systematic development of stability for the context of approximate elementary submodels of a monster metric space, which is not far, but still very distinct from the first order case. In particular we prove the analogue of Morley’s theorem for the classes of complete metric spaces
keywords: M: nec, (mod), (cat)
Sh:840
Shelah, S. (2009). Model theory without choice? Categoricity. J. Symbolic Logic, 74(2), 361–401. arXiv: math/0504196 DOI: 10.2178/jsl/1243948319 MR: 2518563
type: article
abstract: The main result is Łos conjecture: characterizing in ZF of countable first order T categoricity in some uncountable \aleph_\alpha (or every one). If there are \aleph_1 real this is Morley’s theorem, the ZFC one. Otherwise, we get a different theorem. The characterization (and the proof) are different.
keywords: M: cla, M: odm, (mod), (AC)
Sh:841
Shelah, S., & Sági, G. (2005). On topological properties of ultraproducts of finite sets. MLQ Math. Log. Q., 51(3), 254–257. arXiv: math/0404148 DOI: 10.1002/malq.200410024 MR: 2135487
type: article
abstract: Motivated by the model theory of higher order logics, a certain kind of topological spaces had been introduced on ultraproducts. These spaces are called ultratopologies. Ultratopologies provide a natural extra topological structure for ultraproducts and using this extra structure some preservation and characterization theorems had been obtained for higher order logics.

The purely topological properties of ultratopologies seem interesting on their own right. Here we present the solutions of two problems of Gerlits and Sági. More concretely we show that

(1) there are sequences of finite sets of pairwise different cardinality such that in their certain ultraproducts there are homeomorphic ultratopologies and

(2) one can always find a dense set in an ultratopology whose cardinality is strictly smaller than the cardinality of the ultraproduct, provided that the factors of the corresponding ultraproduct are finite.
keywords: M: odm, O: top, (mod), (up)
Sh:843
Mildenberger, H., & Shelah, S. (2007). Increasing the groupwise density number by c.c.c. forcing. Ann. Pure Appl. Logic, 149(1-3), 7–13. arXiv: math/0404147 DOI: 10.1016/j.apal.2007.07.001 MR: 2364193
type: article
abstract: We try to control many cardinal characteristics by working with a notion of orthogonality between two families of forcings. We show that {\mathfrak b}^+<{\mathfrak g} is consistent
keywords: S: for, S: str, (set), (iter), (inv)
Sh:844
Shelah, S., & Usvyatsov, A. (2008). More on SOP_1 and SOP_2. Ann. Pure Appl. Logic, 155(1), 16–31. arXiv: math/0404178 DOI: 10.1016/j.apal.2008.02.003 MR: 2454629
type: article
abstract: This paper continues [DjSh692]. We present a rank function for NSOP_{1} theories and give an example of a theory which is NSOP_{1} but not simple. We also investigate the connection between maximality in the ordering \lhd^* among complete first order theories and the (N)SOP{}_2 property. We complete the proof started in [DjSh692] of the fact that \lhd^*-maximality implies SOP{}_2 and get weaker results in the other direction. The paper provides a step toward the classification of unstable theories without the strict order property.
keywords: M: cla, (mod)
Sh:845
Rosłanowski, A., & Shelah, S. (2007). Universal forcing notions and ideals. Arch. Math. Logic, 46(3-4), 179–196. arXiv: math/0404146 DOI: 10.1007/s00153-007-0037-3 MR: 2306175
type: article
abstract: The main result of this paper is a partial answer to [RoSh:672, Problem 5.5]: a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the \sigma–ideals determined by those universality parameters.
keywords: S: for, S: str, (set), (inv)
Sh:846
Shelah, S. (2008). The spectrum of characters of ultrafilters on \omega. Colloq. Math., 111(2), 213–220. arXiv: math/0612240 DOI: 10.4064/cm111-2-5 MR: 2365799
type: article
abstract: We show the consistency of the statement: “the set of regular cardinals which are the characters of ultrafilters on \omega is not convex". We also deal with the set of \pi-characters of ultrafilters on \omega.
keywords: S: for, S: str, (set), (inv), (cont), (ultraf)
Sh:847
Mildenberger, H., Shelah, S., & Tsaban, B. (2006). Covering the Baire space by families which are not finitely dominating. Ann. Pure Appl. Logic, 140(1-3), 60–71. arXiv: math/0407487 DOI: 10.1016/j.apal.2005.09.008 MR: 2224049
type: article
abstract: We show that the mentioned constellation of three cardinal characteristices is relatively consistent to ZFC
keywords: S: for, S: str, (set), (inv), (cont), (ultraf)
Sh:848
Mildenberger, H., & Shelah, S. (2009). Specializing Aronszajn trees and preserving some weak diamonds. J. Appl. Anal., 15(1), 47–78. DOI: 10.1515/JAA.2009.47 MR: 2537976
type: article
abstract: We show that \diamondsuit({\mathbb R}, {\mathcal L}, \not\in) together with “all Aronszajn trees are special” is consistent relative to ZFC. The weak diamond for the uniformity of Lebegue null sets was the only weak diamond in the Cichoń diagramme for relations whose consistency together with “all Aronszajn trees are special” was not yet settled. We can have CH or 2^{\aleph_0} = \aleph_2. Our techniques give more on coverings by related small sets that are preserved in iterations that are stronger relatives to [Sh:f, Chap. V, sect. 5–7]
keywords: S: for, S: str, (set), (trees), (wd), (aron)
Sh:849
Shelah, S. (2016). Beginning of stability theory for Polish spaces. Israel J. Math., 214(2), 507–537. arXiv: 1011.3578 DOI: 10.1007/s11856-016-1342-5 MR: 3544691
type: article
abstract: We consider stability theory for Polish spaces, and more generally, for definable structures (say with elements of a set of reals). We clarify by proving some equivalent conditions for \aleph_0-stability. We succeed to prove existence of indiscernibles under reasonable conditions; this provides strong evidence that such a theory exists.
keywords: M: cla, S: str, S: dst, (mod), (sta)
Sh:850
Cherlin, G. L., & Shelah, S. (2007). Universal graphs with a forbidden subtree. J. Combin. Theory Ser. B, 97(3), 293–333. arXiv: math/0512218 DOI: 10.1016/j.jctb.2006.05.008 MR: 2305886
type: article
abstract: We show that the problem of the existence of universal graphs with specified forbidden subgraphs can be systematically reduced to certain critical cases by a simple pruning technique which simplifies the underlying structure of the forbidden graphs, viewed as trees of blocks. As an application, we characterize the trees T for which a universal countable T-free graph exists.
keywords: M: odm, (mod), (graph), (univ)
Sh:851
Laskowski, M. C., & Shelah, S. (2006). Decompositions of saturated models of stable theories. Fund. Math., 191(2), 95–124. DOI: 10.4064/fm191-2-1 MR: 2231058
type: article
abstract: We characterize the stable theories T for which the saturated models of T admit decompositions. Additionally, we show that when T is countable the criterion can be weakened.
keywords: M: cla, (mod), (sta)
Sh:852
Kennedy, J. C., & Shelah, S. (2004). More on regular reduced products. J. Symbolic Logic, 69(4), 1261–1266. arXiv: math/0504200 DOI: 10.2178/jsl/1102022222 MR: 2135667
type: article
abstract: The authors show, by means of a finitary version \square^{fin}_{\lambda,D} of the combinatorial principle \square^{b^*}_{\lambda}, the consistency of the failure, relative to the consistency of supercompact cardinals, of the following: for all regular filters D on a cardinal \lambda, if M_i and N_i are elementarily equivalent models of a language of size \le\lambda, then the second player has a winning strategy in the Ehrenfeucht-Fraı̈ssé game of length \lambda^+ on \prod_i M_i/D and \prod_i N_i/D. If in addition 2^{\lambda}=\lambda^+ and i<\lambda implies |M_i|+|N_i|\leq \lambda^+ this means that the ultrapowers are isomorphic.
keywords: M: odm, S: ico, (mod), (up)
Sh:853
Shelah, S. (2005). The depth of ultraproducts of Boolean algebras. Algebra Universalis, 54(1), 91–96. arXiv: math/0406531 DOI: 10.1007/s00012-005-1925-1 MR: 2217966
type: article
abstract: We show that if \mu is a compact cardinal then the depth of ultraproducts of less than \mu many Boolean algebras is at most \mu plus the ultraproduct of the depths of those Boolean algebras
keywords: S: ico, (ba), (large)
Sh:854
Blass, A. R., & Shelah, S. (2005). Ultrafilters and partial products of infinite cyclic groups. Comm. Algebra, 33(6), 1997–2007. arXiv: math/0504199 DOI: 10.1081/AGB-200063355 MR: 2150855
type: article
abstract: We consider, for infinite cardinals \kappa and \alpha\leq\kappa^+, the group \Pi(\kappa,<\alpha) of sequences of integers, of length \kappa, with non-zero entries in fewer than \alpha positions. Our main result tells when \Pi(\kappa,<\alpha) can be embedded in \Pi(\lambda,<\beta). The proof involves some set-theoretic results, one about familes of finite sets and one about families of ultrafilters.
keywords: S: str, O: alg, (set), (ab), (ultraf)
Sh:855
Shelah, S., & Strüngmann, L. H. (2010). Filtration-equivalent \aleph_1-separable abelian groups of cardinality \aleph_1. Ann. Pure Appl. Logic, 161(7), 935–943. arXiv: math/0612241 DOI: 10.1016/j.apal.2009.12.001 MR: 2601022
type: article
abstract: We show that it is consistent with ordinary set theory ZFC and the generalized continuum hypothesis that there exist two aleph_1 separable abelian groups of cardinality \aleph_1 which are filtration-equivalent and one is a Whitehead group but the other is not. This solves one of the open problems of Eklof and Mekler.
keywords: O: alg, (ab), (ba), (stal), (normal)
Sh:856
Rosłanowski, A., & Shelah, S. (2006). How much sweetness is there in the universe? MLQ Math. Log. Q., 52(1), 71–86. arXiv: math/0406612 DOI: 10.1002/malq.200410056 MR: 2195002
type: article
abstract: We continue investigations of forcing notions with strong ccc properties introducing new methods of building sweet forcing notions. We also show that quotients of topologically sweet forcing notions over Cohen reals are topologically sweet.
keywords: S: for, S: str, (set), (creatures)
Sh:857
Kuhlmann, S., & Shelah, S. (2005). \kappa-bounded exponential-logarithmic power series fields. Ann. Pure Appl. Logic, 136(3), 284–296. arXiv: math/0512220 DOI: 10.1016/j.apal.2005.04.001 MR: 2169687
type: article
abstract: In [KKSh:601] it was shown that fields of generalized power series cannot admit an exponential function. In this paper, we construct fields of generalized power series with bounded support which admit an exponential. We give a natural definition of an exponential, which makes these fields into models of real exponentiation. The method allows to construct for every \kappa regular uncountable cardinal, 2^{\kappa} pairwise non-isomorphic models of real exponentiation (of cardinality \kappa), but all isomorphic as ordered fields. Indeed, the 2^{\kappa} exponentials constructed have pairwise distinct growth rates. This method relies on constructing lexicographic chains with many automorphisms.
keywords: S: ico, O: alg
Sh:858
Mildenberger, H., Shelah, S., & Tsaban, B. (2007). The combinatorics of \tau-covers. Topology Appl., 154(1), 263–276. arXiv: math/0409068 DOI: 10.1016/j.topol.2006.04.011 MR: 2271787
type: article
abstract: We compute the critical cardinalities for some topological properties involving \tau-covers. A surprising structure counterpart of one of the computations is that, using Scheepers’ notation:

If X^2 satisfies S_{\rm fin}(\Gamma,T), then X satisfies Hurewicz’ property U_{\rm fin}(\Gamma,\Gamma).

Some results on the additivity of these properties and special elements are also provided.
keywords: S: for, S: str, (set), (inv)
Sh:859
Kellner, J., & Shelah, S. (2011). Saccharinity. J. Symbolic Logic, 76(4), 1153–1183. arXiv: math/0511330 DOI: 10.2178/jsl/1318338844 MR: 2895391
type: article
abstract: We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. We apply this method to construct a forcing (without using an inaccessible or amalgamation) that makes all definable sets of reals measurable with respect to a certain (non-ccc) ideal.
keywords: S: for, S: str, (set), (creatures)
Sh:860
Rosłanowski, A., & Shelah, S. (2006). Reasonably complete forcing notions. In Set theory: recent trends and applications, Vol. 17, Dept. Math., Seconda Univ. Napoli, Caserta, pp. 195–239. arXiv: math/0508272 MR: 2374767
type: article
abstract: We introduce more properties of forcing notions which imply that their \lambda–support iterations are \lambda–proper, where \lambda is an inaccessible cardinal. This paper is a direct continuation of Rosłanowski and Shelah [RoSh:777, §A.2]. As an application of our iteration result we show that it is consistent that dominating numbers associated with two normal filters on \lambda are distinct.
keywords: S: ico, S: for, (set), (ultraf)
Sh:861
Shelah, S. (2007). Power set modulo small, the singular of uncountable cofinality. J. Symbolic Logic, 72(1), 226–242. arXiv: math/0612243 DOI: 10.2178/jsl/1174668393 MR: 2298480
type: article
abstract: Let \mu be singular of uncountable cofinality. If \mu> 2^{{\rm cf}(\mu)}, we prove that in {\mathbb P}=([\mu]^\mu, \supseteq) as a forcing notion we have a natural complete embedding of {\rm Levy}(\aleph_0,\mu^+) (so {\mathbb P} collapses \mu^+ to \aleph_0) and even {\rm Levy}(\aleph_0, {\bf U}_{J^{{\rm bd}}_\kappa}(\mu)). The “natural" means that the forcing (\{p \in [\mu]^\mu:p closed\},\supseteq) is naturally embedded and is equivalent to the Levy algebra. If \mu<2^{{\rm cf}(\mu)} we have weaker results.
keywords: S: ico, (set), (app(pcf))
Sh:862
Baldwin, J. T., & Shelah, S. (2008). Examples of non-locality. J. Symbolic Logic, 73(3), 765–782. DOI: 10.2178/jsl/1230396746 MR: 2444267
type: article
abstract: We use \kappa-free but not Whitehead Abelian groups to construct abstract elementary classes which satisfy the amalgamation property but fail various conditions on the locality of types. We show in fact that in a large class of cases amalgamation can have no positive effect on locality by exhibiting a transformation of aec’s which preserves non-locality but takes any aec satisfying a new property called admitting closures to one with amalgamation.
keywords: M: nec, (mod), (ab), (grp), (aec)
Sh:863
Shelah, S. (2014). Strongly dependent theories. Israel J. Math., 204(1), 1–83. arXiv: math/0504197 DOI: 10.1007/s11856-014-1111-2 MR: 3273451
type: article
abstract: We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing [Sh:783]. If |A|+|T|\le\mu, I\subseteq \mathfrak{C}, |I|\ge \beth_{|T|^+}(\mu) then some J\subseteq I of cardinality \mu^+ is an indiscernible sequence over A.
keywords: M: cla, (mod)
Sh:864
Shelah, S., & Sági, G. (2006). On weak and strong interpolation in algebraic logics. J. Symbolic Logic, 71(1), 104–118. arXiv: math/0612244 DOI: 10.2178/jsl/1140641164 MR: 2210057
type: article
abstract: We show that there is a restriction, or modification of the finite-variable fragments of First Order Logic in which a weak form of Craig’s Interpolation Theorem holds, but a strong form of this theorem does not hold. Translating these results into Algebraic Logic we obtain a finitely axiomatizable subvariety of finite dimensional Representable Cylindric Algebras that has the Strong Amalgamation Property, but does not have the Superamalgamation Property. This settles a conjecture of Pigozzi
keywords: M: odm, (mod)
Sh:865
Doron, M., & Shelah, S. (2007). Relational structures constructible by quantifier free definable operations. J. Symbolic Logic, 72(4), 1283–1298. arXiv: math/0607375 DOI: 10.2178/jsl/1203350786 MR: 2371205
type: article
abstract: We consider three classes of models: models of bounded patch width defined Fisher and Makowsky, models constructible by addition operations that preserve monadic theories defined in [Sh:817], and models monadicaly interpretable in trees defined below. For all three classes we have eventual periodicity of restricted spectrum of MSO sentences. We show that the second and third classes are in some sence equivalent, while the first is essentially smaller.
keywords: M: odm, O: fin, (fmt)
Sh:866
Havlin, C., & Shelah, S. (2007). Existence of EF-equivalent non-isomorphic models. MLQ Math. Log. Q., 53(2), 111–127. arXiv: math/0612245 DOI: 10.1002/malq.200610031 MR: 2308491
type: article
abstract: We prove the existence of pairs of models of the same cardinality \lambda which are very equivalent according to EF games, but not isomorphic. We continue the paper [Sh:836], but we don’t rely on it.
keywords: M: non, M: odm, S: ico
Sh:867
Göbel, R., & Shelah, S. (2006). Generalized E-algebras via \lambda-calculus. I. Fund. Math., 192(2), 155–181. arXiv: 0711.3045 DOI: 10.4064/fm192-2-5 MR: 2283757
type: article
abstract: An R-algebra A is called E(R)–algebra if the canonical homomorphism from A to the endomorphism algebra End_R A of the R-module {}_R A, taking any a\in A to the right multiplication a_r\in End_R A by a is an isomorphism of algebras. In this case {}_R A is called an E(R)–module. E(R)-algebras come up naturally in various topics of algebra, so it’s not surprising that they were investigated thoroughly in the last decade. Despite some efforts it remained an open question whether proper generalized E(R)-algebras exist. These are R–algebras A isomorphic to End_R A but not under the above canonical isomorphism, so not E(R)–algebras. This question was raised about 30 years ago (for R={\mathbb Z}) by Phil Schultz and we will answer it. For PIDs R of characteristic 0 that are neither quotient fields nor complete discrete valuation rings - we will establish the existence of generalized E(R)-algebras. It can be shown that E(R)-algebras over rings R that are complete discrete valuation rings or fields must trivial (copies of R). The main tool is an interesting connection between \lambda-calculus (used in theoretical computer sciences) and algebra. It seems reasonable to divide the work into two parts, in this paper we will work in V=L (Godels universe) hence stronger combinatorial methods make the final arguments more transparent. The proof based entirely on ordinary set theory (the axioms of ZFC) will appear in a subsequent paper.
keywords: O: alg, (stal)
Sh:868
Shelah, S. (2012). When a first order T has limit models. Colloq. Math., 126(2), 187–204. arXiv: math/0603651 DOI: 10.4064/cm126-2-4 MR: 2924249
type: article
abstract: We to a large extent sort out when does a (first order comp- lete theory) T have a superlimit model in a cardinal \lambda. Also we deal with relation notions of being limit.
keywords: M: cla, M: non, (mod), (sta), (cat)
Sh:869
Matet, P., & Shelah, S. (2017). The nonstationary ideal on P_\kappa (\lambda) for \lambda singular. Arch. Math. Logic, 56(7-8), 911–934. arXiv: math/0612246 DOI: 10.1007/s00153-017-0552-9 MR: 3696072
type: article
abstract: Let \kappa be a regular uncountable cardinal and \lambda>\kappa a singular strong limit cardinal. We give a new characterization of the nonstationary subsets of P_\kappa(\lambda) and use this to prove that the nonstationary ideal on P_\kappa(\lambda) is nowhere precipitous.
keywords: S: ico, (set), (normal)
Sh:870
Blass, A. R., & Shelah, S. (2006). Disjoint non-free subgroups of abelian groups. In Set theory: recent trends and applications, Vol. 17, Dept. Math., Seconda Univ. Napoli, Caserta, pp. 1–24. arXiv: math/0509406 MR: 2374760
type: article
abstract: Let G be an abelian group and let \lambda be the smallest rank of any group whose direct sum with a free group is isomorphic to G. If \lambda is uncountable, then G has \lambda pairwise disjoint, non-free subgroups. There is an example where \lambda is countably infinite and G does not have even two disjoint, non-free subgroups.
keywords: S: ico, O: alg, (ab)
Sh:871
Laskowski, M. C., & Shelah, S. (2011). A trichotomy of countable, stable, unsuperstable theories. Trans. Amer. Math. Soc., 363(3), 1619–1629. arXiv: 0711.3043 DOI: 10.1090/S0002-9947-2010-05196-7 MR: 2737280
type: article
abstract: A trichotomy theorem for countable, stable, unsuperstable theories is offered. We develop the notion of a ‘regular ideal’ of formulas and study types that are minimal with respect to such an ideal.
keywords: M: cla, (mod), (sta)
Sh:872
Kellner, J., & Shelah, S. (2009). Decisive creatures and large continuum. J. Symbolic Logic, 74(1), 73–104. arXiv: math/0601083 DOI: 10.2178/jsl/1231082303 MR: 2499421
type: article
abstract: For f>g\in\omega^\omega let c^{\forall}_{f,g} be the minimal number of uniform trees with g-splitting needed to \forall^\infty-cover the uniform tree with f-splitting. c^{\exists}_{f,g} is the dual notion for the \exists^\infty-cover.

Assuming CH and given \aleph_1 many (sufficiently different) pairs (f_\epsilon,g_\epsilon) and cardinals \kappa_\epsilon such that \kappa_\epsilon^{\aleph_0}=\kappa_\epsilon, we construct a partial order forcing that c^{\exists}_{f_\epsilon,g_\epsilon}= c^{\forall}_{f_\epsilon,g_\epsilon}=\kappa_\epsilon.

For this, we introduce a countable support semiproduct of decisive creatures with bigness and halving. This semiproduct satisfies fusion, pure decision and continuous reading of names.
keywords: S: for, S: str, (set)
Sh:873
Shelah, S., & Strüngmann, L. H. (2007). A characterization of \mathrm{Ext}(G,\mathbb Z) assuming (V=L). Fund. Math., 193(2), 141–151. arXiv: math/0609638 DOI: 10.4064/fm193-2-3 MR: 2282712
type: article
abstract: In this paper we complete the characterization of {\rm Ext}(G,{\mathbb Z}) under Gödel’s axiom of constructibility for any torsion-free abelian group G. In particular, we prove in (V=L) that, for a singular cardinal \nu of uncountable cofinality which is less than the first weakly compact cardinal and for every sequence of cardinals (\nu_p : p \in \Pi ) satisfying \nu_p \leq 2^{\nu}, there is a torsion-free abelian group G of size \nu such that \nu_p equals the p-rank of {\rm Ext}(G, {\mathbb Z}) for every prime p and 2^{\nu} is the torsion-free rank of {\rm Ext}(G,{\mathbb Z}).
keywords: O: alg, (ab), (stal), (grp)
Sh:874
Shelah, S., & Strüngmann, L. H. (2007). On the p-rank of \mathrm{Ext}_{\mathbb Z}(G,\mathbb Z) in certain models of ZFC. Algebra Logika, 46(3), 369–397, 403–404. arXiv: math/0609637 DOI: 10.1007/s10469-007-0019-x MR: 2356727
type: article
abstract: We show that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the p-rank of {\rm Ext}_{\mathbb Z}(G,\mathbb Z) is as large as possible for every prime p and any torsion-free abelian group G. Moreover, given an uncountable strong limit cardinal \mu of countable cofinality and a partition of \Pi (the set of primes) into two disjoint subsets \Pi_0 and \Pi_1, we show that in some model which is very close to ZFC there is an almost-free abelian group G of size 2^{\mu}=\mu^+ such that the p-rank of {\rm Ext}_{\mathbb Z}(G,{\mathbb Z}) equals 2^{\mu}=\mu^+ for every p\in\Pi_0 and 0 otherwise, i.e. for p\in\Pi_1.
keywords: O: alg, (ab), (stal), (grp)
Sh:875
Jarden, A., & Shelah, S. (2013). Non-forking frames in abstract elementary classes. Ann. Pure Appl. Logic, 164(3), 135–191. arXiv: 0901.0852 DOI: 10.1016/j.apal.2012.09.007 MR: 3001542
type: article
abstract: The stability theory of first order theories was initiated by Saharon Shelah in 1969. The classification of abstract elementary classes was initiated by Shelah, too. In several papers, he introduced non-forming relations. Later, Shelah (2009) [17, 11] introduced the good non-forking frame, an axiomatization of the non-forking notion. We improve results of Shelah on good non-forming grames, mainly by weakening the stability hypothesis in several important theorems, replacing it by the almost \lambda-stability hypothesis: The number of types over a model of cardinality \lambda is at most \lambda^+. We present conditions on K_\lambda, that imply the existence of a model in K_{\lambda^{+n}} for all n. We do this by providing sufficiently strong conditions on K_\lambda, that they are inherited by a properly chosen subclass of K_{\lambda^+}. What are these conditions? We assume that there is a ‘non-forking’ relation which satisfies the properties of the non-forking relation on superstable first order theories. Note that here we deal with models of fixed cardinality \lambda. While in Shelah (2009) [17,II] we assume stability in \lambda, so we can use brimmed (=limit) models, here we assume almost stability only, but we add an assumption: The conjugation property. In the context of elementary classes, the superstability assumption gives the existence of types with well-defined dimension and the \omega-stability assumption gives the existence and uniqueness of models prime over sets. In our context, the local character assumption is an analog to superstability and the density of the class of uniqueness triples with respect to the relation \preccurlyeq is the analog to omega-stability.
keywords: M: cla, M: nec, (mod), (aec)
Sh:876
Shelah, S. (2008). Minimal bounded index subgroup for dependent theories. Proc. Amer. Math. Soc., 136(3), 1087–1091. arXiv: math/0603652 DOI: 10.1090/S0002-9939-07-08654-6 MR: 2361885
type: article
abstract: For a dependent theory T, in {\mathfrak C}_T for every type definable group G, the intersection of type definable subgroups with bounded index is a type definable subgroup with bounded index.
keywords: M: cla, O: alg, (mod), (grp)
Sh:877
Shelah, S. (2014). Dependent T and existence of limit models. Tbilisi Math. J., 7(1), 99–128. arXiv: math/0609636 DOI: 10.2478/tmj-2014-0010 MR: 3313049
type: article
abstract: We continue [Sh:868] and [Sh:783]. The problem there is when does (first order) T have a model M of cardinality \lambda which is (one of the variants of) a limit model for cofinality \kappa, and the most natural case to try is \lambda=\lambda^{< \lambda}>\kappa={\rm cf}(\kappa)>|T|. The stable theories has one; are there unstable T whnce of limit models AUTHORS: Saharon Shelah ich has such limit models? We find one: the theory T_{\rm ord} of dense linear orders. So does this hold for all unstable T? As T_{\rm ord} is prototypical of dependent theories, it is natural to look for independent theories. A strong, explicit version of T being independent is having the strong independence property. We prove that for such T there are no limit models. We work harder to prove this for every dependent T, i.e., with the independence property though a weaker version. This makes us conjecture that any dependent T has such models. Toward this end we continue the investigation of types for dependent T.
keywords: M: cla, M: non, S: ico, (mod), (linear order)
Sh:878
Garti, S., & Shelah, S. (2008). On Depth and Depth^+ of Boolean algebras. Algebra Universalis, 58(2), 243–248. arXiv: math/0512217 DOI: 10.1007/s00012-008-2065-1 MR: 2386531
type: article
abstract: We show that the {\rm Depth}^+ of an ultraproduct of Boolean Algebras, can not jump over the {\rm Depth}^+ of every component by more than one cardinality. We can have, consequently, similar results for the Depth invariant
keywords: S: ico, O: alg, (set), (ba), (inv(ba))
Sh:879
Eklof, P. C., Fuchs, L., & Shelah, S. (2012). Test groups for Whitehead groups. Rocky Mountain J. Math., 42(6), 1863–1873. arXiv: math/0702293 DOI: 10.1216/RMJ-2012-42-6-1863 MR: 3028765
type: article
abstract: We consider the question of when the dual of a Whitehead group is a test group for Whitehead groups. This turns out to be equivalent to the question of when the tensor product of two Whitehead groups is Whitehead. We investigate what happens in different models of set theory.
keywords: S: for, O: alg, (ab), (stal), (wh)
Sh:880
Göbel, R., & Shelah, S. (2007). Absolutely indecomposable modules. Proc. Amer. Math. Soc., 135(6), 1641–1649. arXiv: 0711.3011 DOI: 10.1090/S0002-9939-07-08725-4 MR: 2286071
type: article
abstract: A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more general result about R-modules over a large class of commutative rings R with endomorphism ring R which remains the same when passing to a generic extension of the universe. It turns out that ‘large’ in this context has the precise meaning, namely being smaller then the first \omega-Erdos cardinal defined below. We will first apply result on large rigid trees with a similar property established by Shelah in 1982, and will prove the existence of related ‘R_\omega-modules’ (R-modules with countably many distinguished submodules) and finally pass to R-modules. The passage through R_\omega-modules has the great advantage that the proofs become very transparent essentially using a few ‘linear algebra’ arguments accessible also for graduate students. The result gives a new construction of indecomposable modules in general using a counting argument.
keywords: O: alg, (ab), (stal)
Sh:881
Shelah, S. (2009). The Erdős-Rado arrow for singular cardinals. Canad. Math. Bull., 52(1), 127–131. arXiv: math/0605385 DOI: 10.4153/CMB-2009-015-8 MR: 2494318
type: article
abstract: We try to prove that if \mathrm{cf}(\lambda) > \aleph_0 and 2^{\mathrm{cf}(\lambda)} < \lambda then \lambda \to(\lambda, \omega +1)^2
keywords: S: ico, (set), (pc)
Sh:882
Kaplan, I., & Shelah, S. (2009). The automorphism tower of a centerless group without choice. Arch. Math. Logic, 48(8), 799–815. arXiv: math/0606216 DOI: 10.1007/s00153-009-0154-2 MR: 2563819
type: article
abstract: For a centerless group G, we can define its automorphism tower. We define G^{\alpha}: G^0=G, G^{\alpha+1}= Aut(G^\alpha) and for limit ordinals G^\delta= \bigcup_{\alpha<\delta}G^\alpha. Let \tau_G be the ordinal when the sequence stabilizes. Thomas’ celebrated theorem says \tau_G< 2^{|G|})^{+} and more. If we consider Thomas’ proof too set theoretical, we have here a shorter proof with little set theory. However, set theoretically we get a parallel theorem without the axiom of choice. We attach to every element in G^\alpha, the \alpha-th member of the automorphism tower of G, a unique quantifier free type over G (whish is a set of words from G* \langle x\rangle). This situation is generalized by defining “(G,A) is a special pair”.
keywords: S: ico, S: dst, O: alg, (set), (grp), (AC)
Sh:883
Shelah, S. (2007). \aleph_n-free abelian group with no non-zero homomorphism to \mathbb Z. Cubo, 9(2), 59–79. arXiv: math/0609634 MR: 2354353
type: article
abstract: For any natural n, we construct an \aleph_n-free abelian groups which have few homomorphisms to \mathbb Z. For this we use “\aleph_n-free (n+1)-dimensional black boxes". The method is relevant to e.g. construction of \aleph_n-free abelian groups with a prescribed endomorphism ring.
keywords: S: ico, O: alg, (set), (grp), (sta)
Sh:884
Goldstern, M., & Shelah, S. (2016). All creatures great and small. Trans. Amer. Math. Soc., 368(11), 7551–7577. arXiv: 0706.1190 DOI: 10.1090/tran/6568 MR: 3546775
type: article
abstract: Let \lambda be an uncountable regular cardinal. Assuming 2^\lambda=\lambda^+, we show that the clone lattice on a set of size \lambda is not dually atomic
keywords: S: ico, O: alg, (sta), (creatures), (ua)
Sh:885
Shelah, S. (2009). A comment on “\mathfrak p<\mathfrak t”. Canad. Math. Bull., 52(2), 303–314. arXiv: math/0404220 DOI: 10.4153/CMB-2009-033-4 MR: 2518968
type: article
abstract: Dealing with the cardinal invariants {\mathfrak p} and {\mathfrak t} of the continuum we prove that {\mathfrak m}\geq {\mathfrak p} = \aleph_2 \Rightarrow {\mathfrak t} = \aleph_1. In other words if {\bf MA}_{\aleph_1} (or a weak version of this) then (of course \aleph_2 \leq {\mathfrak p}\leq {\mathfrak t} and) {\mathfrak p} = \aleph_2 \Rightarrow {\mathfrak p} = {\mathfrak t}. This is based on giving a consequence.
keywords: S: ico, S: str, (set)
Sh:886
Shelah, S. (2017). Definable groups for dependent and 2-dependent theories. Sarajevo J. Math., 13(25)(1), 3–25. arXiv: math/0703045 DOI: 10.5644/SJM.13.1.01 MR: 3666349
type: article
abstract: Let T be a (first order complete) dependent theory, {\mathfrak{C}} a \bar{\kappa}-saturated model of T and G a definable subgroup which is abelian. Among subgroups of bounded index which are the union of <\kappa type definable subsets there is a minimal one, i.e. their intersection has bounded index. In fact, the bound is \leq 2^{|T|}. We then deal with 2-dependent theories, a wider class of first order theories.
keywords: M: cla, (mod), (grp)
Sh:887
Shelah, S. (2008). Groupwise density cannot be much bigger than the unbounded number. MLQ Math. Log. Q., 54(4), 340–344. arXiv: math/0612353 DOI: 10.1002/malq.200710032 MR: 2435897
type: article
abstract: We prove that {\mathfrak g}, (the groupwise density) is smaller or equal to {\mathfrak b}^+, successor of the minimal cardinality of a non-dominated subset of {}^\omega \omega.
keywords: S: ico, S: str, (set)
Sh:888
Rosłanowski, A., & Shelah, S. (2011). Lords of the iteration. In Set theory and its applications, Vol. 533, Amer. Math. Soc., Providence, RI, pp. 287–330. arXiv: math/0611131 DOI: 10.1090/conm/533/10514 MR: 2777755
type: article
abstract: We introduce several properties of forcing notions which imply that their \lambda–support iterations are \lambda–proper. Our methods and techniques refine those studied in [RoSh:655], [RoSh:777], [RoSh:860] and [RoSh:890], covering some new forcing notions (though the exact relation of the new properties to the old ones remains undecided).
keywords: S: ico, S: for, (set), (pure(for)), (ultraf)
Sh:889
Rosłanowski, A., & Shelah, S. (2008). Generating ultrafilters in a reasonable way. MLQ Math. Log. Q., 54(2), 202–220. arXiv: math/0607218 DOI: 10.1002/malq.200610055 MR: 2402629
type: article
abstract: We continue investigations of reasonable ultrafilters on uncountable cardinals defined in Shelah [Sh:830]. We introduce a general scheme of generating a filter on \lambda from filters on smaller sets and we investigate the combinatorics of objects obtained this way.
keywords: S: ico, (set), (ultraf)
Sh:890
Rosłanowski, A., & Shelah, S. (2011). Reasonable ultrafilters, again. Notre Dame J. Form. Log., 52(2), 113–147. arXiv: math/0605067 DOI: 10.1215/00294527-1306154 MR: 2794647
type: article
abstract: We continue investigations of reasonable ultrafilters on uncountable cardinals defined in [Sh:830]. We introduce stronger properties of ultrafilters and we show that those properties may be handled in \lambda–support iterations of reasonably bounding forcing notions. We use this to show that consistently there are reasonable ultrafilters on an inaccessible cardinal \lambda with generating system of size less than 2^\lambda. We also show how reasonable ultrafilters can be killed by forcing notions which have enough reasonable completeness to be iterated with \lambda–supports (and we show the appropriate preservation theorem).
keywords: S: ico, S: for, (set), (ultraf)
Sh:891
Garti, S., & Shelah, S. (2007). Two cardinal models for singular \mu. MLQ Math. Log. Q., 53(6), 636–641. arXiv: math/0612247 DOI: 10.1002/malq.200610053 MR: 2351585
type: article
abstract: We deal here with colorings of the pair (\mu^+,\mu), when \mu is a strong limit and singular cardinal. We show that there exists a coloring c, with no refinement. It follows, that the properties of identities of (\mu^+,\mu) when \mu is singular, differ in an essential way from the case of regular \mu.
keywords: M: odm, S: ico, (set), (mod)
Sh:892
Dror Farjoun, E., Göbel, R., Segev, Y., & Shelah, S. (2007). On kernels of cellular covers. Groups Geom. Dyn., 1(4), 409–419. arXiv: math/0702294 DOI: 10.4171/GGD/20 MR: 2357479
type: article
abstract: In the present paper we continue to examine cellular covers of groups, focusing on the cardinality and the structure of the kernel K of the cellular map G\to M. We show that in general a torsion free reduced abelian group M may have a proper class of non-isomorphic cellular covers. In other words, the cardinality of the kernels is unbounded. In the opposite direction we show that if the kernel of a cellular cover of any group M has certain “freeness” properties, then its cardinality must be bounded.
keywords: O: alg, (ab), (stal)
Sh:893
Shelah, S. (2015). A.E.C. with not too many models. In A. Hirvonen, M. Kesala, J. Kontinen, R. Kossak, & A. Villaveces, eds., Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, Vols. Ontos Mathematical Logic, vol. 5, Berlin, Boston: DeGruyter, pp. 367–402. arXiv: 1302.4841 DOI: 10.1515/9781614516873.367
type: article
abstract: Consider an a.e.c. {\mathfrak K} and the class of {\mathbf C}_{\aleph_0} cardinals of cofinality \aleph_0. A nicely stated consequence of this work is for some closed unbounded class C of cardinals we have

(a)\dot I(\lambda,{\mathfrak K})\geq \lambda for \lambda \in C \cap {\mathbf C_{\aleph_0}}

or

(b)if M \in K_\lambda and \lambda \in {\mathbf C}_{\aleph_0}, then M has \le_{\mathfrak K}-extension (so in {\mathfrak K}) of arbitrarily large cardinals.
keywords: M: nec, M: non, S: ico, (mod), (nni), (aec)
Sh:894
Mildenberger, H., & Shelah, S. (2009). The near coherence of filters principle does not imply the filter dichotomy principle. Trans. Amer. Math. Soc., 361(5), 2305–2317. DOI: 10.1090/S0002-9947-08-04806-X MR: 2471919
type: article
abstract: We show that there is a forcing extension in which any two ultrafilters on \omega are nearly coherent and there is a non-meagre filter that is not nearly ultra. This answers Blass’ longstanding question whether the principle of near coherence of filters is strictly weaker than the filter dichotomy principle.
keywords: S: for, S: str, (set), (ultraf)
Sh:895
Shelah, S. (2010). Large continuum, oracles. Cent. Eur. J. Math., 8(2), 213–234. arXiv: 0707.1818 DOI: 10.2478/s11533-010-0018-3 MR: 2610747
type: article
abstract: Our main theorem is about iterated forcing for making the continuum larger than \aleph_2. We present a generalization of [Sh:669] which is dealing with oracles for random, etc., replacing \aleph_1,\aleph_2 by \lambda,\lambda^+ (starting with \lambda = \lambda^{< \lambda} > \aleph_1). Instead of properness we demand absolute c.c.c. So we get, e.g. the continuum is \lambda^+ but we can get cov(meagre) =\lambda. We also give some applications related to peculiar cuts of [Sh:885].
keywords: S: for, S: str, (set), (cont)
Sh:896
Kellner, J., Pauna, M., & Shelah, S. (2007). Winning the pressing down game but not Banach-Mazur. J. Symbolic Logic, 72(4), 1323–1335. arXiv: math/0609655 DOI: 10.2178/jsl/1203350789 MR: 2371208
type: article
abstract: Let S be the set of those \alpha\in\omega_2 that have cofinality \omega_1. It is consistent relative to a measurable that player II (the nonempty player) wins the pressing down game of length \omega_1, but not the Banach Mazur game of length \omega+1 (both starting with S).
keywords: S: for, (normal)
Sh:897
Shelah, S. (2008). Theories with Ehrenfeucht-Fraïssé equivalent non-isomorphic models. Tbil. Math. J., 1, 133–164. arXiv: math/0703477 MR: 2563810
type: article
abstract: Our “large scale” aim is to characterize the first order T (at least the countable ones) such that: for every ordinal \alpha there \lambda,M_1,M_2 such that M_1,M_2 are non-isomorphic models of T of cardinality \lambda which are EF_{\alpha,\lambda}-equivalent. We expect that as in the main gap ([Sh:c,XII]) we get a strong dichotomy, so in the non-structure side we have more, better example, and in the structure side we have a parallel of [Sh:c,XIII]. We presently prove the consistency of the non-structure side for T which is \aleph_0-independent (= not strongly dependent) or just not strongly stable, even for PC(T_1,T) and more for unstable T (see [Sh:c,VII] or [Sh:h]) and infinite linear order I.
keywords: M: cla, M: non, (mod), (sta)
Sh:898
Shelah, S. (2013). Pcf and abelian groups. Forum Math., 25(5), 967–1038. arXiv: 0710.0157 DOI: 10.1515/forum-2013-0119 MR: 3100959
type: article
abstract: We deal with some pcf investigations mostly motivated by abelian group theory problems and deal their applications to test problems (we expect reasonably wide applications). We prove almost always the existence of \aleph_\omega-free abelian groups with trivial dual, i.e. no non-trivial homomorphisms to the integers. This relies on investigation of pcf; more specifically, for this we prove that “almost always” there are {\mathcal F} \subseteq {}^\kappa \lambda which are quite free and has black boxes. The “almost always” means that there are strong restrictions on cardinal arithmetic if the universe fails this, this restriction are “everywhere”. Those are irrating results; we replace Abelian groups by R-modules, so in some sense our advantage over earlier results becomes clearer.
keywords: S: pcf, O: alg, (BB), (ab), (stal)
Sh:899
Juhász, I., & Shelah, S. (2008). Hereditarily Lindelöf spaces of singular density. Studia Sci. Math. Hungar., 45(4), 557–562. arXiv: math/0702295 DOI: 10.1556/SScMath.2007.1037 MR: 2641451
type: article
abstract: A cardinal \lambda is called \omega-inaccessible if for all \mu<\lambda we have \mu^\omega < \lambda. We show that for every \omega-inaccessible cardinal \lambda there is a CCC (hence cardinality and cofinality preserving) forcing that adds a hereditarily Lindel'́ of regular space of density \lambda. This extends an analogous earlier result of ours that only worked for regular \lambda.
keywords: S: for, (set), (gt)
Sh:900
Shelah, S. (2015). Dependent theories and the generic pair conjecture. Commun. Contemp. Math., 17(1), 1550004, 64. arXiv: math/0702292 DOI: 10.1142/S0219199715500042 MR: 3291978
type: article
abstract: On the one hand we try to understand complete types over somewhat saturated model of a complete first order theory which is dependent, by “decomposition theorems for such types”. Our thesis is that the picture of dependent theory is the combination of the one for stable theories and the one for the theory of dense linear order or trees (and first we should try to understand the quite saturated case). On the other hand as a measure of our progress, we give several applications considering some test questions; in particular we try to prove the generic pair conjecture and do it for measurable cardinals. The order of the sections is by their conceptions, so there are some repetitions.
keywords: M: cla, (mod), (ua)
Sh:901
Juhász, I., Shelah, S., & Soukup, L. (2009). Resolvability vs. almost resolvability. Topology Appl., 156(11), 1966–1969. arXiv: math/0702296 DOI: 10.1016/j.topol.2009.03.019 MR: 2536179
type: article
abstract: A space X is \kappa-resolvable (resp. almost \kappa-resolvable) if it contains \kappa dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets of X).

Answering a problem raised by Juhász, Soukup, and Szentmiklóssy, and improving a consistency result of Comfort and Hu, we prove, in ZFC, that for every infinite cardinal {\kappa} there is an almost 2^{\kappa}-resolvable but not {\omega}_1-resolvable space of dispersion character {\kappa}.
keywords: S: ico
Sh:902
Larson, P. B., & Shelah, S. (2008). The stationary set splitting game. MLQ Math. Log. Q., 54(2), 187–193. arXiv: 1003.2425 DOI: 10.1002/malq.200610054 MR: 2402627
type: article
abstract: The stationary set splitting game is a game of perfect information of length \omega_{1} between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to \Sigma^{2}_{2} maximality with a predicate for the nonstationary ideal on \omega_{1}, and an example of a consistently undetermined game of length \omega_{1} with payoff definable in the second-order monadic logic of order. We also show that the determinacy of the game is consistent with Martin’s Axiom but not Martin’s Maximum.
keywords: S: ico, (set), (normal)
Sh:903
Machura, M., Shelah, S., & Tsaban, B. (2010). Squares of Menger-bounded groups. Trans. Amer. Math. Soc., 362(4), 1751–1764. arXiv: math/0611353 DOI: 10.1090/S0002-9947-09-05169-1 MR: 2574876
type: article
abstract: Using a portion of the Continuum Hypothesis, we prove that there is a Menger-bounded (also called o-bounded) subgroup of the Baer-Specker group \mathbb{Z}^{\mathbb{N}}, whose square is not Menger-bounded. This settles a major open problem concerning boundedness notions for groups and implies that Menger-bounded groups need not be Scheepers-bounded. This also answers some questions of Banakh, Nickolas, and Sanchis.
keywords: S: ico, S: str, (set)
Sh:904
Shelah, S. (2010). Reflexive abelian groups and measurable cardinals and full MAD families. Algebra Universalis, 63(4), 351–366. arXiv: math/0703493 DOI: 10.1007/s00012-010-0086-z MR: 2734302
type: article
abstract: Answering problem (DG) of [EM90], [EM02], we show that there is a reflexive group of cardinality \ge first measurable.
keywords: S: ico, S: pcf, (set), (ab), (stal)
Sh:905
Kellner, J., & Shelah, S. (2010). A Sacks real out of nowhere. J. Symbolic Logic, 75(1), 51–76. arXiv: math/0703302 DOI: 10.2178/jsl/1264433909 MR: 2605882
type: article
abstract: There is a proper countable support iteration of length \omega adding no new reals at finite stages and adding a Sacks real in the limit.
keywords: S: for, S: str, (set)
Sh:906
Shelah, S. (2011). No limit model in inaccessibles. In Models, logics, and higher-dimensional categories, Vol. 53, Amer. Math. Soc., Providence, RI, pp. 277–290. arXiv: 0705.4131 MR: 2867976
type: article
abstract: Our aim is to improve the negative results i.e. non-existence of limit models, and the failure of the generic pair property from [Sh:877] to inaccessible \lambda as promised there. The motivation is that in [Sh:F756] the positive results are for \lambda measurable hence inaccessible, whereas in [Sh:877] in the negative results obtained only on non-strong limit cardinals.
keywords: M: cla, (mod)
Sh:907
Shelah, S. (2008). EF-equivalent not isomorphic pair of models. Proc. Amer. Math. Soc., 136(12), 4405–4412. arXiv: 0705.4126 DOI: 10.1090/S0002-9939-08-09362-3 MR: 2431056
type: article
abstract: We construct non-isomorphic models M, N, e.g. of cardinality \aleph_1 such that in the Ehrenfeucht-Fraissé game of length \zeta < \omega_1 the isomorphism player wins
keywords: S: ico, (mod)
Sh:908
Shelah, S. (2010). On long increasing chains modulo flat ideals. MLQ Math. Log. Q., 56(4), 397–399. arXiv: 0705.4130 DOI: 10.1002/malq.200910010 MR: 2681343
type: article
abstract: We prove that e.g. there is no \omega_4—sequence in (\omega_3)^{\omega_3} increasing mod countable.
keywords: S: ico, (set), (normal)
Sh:909
Gruenhut, E., & Shelah, S. (2011). Uniforming n-place functions on well founded trees. In Set theory and its applications, Vol. 533, Amer. Math. Soc., Providence, RI, pp. 267–280. arXiv: 0906.3055 DOI: 10.1090/conm/533/10512 MR: 2777753
type: article
abstract: In this paper the Erdős-Rado theorem is generalized to the class of well founded trees. We define an equivalence relation on the class {\rm rs}(\infty)^{<\aleph_0} (finite sequences of decreasing sequences of ordinals) with \aleph_0 equivalence classes, and for n<\omega a notion of n-end-uniformity for a colouring of {\rm rs}(\infty)^{<\aleph_0} with \mu colours. We then show that for every ordinal \alpha, n<\omega and cardinal \mu there is an ordinal \lambda so that for any colouring c of T={\rm rs}(\lambda)^{<\aleph_0} with \mu colours, T contains S isomorphic to {\rm rs}(\alpha) so that c\restriction S^{<\aleph_0} is n-end uniform. For c with domain T^n this is equivalent to finding S\subseteq T isomorphic to {\rm rs}(\alpha) so that c\upharpoonright S^{n} depends only on the equivalence class of the defined relation, so in particular T\rightarrow({\rm rs}(\alpha))^n_{\mu,\aleph_0}. We also draw a conclusion on colourings of n-tuples from a scattered linear order.
keywords: S: ico, (set)
Sh:910
Blass, A. R., & Shelah, S. (2008). Basic subgroups and freeness, a counterexample. In Models, modules and abelian groups, Walter de Gruyter, Berlin, pp. 63–73. arXiv: 0711.3031 DOI: 10.1515/9783110203035.63 MR: 2513227
type: article
abstract: We construct a non-free but \aleph_1-separable, torsion-free abelian group G with a pure free subgroup B such that all subgroups of G disjoint from B are free and such that G/B is divisible. This answers a question of Irwin and shows that a theorem of Blass and Irwin cannot be strengthened so as to give an exact analog for torsion-free groups of a result proved for p-groups by Benabdallah and Irwin.
keywords: O: alg, (ab), (pc)
Sh:911
Garti, S., & Shelah, S. (2011). Depth of Boolean algebras. Notre Dame J. Form. Log., 52(3), 307–314. arXiv: 0802.4185 DOI: 10.1215/00294527-1435474 MR: 2822491
type: article
abstract: We show under the assumption \bf {V} = \bf {L} that the {\rm Depth} of an ultraproduct of Boolean Algebras cannot jump over the {\rm Depth} of every component by more than one cardinal. This is done for every cardinal. We also get better results for singular cardinals with countable cofinality.
keywords: S: ico, (set), (ba)
Sh:912
Kennedy, J. C., Shelah, S., & Väänänen, J. A. (2008). Regular ultrafilters and finite square principles. J. Symbolic Logic, 73(3), 817–823. DOI: 10.2178/jsl/1230396748 MR: 2444269
type: article
abstract: We show that many singular cardinals \lambda above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle \square^{fin}_{\lambda, D} introduced in [3]. For such ultrafilters D and cardinals \lambda there are models of size \lambda for which M^{\lambda}/D is not \lambda^{++}-universal and elementarily equivalent models M and N of size \lambda for which M^\lambda/D and N^\lambda/D are non-isomorphic. The question of the existence of such ultrafilters and models was raised in [1].
keywords: S: ico, (set)
Sh:913
Kaplan, I., & Shelah, S. (2012). Automorphism towers and automorphism groups of fields without choice. In Groups and model theory, Vol. 576, Amer. Math. Soc., Providence, RI, pp. 187–203. arXiv: 1004.1810 DOI: 10.1090/conm/576/11337 MR: 2962885
type: article
abstract: The main result (already achieved in [FriedKollar]), is that any group can be represented as an automorphism group of a field. We introduce a new proof, using a simple construction. We conclude \tau_{\kappa}^{nlg}\leq\tau_{\kappa} without choice.
keywords: S: ods, O: alg, (mod), (auto), (grp)
Sh:914
Shelah, S. (2011). The first almost free Whitehead group. Tbil. Math. J., 4, 17–30. arXiv: 0708.1980 MR: 2886755
type: article
abstract: Assume G.C.H. and \kappa is the first uncountable cardinal such that there is a \kappa-free abelian group which is not a Whitehead (abelian) group. We prove that \kappa is necessarily an inaccessible cardinal.
keywords: S: str, O: alg, (ab), (pc)
Sh:915
Shelah, S. (2011). The character spectrum of \beta(\mathbb N). Topology Appl., 158(18), 2535–2555. arXiv: 1004.2083 DOI: 10.1016/j.topol.2011.08.014 MR: 2847327
type: article
abstract: We show the consistency of: the set of regular cardinals which are the character of some ultrafilter on \omega can be quite chaotic, in particular not only can be not convex but can have many gaps. We also deal with the set of \pi-characters of ultrafilters on \omega
keywords: S: str, (set), (ultraf)
Sh:916
Dow, A. S., & Shelah, S. (2008). Tie-points and fixed-points in \mathbb N^*. Topology Appl., 155(15), 1661–1671. arXiv: 0711.3037 DOI: 10.1016/j.topol.2008.05.002 MR: 2437015
type: article
abstract: A point x is a (bow) tie-point of a space X if X\setminus \{x\} can be partitioned into (relatively) clopen sets each with x in its closure. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of \beta{\mathbb N} \setminus {\mathbb N} (e.g. [veli.oca, ShSt735]) and in the recent study of (precisely) 2-to-1 maps on \beta{\mathbb N} \setminus {\mathbb N}. In these cases the tie-points have been the unique fixed point of an involution on \beta{\mathbb N} \setminus {\mathbb N}. This paper is motivated by the search for 2-to-1 maps and obtaining tie-points of strikingly differing characteristics.
keywords: S: str
Sh:917
Dow, A. S., & Shelah, S. (2009). More on tie-points and homeomorphism in \mathbb N^\ast. Fund. Math., 203(3), 191–210. arXiv: 0711.3038 DOI: 10.4064/fm203-3-1 MR: 2506596
type: article
abstract: A point x is a (bow) tie-point of a space X if X\setminus \{x\} can be partitioned into (relatively) clopen sets each with x in its closure. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of \beta{\mathbb N} \setminus {\mathbb N}=\mathbb N^* and in the recent study of (precisely) 2-to-1 maps on \mathbb N^*. In these cases the tie-points have been the unique fixed point of an involution on \mathbb N^*. One application of the results in this paper is the consistency of there being a 2-to-1 continuous image of \mathbb N^* which is not a homeomorph of \mathbb N^*.
keywords: S: str, (auto)
Sh:918
Shelah, S. (2012). Many partition relations below density. Israel J. Math., 191(2), 507–543. arXiv: 0902.0440 DOI: 10.1007/s11856-012-0027-y MR: 3011486
type: article
abstract: We force 2^\lambda to be large and for many pairs in the interval (\lambda,2^\lambda) a stronger version of the polarized partition relations hold. We apply this toproblem in general topology
keywords: S: ico, S: for, (set)
Sh:919
Cohen, M., & Shelah, S. (2016). Stable theories and representation over sets. MLQ Math. Log. Q., 62(3), 140–154. arXiv: 0906.3050 DOI: 10.1002/malq.200920105 MR: 3509699
type: article
abstract: In this paper we give a characterization of the class of first order stable theories using.
keywords: M: cla, (mod), (sta)
Sh:920
Göbel, R., & Shelah, S. (2009). \aleph_n-free modules with trivial duals. Results Math., 54(1-2), 53–64. DOI: 10.1007/s00025-009-0382-0 MR: 2529626
type: article
abstract: In the first part of this paper we introduce a simplified version of a new Black Box from Shelah [Sh:883] which can be used to construct complicated \aleph_n-free abelian groups for any natural number n\in N. In the second part we apply this prediction principle to derive for many commutative rings R the existence of \aleph_n-free R-modules M with trivial dual M^*=0, where M^*={\rm Hom}(M,R). The minimal size of the \aleph_n-free abelian groups constructed below is \beth_n, and this lower bound is also necessary as can be seen immediately if we apply GCH.
keywords: O: alg, (ab), (pc)
Sh:921
Shelah, S., & Tsaban, B. (2010). On a problem of Juhász and van Mill. Topology Proc., 36, 385–392. arXiv: 0710.2768 MR: 2646986
type: article
abstract: A 27 years old and still open problem of Juhász and van Mill asks whether there exists a cardinal \kappa such that every regular dense in itself countably compact space has a dense in itself subset of cardinality at most \kappa. We give a negative answer for the analogous question where regular is weakened to Hausdorff, and coutnably compact is strengthened to sequentially compact.
keywords: S: str
Sh:922
Shelah, S. (2010). Diamonds. Proc. Amer. Math. Soc., 138(6), 2151–2161. arXiv: 0711.3030 DOI: 10.1090/S0002-9939-10-10254-8 MR: 2596054
type: article
abstract: We prove, e.g. that if \lambda=\chi^+=2^\chi and S \subseteq \{\delta<\lambda:cf(\delta)\ne cf(\chi)\} is stationary then \diamondsuit_\lambda.
keywords: S: ico, (set)
Sh:923
Ardal, H., Maňuch, J., Rosenfeld, M., Shelah, S., & Stacho, L. (2009). The odd-distance plane graph. Discrete Comput. Geom., 42(2), 132–141. DOI: 10.1007/s00454-009-9190-2 MR: 2519871
type: article
abstract: The vertices of the odd-distance graph are the points of the plane \mathbb{R}^2. Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chormatic number of this graph is at least five. We also prove that the odd-distance graph in \mathbb{R}^2 is countably choosable, which such a graph in \mathbb{R}^3 is not.
keywords: S: ico, (graph)
Sh:924
Shelah, S. (2015). Models of PA: when two elements are necessarily order automorphic. MLQ Math. Log. Q., 61(6), 399–417. arXiv: 1004.3342 DOI: 10.1002/malq.200920124 MR: 3433640
type: article
abstract: We are interested in the question of how much the order of a non-standard model of PA can determine the model. In particular, for a model M, we want to characterize the complete types p(x,y) of non-standard elements (a,b) such that the linear orders \{x:x< a\} and \{x:x < b\} are necessarily isomorphic. It is proved that this set includes the complete types p(x,y) such that if the pair (a,b) realizes it (in M) then there is an element c such that for all standard n,c^n < a,c^n < b,a < bc and b < ac. We prove that this is optimal, because if \diamondsuit_{\aleph_1} holds, then there is M of cardinality \aleph_1 for which we get equality. We also deal with how much the order in a model of PA may determine the addition.
keywords: M: odm, (mod)
Sh:925
Larson, P. B., & Shelah, S. (2009). Splitting stationary sets from weak forms of choice. MLQ Math. Log. Q., 55(3), 299–306. arXiv: 1003.2477 DOI: 10.1002/malq.200810011 MR: 2519245
type: article
abstract: We consider splitting stationary sets under the assumption of ZF + DC plus the existence of a ladder system for ordinals of countable cofinality. This is a continuation of [Sh835].
keywords: S: ods, (set), (normal), (AC)
Sh:926
Bartoszyński, T., & Shelah, S. (2010). Dual Borel conjecture and Cohen reals. J. Symbolic Logic, 75(4), 1293–1310. DOI: 10.2178/jsl/1286198147 MR: 2767969
type: article
abstract: We construct a model of ZFC satisfying the Dual Borel Conjecture in which there is a set of size \aleph_1 that does not have measure zero.
keywords: S: for, S: str, (set)
Sh:927
Baldwin, J. T., Kolesnikov, A. S., & Shelah, S. (2009). The amalgamation spectrum. J. Symbolic Logic, 74(3), 914–928. DOI: 10.2178/jsl/1245158091 MR: 2548468
See [Sh:927a]
type: article
abstract: For every natural number k^*, there is a class {\mathbf{K}}_* defined by a sentence in L_{\omega_1,\omega} that has no models of cardinality > \beth_{k^*+1}, but {\mathbf{K}}_* has the d isjoint amalgamation property on models of cardinality \leq \aleph_{{k^*}-3} and has models of cardinality \aleph_{{k^*}-1}. More strongly, For every countable ordinal \alpha^*, there is a class {\mathbf{K}}_* defined by a sentence in L_{\omega_1,\omega} that has no models of cardinality > \beth_{\alpha}, but {\mathbf{K}}_* has the disjoint amalgamation property on models of cardinality \leq \aleph_{\alpha}. Similar results hold for arbitrary \kappa and L_{\kappa^+,\omega}.
keywords: M: smt, (mod), (aec)
Sh:929
Herden, D., & Shelah, S. (2010). \kappa-fold transitive groups. Forum Math., 22(4), 627–640. DOI: 10.1515/FORUM.2010.034 MR: 2661440
type: article
abstract: A group G of type 0 is called \kappa-transitive for some cardinal \kappa >0 if for any ordered pair of pure elements x,y \in G there exist exactly \kappa-many \varphi \in {\rm Aut} G such that x\varphi =y. We shows the existence of large \kappa-transitive groups for every \kappa\ge \aleph_0 assuming V=L and ZFC respectively.
keywords: M: non, (grp)
Sh:930
Herden, D., & Shelah, S. (2009). An upper cardinal bound on absolute E-rings. Proc. Amer. Math. Soc., 137(9), 2843–2847. DOI: 10.1090/S0002-9939-09-09842-6 MR: 2506440
type: article
abstract: We show that for every abelian group A of cardinality \ge\kappa(\omega) there exists a generic extension of the universe, where A is countable with 2^{\aleph_O} injective endomorphisms. As an immediate consequence of this result there are no absolute E-rings of cardinality \ge \kappa (\omega). This paper does not require any specific prior knowledge of forcing or model theory and can be considered accessible also for graduate students.
keywords: M: non
Sh:931
Shelah, S., & Steprāns, J. (2011). Masas in the Calkin algebra without the continuum hypothesis. J. Appl. Anal., 17(1), 69–89. DOI: 10.1515/JAA.2011.004 MR: 2805847
type: article
abstract: Methods for constructing masas in the Calkin algebra without assuming the Continuum Hypothesis are developed
keywords: S: ico, S: str, (set)
Sh:933
Laskowski, M. C., & Shelah, S. (2015). \mathbf P-NDOP and \mathbf P-decompositions of \aleph_\epsilon-saturated models of superstable theories. Fund. Math., 229(1), 47–81. arXiv: 1206.6028 DOI: 10.4064/fm229-1-2 MR: 3312115
type: article
abstract: Assume a complete first order theory T is superstable. We generalize revise [Sh:401] in two respects, so do not depend on it. First issue we deal with a more general case. Let \mathbf P be a class of regular types in \mathfrak{C}, closed under automorphisms and under \pm. We generalize [Sh:401] to this context to \mathbf P^\pm-saturated M’s, assuming \mathbf P-NDOP which is weaker than NDOP. Second issue, in this content it is more delicate to find sufficient condition on two \mathbf P-decomposition trees to give non-isomorphic models. For this we investigate natural structures on the set of regular types mod \pm in M. Actually it suffices to deal with the case M is \aleph_\varepsilon-saturated \mathfrak{d}_\ell = \langle M^\ell_\eta,a_\eta:\eta \in I_\ell\rangle is a \mathbf P-decomposition of M for \ell=1,2 and \{p^{\mathfrak{d}_\ell}_\eta:\eta \in I_\ell\}/\pm = ({\mathcal P} \cap \mathbf S(M))/\pm and show the two trees are quite similar (or isomorphic).
keywords: M: cla, (mod)
Sh:934
Hall, E. J., & Shelah, S. (2013). Partial choice functions for families of finite sets. Fund. Math., 220(3), 207–216. arXiv: 0808.0535 DOI: 10.4064/fm220-3-2 MR: 3040670
type: article
abstract: Let p be a prime. We show that ZF + “Every countable set of p-element sets has an infinite partial choice function” is not strong enough to prove that every countable set of p-element sets has a choice function, answering an open question from [1]. The independence result is obtained by way of a permutation (Fraenkel-Mostowski) model in which the set of atoms has the structure of a vector space over the field of p elements. By way of comparison, some simpler permutation models are considered in which some countable families of p-element sets fail to have infinite partial choice functions.
keywords: S: ods, (set), (AC)
Sh:935
Shelah, S. (2011). MAD saturated families and SANE player. Canad. J. Math., 63(6), 1416–1435. arXiv: 0904.0816 DOI: 10.4153/CJM-2011-057-1 MR: 2894445
type: article
abstract: We throw some light on the question: is there a MAD family (= a family of infinite subsets of \mathbb N, the intersection of any two is finite) which is completely separable (i.e. any X \subseteq \mathbb N is included in a finite union of members of the family include a member of the family). We prove that it is hard to prove the consistency of the negation: “(a)” if 2^{\aleph_0} < \aleph_\omega, then there is such a family “(b)” if there is no such families then some situation related to pcf holds whose consistency is large.
keywords: S: pcf, S: str, (set)
Sh:936
Enayat, A., & Shelah, S. (2011). An improper arithmetically closed Borel subalgebra of \mathcal P(\omega)\bmod\mathrm{FIN}. Topology Appl., 158(18), 2495–2502. DOI: 10.1016/j.topol.2011.08.006 MR: 2847322
type: article
abstract: We show the existence of a subalgebra {\mathcal A}\subset {\mathcal P}(\omega) that satisfies the following three conditions.

{\mathcal A} is Borel (when {\mathcal P}(\omega) is identified with 2^\omega).

{\mathcal A} is arithmetically closed (i.e., {\mathcal A} is closed under the Turing jump, and Turing reducibility).

The forcing notion ({\mathcal A}, \subset) modulo the ideal FIN of finite sets collapses the continuum to \aleph_0.
keywords: S: str, (set), (mod)
Sh:937
Shelah, S. (2011). Models of expansions of \mathbb N with no end extensions. MLQ Math. Log. Q., 57(4), 341–365. arXiv: 0808.2960 DOI: 10.1002/malq.200910129 MR: 2832642
type: article
abstract: We deal with models of Peano arithmetic (specifically with a question of Ali Enayat). The methods are from creature forcing. We find an expansion of \mathbb N such that its theory has models with no (elementary) end extensions. In fact there is a Borel uncountable set of subsets of \mathbb N such that expanding \mathbb N by any uncountably many of them suffice. Also we find arithmetically closed {\mathcal A} with no definably closed ultrafilter on it
keywords: S: for, (set), (mod)
Sh:938
Shelah, S. (2012). PCF arithmetic without and with choice. Israel J. Math., 191(1), 1–40. arXiv: 0905.3021 DOI: 10.1007/s11856-012-0026-z MR: 2970861
type: article
abstract: We deal with relatives of GCH which are provable. In particular we deal with rank version of the revised GCH. Our motivation was to find such results when only weak versions of the axiom of choice are assumed but some of the results gives us additional information even in ZFC.
keywords: S: for, S: ods, (set), (AC)
Sh:939
Kellner, J., & Shelah, S. (2011). More on the pressing down game. Arch. Math. Logic, 50(3-4), 477–501. arXiv: 0905.3913 DOI: 10.1007/s00153-011-0227-x MR: 2786767
type: article
abstract: We compare the pressing down and the Banach Mazur games and show: Consistently relative to a supercompact there is a nowhere precipitous normal ideal I on \aleph_2 such that for every I-positive set A nonempty wins the pressing down game of length \aleph_1 on I starting with A.
keywords: S: for, (set), (normal)
Sh:941
Rosłanowski, A., Shelah, S., & Spinas, O. (2012). Nonproper products. Bull. Lond. Math. Soc., 44(2), 299–310. arXiv: 0905.0526 DOI: 10.1112/blms/bdr094 MR: 2914608
type: article
abstract: We show that there exist two proper creature forcings having a simple (Borel) definition, whose product is not proper. We also give a new condition ensuring properness of some forcings with norms.
keywords: S: for, (set)
Sh:942
Rosłanowski, A., & Shelah, S. (2013). More about \lambda-support iterations of (<\lambda)-complete forcing notions. Arch. Math. Logic, 52(5-6), 603–629. arXiv: 1105.6049 DOI: 10.1007/s00153-013-0334-y MR: 3072781
type: article
abstract: This article continues Rosłanowski and Shelah [RoSh:655, RoSh:860, RoSh:777, RoSh:888, RoSh:890] and we introduce here a new property of ({<}\lambda)–strategically complete forcing notions which implies that their \lambda–support iterations do not collapse \lambda^+ (for a strongly inaccessible cardinal \lambda).
keywords: S: for, (set)
Sh:943
Göbel, R., Herden, D., & Shelah, S. (2009). Skeletons, bodies and generalized E(R)-algebras. J. Eur. Math. Soc. (JEMS), 11(4), 845–901. DOI: 10.4171/JEMS/169 MR: 2538507
type: article
abstract: fill in
keywords: O: alg, (ab), (stal)
Sh:944
Shelah, S. (2018). Models of PA: standard systems without minimal ultrafilters. Sarajevo J. Math., 14(27)(1), 3–11. arXiv: 0901.1499 MR: 3858024
type: article
abstract: We prove that \mathbb N has an uncountable elementary extension N such that there is no ultrafilter on the Boolean Algebra of subsets of \mathbb N represented in N which is so called minimal.
keywords: S: for, (ultraf)
Sh:945
Shelah, S. (2020). On \mathrm{con}(\mathfrak{d}_\lambda>\mathrm{cov}_\lambda(\mathrm{meagre})). Trans. Amer. Math. Soc., 373(8), 5351–5369. arXiv: 0904.0817 DOI: 10.1090/tran/7948 MR: 4127879
type: article
abstract: We prove the consistency of: for suitable strongly inaccessible cardinal \lambda the dominating number, i.e. the cofinaty of {}^\lambda \lambda is strictly bigger than cov(meagre_\lambda), i.e. the minimal number of no-where-dense subsets of {}^\lambda 2 needed to cover it. This answers a question of Matet.
keywords: S: for, (set)
Sh:946
Kaplan, I., & Shelah, S. (2014). Examples in dependent theories. J. Symb. Log., 79(2), 585–619. arXiv: 1009.5420 DOI: 10.1017/jsl.2013.11 MR: 3224981
type: article
abstract: We show a counterexample to a conjecture by Shelah regarding the existence of indiscernible sequences in dependent theories
keywords: M: cla, (mod)
Sh:947
Larson, P. B., Neeman, I., & Shelah, S. (2010). Universally measurable sets in generic extensions. Fund. Math., 208(2), 173–192. arXiv: 1003.2479 DOI: 10.4064/fm208-2-4 MR: 2640071
type: article
abstract: A subset of a topological space is said to be universally measurable if it is measurable with respect to every complete, countably additive \sigma-finite measure on the space, and universally null if it has measure zero for each such atomless measure. In 1934, Hausdorff proved that there exist universally null sets of cardinality \aleph_1, and thus that there exist a least 2^{\aleph_1} such sets. Laver showed in the 1970’s that consistently there are just continuum many universally null sets. The question of whether there exist more than continuum many universally measurable sets was asked by Mauldin no later than 1984. We show that consistently there exist only continuum many universally measurable sets.
keywords: S: for, S: str, (set)
Sh:948
Göbel, R., Herden, D., & Shelah, S. (2011). Absolute E-rings. Adv. Math., 226(1), 235–253. DOI: 10.1016/j.aim.2010.06.019 MR: 2735757
type: article
abstract: A ring R with 1 is called an E-ring if End_{\mathbb Z} R is ring-isomorphic to R under the canonical homomorphism taking the value 1 \sigma for any \sigma \in End_{\mathbb Z} R. Moreover R is an absolute E-ring if it remains an E-ring in every generic extension of the universe. E-rings are an important tool for algebraic topology as explained in the introduction. The existence of an E-ring R of each cardinality of the form \lambda^{\aleph_0} was shown by Dugas, Mader and Vinsonhaler [DMV]. We want to show the existence of absolute E-rings. It turns out that there is a precise cardinal-barrier \kappa(\omega) for this problem: (The cardinal \kappa(\omega) is the first \omega-Erdős cardinal defined in the introduction. It is a relative of measurable cardinals.) We will construct absolute E-rings of any size \lambda <\kappa(\omega). But there are no absolute E-rings of cardinality \ge\kappa(\omega). The non-existence of huge, absolute E-rings \ge \kappa(\omega) follows from a recent paper by Herden and Shelah [HS] and the construction of absolute E-rings R is based on an old result by Shelah [S] where families of absolute, rigid colored trees (with no automorphism between any distinct members) are constructed. We plant these trees into our potential E-rings with the aim to prevent unwanted endomorphisms of their additive group to survive. Endomorphisms will recognize the trees which will have branches infinitely often divisible by primes. Our main result provides the existence of absolute E-rings for all infinite cardinals \lambda <\kappa(\omega), i.e. these E-rings remain E-rings in all generic extensions of the universe (e.g. using forcing arguments). Indeed all previously known E-rings ([DMV,GT]) of cardinality \ge 2^{\aleph_0} have a free additive group R^+ in some extended universe, thus are no longer E-rings, as explained in the introduction. Our construction also fills all cardinal-gaps of the earlier constructions (which have only sizes \lambda^{\aleph_0}). These E-rings are domains and as a by-product we obtain the existence of absolutely indecomposable abelian groups, compare [GS2].
keywords: S: ico, O: alg
Sh:949
Garti, S., & Shelah, S. (2012). A strong polarized relation. J. Symbolic Logic, 77(3), 766–776. arXiv: 1103.0350 DOI: 10.2178/jsl/1344862161 MR: 2987137
type: article
abstract: We prove that the strong polarized relation {{\mu^+}\choose {\mu}}\rightarrow {{\mu^+}\choose {\mu}}^{1,1}_2 is consistent with ZFC.
keywords: S: ico, (set)
Sh:951
Mildenberger, H., & Shelah, S. (2011). Proper translation. Fund. Math., 215(1), 1–38. DOI: 10.4064/fm215-1-1 MR: 2851699
type: article
abstract: We continue our work on weak diamonds [MdSh:848]. We show that 2^\omega= \aleph_2 together with the weak diamond for covering by thin trees, the weak diamond for covering by meagre sets, the weak diamond for covering by null sets, and “all Aronszajn trees are special” is consistent relative to ZFC. We iterate alternately forcings specialising Aronszajn trees without adding reals (the NNR forcing from [Sh:f, Ch. IV]) and <\omega_1-proper {}^\omega \omega-bounding forcings adding reals. We show that over a tower of elementary submodels there is a sort of a reduction (“proper translation”) of our iteration to the c.s. iteration of simpler iterands. If we use only Sacks iterands and NNR iterands, this allows us to guess the values of Borel functions into small trees and thus derive the mentionedweak diamonds.
keywords: S: for, S: str, (set)
Sh:952
Shelah, S., & Zapletal, J. (2011). Ramsey theorems for product of finite sets with submeasures. Combinatorica, 31(2), 225–244. DOI: 10.1007/s00493-011-2677-5 MR: 2848252
type: article
abstract: We prove parametrized partition theorem on products of finite sets equipped with submeasures, improving the results of DiPrisco, Llopis, and Todorcevic
keywords: S: for, S: str, (set), (creatures)
Sh:953
Doron, M., & Shelah, S. (2010). Hereditary zero-one laws for graphs. In Fields of logic and computation, Vol. 6300, Springer, Berlin, pp. 581–614. arXiv: 1006.2888 DOI: 10.1007/978-3-642-15025-8_29 MR: 2756404
type: article
abstract: We consider the random graph M^n_{\bar{p}} on the set [n], were the probability of \{x,y\} being an edge is p_{|x-y|}, and \bar{p}=(p_1,p_2,p_3,...) is a series of probabilitie. We consider the set of all \bar{q} derived from \bar{p} by inserting 0 probabilities to \bar{p}, or alternatively by decreasing some of the p_i. We say that \bar{p} hereditarily satisfies the 0-1 law if the 0-1 law (for first order logic) holds in M^n_{\bar{q}} for any \bar{q} derived from \bar{p} in the relevant way described above. We give a necessary and sufficient condition on \bar{p} for it to hereditarily satisfy the 0-1 law.
keywords: M: odm, O: fin, (fmt)
Sh:954
Farah, I., & Shelah, S. (2010). A dichotomy for the number of ultrapowers. J. Math. Log., 10(1-2), 45–81. arXiv: 0912.0406 DOI: 10.1142/S0219061310000936 MR: 2802082
type: article
abstract: We prove a strong dichotomy for the number of ultrapowers of a given model of cardinality \le c associated with nonprincipal ultrafilters on \mathbb{N}. They are either all isomorphic, or else there are 2^c many. We prove the analogous result for metric structures, including C^*-algebras and II_1 factors, as well as their relative commutants
keywords: M: non, (mod), (nni)
Sh:955
Shelah, S. (2014). Pseudo PCF. Israel J. Math., 201(1), 185–231. arXiv: 1107.4625 DOI: 10.1007/s11856-014-1063-6 MR: 3265284
type: article
abstract: We continue our investigation on pcf with weak forms of the axiom of choice. Characteristically, we assume DC + {\mathcal P}(Y) when looking at \prod\limits_{s \in Y} \delta_s. We get more parallels of pcf theorems.
keywords: S: pcf, (set), (AC)
Sh:956
Garti, S., & Shelah, S. (2012). (\kappa,\theta)-weak normality. J. Math. Soc. Japan, 64(2), 549–559. arXiv: 1104.1491 http://projecteuclid.org/euclid.jmsj/1335444403 MR: 2916079
type: article
abstract: We deal with the property of weak normality (for non-principal ultrafilters). We characterize the situation of |\Pi_{i<\kappa} \lambda_i / D| = \lambda. We have an application ofr a question of Depth in Boolean algebras.
keywords: S: ico, (set)
Sh:957
Rosłanowski, A., & Shelah, S. (2013). Partition theorems from creatures and idempotent ultrafilters. Ann. Comb., 17(2), 353–378. arXiv: 1005.2803 DOI: 10.1007/s00026-013-0184-7 MR: 3056773
type: article
abstract: We show a general scheme of Ramsey-type results for partitions of countable sets of finite functions, where “one piece is big” is interpreted in the language originating in creature forcing. The heart of our proofs follows Glazer’s proof of the Hindman Theorem, so we prove the existence of idempotent ultrafilters with respect to suitable operation. Then we deduce partition theorems related to creature forcings.
keywords: S: ico, (set), (creatures)
Sh:958
Baldwin, J. T., & Shelah, S. (2011). A Hanf number for saturation and omission. Fund. Math., 213(3), 255–270. DOI: 10.4064/fm213-3-5 MR: 2822421
type: article
abstract: Suppose \mathbf T =(T,T_1,p) is a triple of two countable theories in languages \tau \subset \tau_1 and a \tau_1-type p over the empty set. We show the Hanf number for the property: There is a model M_1 of T_1 which omits p, but M_1 \restriction \tau is saturated is at least the Löwenheim number of second order logic.
keywords: M: smt, (mod)
Sh:959
Baldwin, J. T., & Shelah, S. (2012). The stability spectrum for classes of atomic models. J. Math. Log., 12(1), 1250001, 19. DOI: 10.1142/S0219061312500018 MR: 2950191
type: article
abstract: We prove two results on the stability spectrum for L_{\omega_1,\omega}. Here S^m_i(M) denotes an appropriate notion ({\rm at}(+ 1 962) or {\rm mod}) of Stone space of m-types over M. Theorem A. Suppose that for some positive integer m and for every \alpha< \delta(T), there is an M \in \mathbf K with |S^m_i(M)| > |M|^{\beth_\alpha(|T|)}. Then for every \lambda \geq |T|, there is an M with |S^m_i(M)| > |M|. Theorem B. Suppose that for every \alpha<\delta(T), there is M_\alpha \in \mathbf K such that \lambda_\alpha = |M_{\alpha}| \geq \beth_\alpha and |S^m_{i}(M_\alpha)| > \lambda_\alpha. Then for any \mu with \mu^{\aleph_0}>\mu \mathbf K is not i-stable in \mu. These results provide a new kind of sufficient condition for the unstable case and shed some light on the spectrum of strictly stable theories in this context. The methods avoid the use of compactness in the theory under study. In the Section [treeindis], we expound the construction of tree indiscernibles for sentences of L_{\omega_1,\omega}.
keywords: M: nec, (mod)
Sh:960
Shelah, S. (2017). Preserving old ([\omega]^{\aleph_0},\supseteq^*) is proper. Sarajevo J. Math., 13(26)(2), 141–154. arXiv: 1005.2802 MR: 3744785
type: article
abstract: We give some sufficient and necessary conditions on a forcing notion Q for preserving the forcing notion ([\omega]^{\aleph_0},\supseteq) is proper. They cover many reasonable forcing notions.
keywords: S: for
Sh:961
Kellner, J., & Shelah, S. (2012). Creature forcing and large continuum: the joy of halving. Arch. Math. Logic, 51(1-2), 49–70. arXiv: 1003.3425 DOI: 10.1007/s00153-011-0253-8 MR: 2864397
type: article
abstract: (none)
keywords: S: for, S: str, (set), (creatures)
Sh:962
Garti, S., & Shelah, S. (2012). Combinatorial aspects of the splitting number. Ann. Comb., 16(4), 709–717. arXiv: 1007.2266 DOI: 10.1007/s00026-012-0154-5 MR: 3000439
type: article
abstract: We prove that the existence of strong splitting families is equivalent to the failure of the polarized relation {\mathfrak{s}\choose\omega} \rightarrow {\mathfrak{s}\choose {\omega}^{1,1}_2}. We show that both directions are consistent with ZFC, and that the strong splitting number equals to the splitting number, when exists. Consequently, we can put some restriction on the possibility that \mathfrak{s} is singular.
keywords: S: str, (set)
Sh:963
Cummings, J., Džamonja, M., Magidor, M., Morgan, C., & Shelah, S. (2017). A framework for forcing constructions at successors of singular cardinals. Trans. Amer. Math. Soc., 369(10), 7405–7441. arXiv: 1403.6795 DOI: 10.1090/tran/6974 MR: 3683113
type: article
abstract: We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal \kappa of uncountable cofinality, while \kappa^+ enjoys various combinatorial properties.

As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal \kappa of uncountable cofinality where SCH fails and such that there is a collection of size less than 2^{\kappa^+} of graphs on kappa^+ such that any graph on \kappa^+ embeds into one of the graphs in the collection.
keywords: S: for, (set), (univ)
Sh:964
Garti, S., & Shelah, S. (2012). Strong polarized relations for the continuum. Ann. Comb., 16(2), 271–276. arXiv: 1103.5195 DOI: 10.1007/s00026-012-0130-0 MR: 2927607
type: article
abstract: sufficient conditions for the bipartite (2^\kappa, \kappa) ------> (2^\kappa , \kappa) when \mu - 2 \kappa is singular eg cf(\mu) < \mathfrak{s} <\mu, generalize for \lambda instead {\aleph_0}
keywords: S: ico, (pc)
Sh:965
Larson, P. B., Matteo, N., & Shelah, S. (2012). Majority decisions when abstention is possible. Discrete Math., 312(7), 1336–1352. arXiv: 1003.2756 DOI: 10.1016/j.disc.2011.12.024 MR: 2885917
type: article
abstract: Suppose we are given a family of choice functions on pairs from a given finite set. The set is considered as a set of alternatives (say candidates for an office) and the functions as potential “voters.” The question is, what choice functions agree, on every pair, with the majority of some finite subfamily of the voters? For the problem as stated, a complete characterization was given in [Sh:816], but here we allow each voter to abstain. There are four cases.
keywords: O: fin, O: odo
Sh:967
Mildenberger, H., & Shelah, S. (2011). The minimal cofinality of an ultrapower of \omega and the cofinality of the symmetric group can be larger than \mathfrak b^+. J. Symbolic Logic, 76(4), 1322–1340. DOI: 10.2178/jsl/1318338852 MR: 2895398
type: article
abstract: We prove the statement in the title
keywords: S: for, S: str
Sh:968
Shelah, S., & Strüngmann, L. H. (2014). On the p-rank of \mathrm{Ext}(A,B) for countable abelian groups A and B. Israel J. Math., 199(2), 567–572. arXiv: 1401.5317 DOI: 10.1007/s11856-013-0039-2 MR: 3219548
type: article
abstract: In this note we show that the p-rank of {\rm Ext}(A,B) for countable torsion-free abelian groups A and B is either countable or the size of the continuum.
keywords: S: dst, O: alg, (ab), (stal)
Sh:969
Goldstern, M., Kellner, J., Shelah, S., & Wohofsky, W. (2014). Borel conjecture and dual Borel conjecture. Trans. Amer. Math. Soc., 366(1), 245–307. arXiv: 1105.0823 DOI: 10.1090/S0002-9947-2013-05783-2 MR: 3118397
See [Sh:969a]
type: article
abstract: We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.
keywords: S: for, S: str, (inv)
Sh:970
Göbel, R., Herden, D., & Shelah, S. (2014). Prescribing endomorphism algebras of \aleph_n-free modules. J. Eur. Math. Soc. (JEMS), 16(9), 1775–1816. DOI: 10.4171/JEMS/475 MR: 3273308
type: article
abstract: (none)
keywords: O: alg, (ab), (stal)
Sh:971
Khelif, A., & Shelah, S. (2010). Équivalence élémentaire de puissances cartésiennes d’un même groupe. C. R. Math. Acad. Sci. Paris, 348(23-24), 1241–1244. DOI: 10.1016/j.crma.2010.10.034 MR: 2745331
type: article
abstract: We prove that if I and J are infinite sets and G an abelian torsion group the groups G^I and G^J are elementarily equivalent for the logic L_{\infty\omega}. The proof is based on a new and simple property with a Cantor-Bernstein flavour. A criterion applying to non commutative groups allows us to exhibit various groups (free or soluble or nilpotent or ...) G such that for I infinite countable and J uncountable the groups G^I and G^J are not even elementarily equivalent for the L_{\omega_I \omega} logic. Another argument leads to a countable commutative group having the same property.
keywords: S: str, O: alg
Sh:972
Rosłanowski, A., & Shelah, S. (2014). Monotone hulls for \mathcal N\cap\mathcal M. Period. Math. Hungar., 69(1), 79–95. arXiv: 1007.5368 DOI: 10.1007/s10998-014-0042-3 MR: 3269711
type: article
abstract: Using the method of decisive creatures ([KrSh:872]) we show the consistency of “there is no increasing \omega_2–chain of Borel sets and {\rm non}({\mathcal N})= {\rm non}({\mathcal M})=\omega_2=2^\omega”. Hence, consistently, there are no monotone hulls for the ideal {\mathcal M}\cap {\mathcal N}. This answers Balcerzak and Filipczak. Next we use FS iteration with partial memory to show that there may be monotone Borel hulls for the ideals {\mathcal M}, {\mathcal N} even if they are not generated by towers.
keywords: S: for, S: str
Sh:973
Mildenberger, H., & Shelah, S. (2014). Many countable support iterations of proper forcings preserve Souslin trees. Ann. Pure Appl. Logic, 165(2), 573–608. arXiv: 1309.0196 DOI: 10.1016/j.apal.2013.08.002 MR: 3129729
type: article
abstract: We show that there are many models of \rm{cov} {\mathcal M}= \aleph_1 and {\rm cof}{{{\mathcal M}}} = \aleph_2 in which the club principle holds and there are Souslin trees. The proof consists of the following main steps:
1. We give some iterable and some non-iterable conditions on a forcing in terms of games that imply that the forcing is (T,Y,\mathcal{S})-preserving. A special case of (T,Y,\mathcal{S})-preserving is preserving the Souslinity of an \omega_1-tree.
2. We show that some tree-creature forcings from [RoSh:470] satisfy the sufficient condition for one of the strongest games.
3. Without the games, we show that some linear creature forcings from [RoSh:470] are (T,Y,\mathcal{S})-preserving. There are non-Cohen preserving examples.
4. For the wider class of non-elementary proper forcings we show that \omega-Cohen preserving for certain candidates implies (T,Y,\mathcal{S})-preserving.
5. (+ 1 978)We give a less general but hopefully more easily readable presentation of a result from [Sh:f, Chapter 18, §3]: If all iterands in a countable support iteration are proper and (T,Y,\mathcal{S})-preserving, then also the iteration is (T,Y,\mathcal{S})-preserving. This is a presentation of the so-called case A in which a division in forcings that add reals and those who do not is not needed.
In [Mi:clubdistr] we showed: Many proper forcings from [RoSh:470] with finite or countable {\rm{H}}(n) (see Section 2.1) force over a ground model with \diamondsuit_{\omega_1} in a countable support iteration the club principle. After \omega_1 iteration steps the diamond holds anyway.
keywords: S: for, (inv)
Sh:974
Garti, S., & Shelah, S. (2019). Depth^+ and Length^+ of Boolean algebras. Houston J. Math., 45(4), 953–963. arXiv: 1303.3704 MR: 4102862
type: article
abstract: Suppose \kappa=\textrm{cf}(\kappa), \lambda>\textrm{cf}(\lambda)=\kappa^+ and \lambda=\lambda^\kappa. We prove that there exist a sequence \langle{\mathbf{B}}_i:i<\kappa\rangle of Boolean algebras and an ultrafilter D on \kappa so that \lambda=\prod\limits_{i<\kappa} {\rm Depth}^+({\mathbf{B}}_i)/D< {\rm Depth}^+(\prod\limits_{i< \kappa}{\mathbf B}_i/D)= \lambda^+. An identical result holds also for {\rm Length}^+. The proof is carried in ZFC and it holds even above large cardinals.
keywords: S: ico, (ba)
Sh:975
Kaplan, I., & Shelah, S. (2014). A dependent theory with few indiscernibles. Israel J. Math., 202(1), 59–103. arXiv: 1010.0388 DOI: 10.1007/s11856-014-1067-2 MR: 3265314
type: article
abstract: We continue the work started in [KpSh:946], and prove the following theorem: For every \theta there is a dependent theory T of size \theta such that for all \kappa and \delta, \kappa\to\left(\delta\right)_{T,1} iff \kappa\to\left(\delta\right)_{\theta}^{<\omega}.
keywords: M: cla, S: ico
Sh:976
Shelah, S., & Strüngmann, L. H. (2011). Kulikov’s problem on universal torsion-free abelian groups revisited. Bull. Lond. Math. Soc., 43(6), 1198–1204. DOI: 10.1112/blms/bdr055 MR: 2861541
type: article
abstract: Let T be a torsion abelian group. We consider the class of all torsion-free abelian groups G satisfying {\rm Ext}(G,T)=0 and search for \lambda-universal objects in this class. We show that for certain T there is no \omega-universal group. However, for uncountable cardinals \lambda there is always a \lambda-universal group if we assume (V=L). Together with results by the second author this solves completely a problem by Kulikov.
keywords: O: alg, (ab), (stal)
Sh:977
Shelah, S. (2012). Modules and infinitary logics. In Groups and model theory, Vol. 576, Amer. Math. Soc., Providence, RI, pp. 305–316. arXiv: 1011.3581 DOI: 10.1090/conm/576/11376 MR: 2962893
type: article
abstract: We prove that the theory of abelian groups and R-modules even in infinitary logic is stable and understood to some extent.
keywords: M: smt, O: alg, (ab)
Sh:978
Malliaris, M., & Shelah, S. (2014). Regularity lemmas for stable graphs. Trans. Amer. Math. Soc., 366(3), 1551–1585. arXiv: 1102.3904 DOI: 10.1090/S0002-9947-2013-05820-5 MR: 3145742
type: article
abstract: Let G be a finite graph with the non-k_*-order property (essentially, a uniform finite bound on the size of an induced sub-half-graph). The main result of the paper applies model-theoretic arguments to obtain a stronger version of Szemerédi’s regularity lemma for such graphs:

Theorem: Let k_* and therefore k_{**} (a constant \leq 2^{k_*+2}) be given. Let G be a graph with the non-k_*-order property. Then for any \epsilon>0 there exists m=m(\epsilon) such that if G is sufficiently large, there is a partition \langle A_i : i<i(*)\leq m\rangle of G into at most m pieces, where:

1. m \leq \frac{1}{\epsilon_0^{4k_{**}}}, for \epsilon_0 = {\rm{min}}\{\epsilon, \frac{1}{2^{4k_{**}+1}}\} 2. for all i,j<i(*), ||A_i|-|A_j||\leq 1 3. all of the pairs (A_i, A_j) are (\epsilon, \epsilon)-uniform, meaning that for some truth value {\textbf{t}}= {\textbf{t}}(A_i,A_j) \in \{0,1\}, for all but <\epsilon|A_i| of the elements of |A_i|, for all but < \zeta|A_j| of the elements of A_j, (aRb) \equiv ({\textbf{t}}(A_i,A_j)=1)

4. all of the pieces A_i are \epsilon-excellent (an indivisibility condition, Definition (6.2) below)

We also give several other partition theorems for this class of graphs, described in the Introduction, and discuss extensions to graphs without the independence property. Motivation for this work comes from a coincidence of model-theoretic and graph-theoretic ideas. Namely, it was known that the “irregular pairs” in the statement of Szemerédi’s regularity lemma cannot be eliminated, due to the counterexample of half-graphs. The results of this paper show in what sense this counterexample is the essential difficulty. The proof is largely model-theoretic: arbitrarily large half-graphs coincide with model-theoretic instability, so in their absence, structure theorems and technology from stability theory apply.
keywords: O: fin, (stal)
Sh:979
Shelah, S., & Simon, P. (2012). Adding linear orders. J. Symbolic Logic, 77(2), 717–725. arXiv: 1103.0206 DOI: 10.2178/jsl/1333566647 MR: 2963031
type: article
abstract: We address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)=A for all A, then this is true. Otherwise, we give counterexamples. More precisely, there is a totally categorical theory for which every expansion by a linear order has IP. There is also an \omega-stable NDOP theory for which every expansion by a linear order interprets bounded arithmetic.
keywords: M: odm
Sh:980
Shelah, S. (2023). Nice \aleph_1 generated non-P-points, Part I. MLQ Math. Log. Q., 69(1), 117–129. arXiv: 1112.0819 DOI: 10.1002/malq.202200070 MR: 4606452
type: article
abstract: We define a family of a (non-principal) ultrafilter on {\mathbb N}, i.e. a point which are very far from P-points. We first under reasonable conditions, prove its existence. In a continuation we shall prove that such a point may exist while no P-point exists.
keywords: S: ico, S: for, S: str
Sh:981
Göbel, R., Shelah, S., & Strüngmann, L. H. (2013). \aleph_n-free modules over complete discrete valuation domains with almost trivial dual. Glasg. Math. J., 55(2), 369–380. DOI: 10.1017/S0017089512000614 MR: 3040868
type: article
abstract: Let M^*Hom_R(M,R) be the dual module of M for any commutative ring R. In [GS] we applied a recent prediction principle from Shelah [Sh] to find \aleph_n-free R-modules M with trivial dual M^*=0 for each natural number n. This can be achieved for a large class of rings R including all countable, principle ideal domains which are not fields. (Recall that an R-module M is \kappa-free for some infinite cardinal \kappa if all its submodules generated by <\kappa elements are contained in a free R-submodule.) However, the result fails if R is uncountable, as can be seen from Kaplansky’s [Ka] well-known splitting theorems for modules over the ring J_p of p-adic integers. Nevertheless we want to extend the main result from [GS] to complete discrete valuation domains (DVDs) R, in particular to p-adic modules R and define a duality-test which circumvents Kaplansky’s counterexamples. We say that M has almost a trivial dual if there is no homomorphism from M onto a free R-module of countable (infinite) rank. In the first part of this paper we must strengthen and adjust the new combinatorial principle (called the \aleph_n-Black Box) and in the second part we will apply it to find arbitrarily large \aleph_n-free R-modules over complete DVD with almost trivial dual. A corresponding result for torsion modules is obtained as well. Also observe, that the existence of such modules can easily be established assuming GCH (see e.g. Eklof-Mekler [EM] on diamonds), so the problem rests on the fact that we want to stay in ordinary set theory.
keywords: O: alg, (ab), (stal)
Sh:982
Lücke, P., & Shelah, S. (2012). External automorphisms of ultraproducts of finite models. Arch. Math. Logic, 51(3-4), 433–441. DOI: 10.1007/s00153-012-0271-1 MR: 2899700
type: article
abstract: Let \mathcal{L} be a finite first-order language and \langle{\mathcal{M}_n}{n<\omega}\rangle be a sequence of finite \mathcal{L}-models containing models of arbitrarily large finite cardinality. If the intersection of less than continuum-many dense open subsets of Cantor Space {}^\omega 2 is non-empty, then there is a non-principal ultrafilter \mathcal{U} over \omega such that the corresponding ultraproduct \prod_{\mathcal{U}} \mathcal{M}_n is infinite and has an automorphism that is not induced by an element of \prod_{n<\omega} {\rm Aut}{\mathcal{M}_n}.
keywords: M: odm
Sh:983
Pinsker, M., & Shelah, S. (2013). Universality of the lattice of transformation monoids. Proc. Amer. Math. Soc., 141(9), 3005–3011. arXiv: 1107.3946 DOI: 10.1090/S0002-9939-2013-11566-2 MR: 3068953
type: article
abstract: The set of all transformation monoids on a fixed set of infinite cardinality \lambda, equipped with the order of inclusion, forms an algebraic lattice Mon(\lambda) with 2^\lambda compact elements. We show that this lattice is universal, i.e., every algebraic lattice with at most 2^\lambda compact elements is isomorphic to a complete sublattice of Mon(\lambda).
keywords: S: ico, O: alg
Sh:984
Dow, A. S., & Shelah, S. (2013). An Efimov space from Martin’s axiom. Houston J. Math., 39(4), 1423–1435. MR: 3164725
type: article
abstract: We show from a weak set-theoretic hypothesis that there is an Efimov space.
keywords: S: for, S: str
Sh:985
Dow, A. S., & Shelah, S. (2012). Martin’s axiom and separated mad families. Rend. Circ. Mat. Palermo (2), 61(1), 107–115. DOI: 10.1007/s12215-011-0078-7 MR: 2897749
type: article
abstract: Two families \mathcal A, \mathcal B of subsets of \omega are said to be separated if there is a subset of \omega which mod finite contains every member of \mathcal A and is almost disjoint from every member of \mathcal B. If \mathcal A and \mathcal B are countable disjoint subsets of an almost disjoint family, then they are separated. Luzin gaps are well-known examples of of \omega_1-sized subfamilies of an almost disjoint family which can not be separated. An almost disjoint family will be said to be \omega_1-separated if any disjoint pair of {\leq}\omega_1-sized subsets are separated. It is known that the proper forcing axiom (PFA) implies that no maximal almost disjoint family is {\leq}\omega_1-separated. We prove that this does not follow from Martin’s Axiom.
keywords: S: ico, S: str
Sh:987
Farah, I., & Shelah, S. (2014). Trivial automorphisms. Israel J. Math., 201(2), 701–728. arXiv: 1112.3571 DOI: 10.1007/s11856-014-1048-5 MR: 3265300
type: article
abstract: We prove that the statement ‘For all Borel ideals {\mathcal I} and {\mathcal J} on \omega, every isomorphism between Boolean algebras {\mathcal P}(\omega)/{\mathcal I} and {\mathcal P}(\omega)/{\mathcal J} has a continuous representation’ is relatively consistent with ZFC. In a model of this statement we have that for a number of Borel ideals {\mathcal I} on \omega every isomorphism between {\mathcal P} (\omega)/{\mathcal I} and any other quotient {\mathcal P}(\omega)/{\mathcal J} over a Borel ideal is trivial.

We can also assure that in this model the dominating number, {\mathfrak d}, is equal to \aleph_1 and that 2^{\aleph_1} is arbitrarily large. In this model Calkin algebra has outer automorphisms while all automorphisms of {\mathcal P}(\omega)/{\mathop{Fin}} are trivial.
keywords: S: str
Sh:988
Mildenberger, H., & Shelah, S. (2019). Specializing Aronszajn trees with strong axiom A and halving. Notre Dame J. Form. Log., 60(4), 587–616. DOI: 10.1215/00294527-2019-0021 MR: 4019863
type: article
abstract: We show that there are proper forcings based upon countable trees of creatures that specialise a given Aronszajn tree. We try to get the club.
keywords: S: for, S: str
Sh:989
Goldstern, M., Shelah, S., & Sági, G. (2013). Very many clones above the unary clone. Algebra Universalis, 69(4), 387–399. arXiv: 1108.2061 DOI: 10.1007/s00012-013-0236-1 MR: 3061094
type: article
abstract: Let \mathfrak c:= 2^{\aleph_0}. We give a family of pairwise incomparable clones on \mathbb{N} with 2^{\mathfrak c} members, all with the same unary fragment, namely the set of all unary operations.

We also give, for each n, a family of 2^{\mathfrak c} clones all with the same n-ary fragment, and all containing the set of all unary operations.
keywords: S: ico, O: alg
Sh:990
Shelah, S., & Steprāns, J. (2015). Non-trivial automorphisms of \mathcal P(\mathbb N)/[\mathbb N]^{<\aleph_0} from variants of small dominating number. Eur. J. Math., 1(3), 534–544. DOI: 10.1007/s40879-015-0058-0 MR: 3401904
See [Sh:990a]
type: article
abstract: It is shown that if various cardinal invariants of the continuum related to \mathfrak d are equal to \aleph_1 then there is a nontrivial automorphism of \mathcal{P}(\mathbb{N})/ [\mathbb{N}]^{<\aleph_0}. Some of these results extend to automorphisms of \mathcal{P}(\kappa)/[\kappa]^{<\kappa} if \kappa is inaccessible
keywords: S: for, S: str
Sh:991
Raghavan, D., & Shelah, S. (2012). Comparing the closed almost disjointness and dominating numbers. Fund. Math., 217(1), 73–81. arXiv: 1110.6690 DOI: 10.4064/fm217-1-6 MR: 2914923
type: article
abstract: We prove that if there is a dominating family of size {\aleph}_{1}, then there is are {\aleph}_{1} many compact subsets of \omega^{\omega} whose union is a maximal almost disjoint family of functions that is also maximal with respect to infinite partial functions.
keywords: S: for, S: str
Sh:992
Baldwin, J. T., & Shelah, S. (2014). A Hanf number for saturation and omission: the superstable case. MLQ Math. Log. Q., 60(6), 437–443. DOI: 10.1002/malq.201300022 MR: 3274973
type: article
abstract: Suppose {\bf t} =(T,T_1,p) is a triple of two theories in vocabularies \tau \subset \tau_1 with cardinality \lambda and a \tau_1-type p over the empty set. We show the Hanf number for the property: There is a model M_1 of T_1 which omits p, but M_1 \restriction \tau is saturated is less than \beth_{({2^{{(2^{ \lambda})}^+}}){}^{^+}} if T is superstable. If T is required only to be stable, the Hanf number is bounded by the Hanf number of L_{(2^\lambda)^+,\kappa(T)}.

We showed in an earlier paper that without the stability restriction the Hanf number is essentially equal to the Löwenheim number of second order logic.
keywords: M: cla, M: smt
Sh:993
Kaplan, I., & Shelah, S. (2013). Chain conditions in dependent groups. Ann. Pure Appl. Logic, 164(12), 1322–1337. arXiv: 1112.0807 DOI: 10.1016/j.apal.2013.06.014 MR: 3093393
type: article
abstract: In this short note we prove and disprove some chain conditions in dependent, strongly dependent and strongly^{2} dependent theories, focusing on type definable groups.
keywords: M: cla, O: alg
Sh:994
Goldstern, M., Pinsker, M., & Shelah, S. (2013). A closed algebra with a non-Borel clone and an ideal with a Borel clone. Internat. J. Algebra Comput., 23(5), 1115–1125. arXiv: 1112.0774 DOI: 10.1142/S0218196713500197 MR: 3096314
type: article
abstract: (none)
keywords: S: ico, O: alg
Sh:995
Garti, S., & Shelah, S. (2014). Partition calculus and cardinal invariants. J. Math. Soc. Japan, 66(2), 425–434. arXiv: 1112.5772 DOI: 10.2969/jmsj/06620425 MR: 3201820
type: article
abstract: (none)
keywords: S: str, (pc), (inv)
Sh:996
Malliaris, M., & Shelah, S. (2015). Constructing regular ultrafilters from a model-theoretic point of view. Trans. Amer. Math. Soc., 367(11), 8139–8173. arXiv: 1204.1481 DOI: 10.1090/S0002-9947-2015-06303-X MR: 3391912
type: article
abstract: This paper, the first of two, contributes to the set-theoretic side of understanding Keisler’s order. We deal with properties of ultrafilters which affect saturation of unstable theories: the lower cofinality {\mathrm{lcf}}(\aleph_0, \mathcal{D}) of \aleph_0 modulo \mathcal{D}, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, known to be detected by non-low theories. Assuming \kappa > \aleph_0 is measurable, we construct a regular ultrafilter on \lambda \geq 2^\kappa which is flexible (thus: ok) but not good, and which moreover has large {\mathrm{lcf}} (\aleph_0) but does not even saturate models of the random graph. This implies (a) that flexibility alone cannot characterize saturation of any theory, however (b) by separating flexibility from goodness, we remove a main obstacle to proving non-low does not imply maximal, and (c) from a set-theoretic point of view, consistently, ok need not imply good, answering a question from Dow 1985. Under no additional assumptions, we prove that there is a loss of saturation in regular ultrapowers of unstable theories, and give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. More precisely, for \mathcal{D} regular on \kappa and M a model of an unstable theory, M^\kappa/ \mathcal{D} is not (2^\kappa)^+-saturated; and for M a model of a non-simple theory and \lambda = \lambda^{<\lambda}, M^\lambda/\mathcal{D} is not \lambda^{++}-saturated. Finally, we investigate realization and omission of symmetric cuts, significant both because of the maximality of the strict order property in Keisler’s order, and by recent work of the authors on SOP_2. We prove that if \mathcal{D} is a \kappa- complete ultrafilter on \kappa, any ultrapower of a sufficiently saturated model of linear order will have no (\kappa, \kappa)-cuts, and that if \mathcal{D} is also normal, it will have a (\kappa^+, \kappa^+)-cut. We apply this to prove that for any n < \omega, assuming the existence of n measurable cardinals below \lambda, there is a regular ultrafilter D on \lambda such that any D-ultrapower of a model of linear order will have n alternations of cuts, as defined below. Moreover, D will \lambda^+-saturate all stable theories but will not (2^{\kappa})^+-saturate any unstable theory, where \kappa is the smallest measurable cardinal used in the construction.
keywords: M: cla, S: ico
Sh:997
Malliaris, M., & Shelah, S. (2014). Model-theoretic properties of ultrafilters built by independent families of functions. J. Symb. Log., 79(1), 103–134. arXiv: 1208.2579 DOI: 10.1017/jsl.2013.28 MR: 3226014
type: article
abstract: In this paper, the second of two, we continue our investigations of model-theoretic properties of ultrafilters: \mu(\mathcal{D}), the minimum size of a product of an unbounded sequence of natural numbers modulo \mathcal{D}; \mathrm{lcf}(\aleph_0, \mathcal{D}) the lower cofinality (coinitiality) of \aleph_0 modulo \mathcal{D};flexibility, discussed extensively in the paper; realization of symmetric cuts; and goodness. We work in ZFC except where noted. Our main results are as follows. First, we prove that any ultrafilter \mathcal{D} which is \lambda-flexible (thus: \lambda^+-o.k.) must have \mu(\mathcal{D}) = 2^\lambda. Thus, a fortiori, \mathcal{D} will saturate any stable theory. This is the strongest possible statement about the saturation power of flexibility alone, in light of our proof in the companion paper (I) that consistently, flexibility does not imply saturation of the random graph. In the remainder of the paper, we focus on the method of constructing ultrafilters via families of independent functions. Our second result is a constraint, that is, a tool for building ultrafilters which are not flexible. Specifically, we prove that if, at any point in a construction by independent functions the cardinality of the range of the remaining independent family is strictly smaller than the index set, then essentially no subsequent ultrafilter can be flexible. This is a useful point of leverage since any ultrafilter which is not flexible will fail to saturate any non-low theory. The third and fourth results are ultrafilter constructions. Third, assuming the existence of a measurable cardinal \kappa (to obtain a \kappa-complete ultrafilter), we prove that on any \lambda \geq \kappa^+ there is a regular ultrafilter which is flexible but not good. This gives a second proof, of independent interest, of a question from Dow [1985], complementing the proof in the companion paper (I). Fourth, assuming the existence of a weakly compact cardinal \kappa, we prove that for \aleph_0 < \theta = \textrm{cf}(\theta) < \kappa \leq \lambda there is a regular ultrafilter \mathcal{D} on \lambda such that \mathrm{lcf}(\aleph_0, \mathcal{D}) = \theta but (\mathbb{N}, <)^\lambda/\mathcal{D} has no (\kappa, \kappa)-cuts. This appears counter to model-theoretic intuition, since it shows some families of cuts in linear order can be realized without saturating the minimum unstable theory. We give several extensions of this last result, and show how to eliminate the large cardinal hypothesis in the case of asymmetric cuts.
keywords: M: cla, S: ico
Sh:998
Malliaris, M., & Shelah, S. (2016). Cofinality spectrum theorems in model theory, set theory, and general topology. J. Amer. Math. Soc., 29(1), 237–297. arXiv: 1208.5424 DOI: 10.1090/jams830 MR: 3402699
type: article
abstract: We solve a set-theoretic problem (p=t) and show it has consequences for the classification of unstable theories. That is, we consider what pairs of cardinals (\kappa_1, \kappa_2) may appear as the cofinalities of a cut in a regular ultrapower of linear order, under the assumption that all symmetric pre-cuts of cofinality no more than the size of the index set are realized. We prove that the only possibility is (\kappa, \kappa^+) where \kappa is regular and \kappa^+ is the cardinality of the index set I. This shows that unless |I| is the successor of a regular cardinal, any such ultrafilter must be |I|^+-good. We then connect this work to the problem of determining the boundary of the maximum class in Keisler’s order. Currently, SOP_3 is known to imply maximality. Here, we show that the property of realizing all symmetric pre-cuts characterizes the existence of paths through trees and thus realization of types with SOP_2 (it was known that realizing all pre-cuts characterizes realization of types with SOP_3). Thus whenever \lambda is not the successor of a regular cardinal, SOP_2 is \lambda-maximal in Keisler’s order. Moreover, the question of the full maximality of SOP_2 is reduced to either constructing a regular ultrafilter admitting the single asymmetric cut described, or showing one cannot exist.
keywords: M: cla, S: str, (inv)
Sh:999
Malliaris, M., & Shelah, S. (2013). A dividing line within simple unstable theories. Adv. Math., 249, 250–288. arXiv: 1208.2140 DOI: 10.1016/j.aim.2013.08.027 MR: 3116572
type: article
abstract: We give the first (ZFC) dividing line in Keisler’s order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal \lambda for which there is \mu < \lambda \leq 2^\mu, we construct a regular ultrafilter \mathcal{D} on \lambda so that (i) for any model M of a stable theory or of the random graph, M^\lambda/\mathcal{D} is \lambda^+-saturated but (ii) if Th(N) is not simple or not low then N^\lambda/\mathcal{d} is not \lambda^+-saturated. The non-saturation result relies on the notion of flexible ultrafilters. To prove the saturation result we develop a property of a class of simple theories, called Qr_1, generalizing the fact that whenever B is a set of parameters in some sufficiently saturated model of the random graph, |B| = \lambda and \mu < \lambda \leq 2^\mu, then there is a set A with |A| = \mu so that any nonalgebraic p \in S(B) is finitely realized in A. In addition to giving information about simple unstable theories, our proof reframes the problem of saturation of ultrapowers in several key ways. We give a new characterization of good filters in terms of “excellence,” a measure of the accuracy of the quotient Boolean algebra. We introduce and develop the notion of moral ultrafilters on Boolean algebras. We prove a so-called “separation of variables” result which shows how the problem of constructing ultrafilters to have a precise degree of saturation may be profitably separated into a more set-theoretic stage, building an excellent filter, followed by a more model-theoretic stage: building so-called moral ultrafilters on the quotient Boolean algebra, a process which highlights the complexity of certain patterns, arising from first-order formulas, in certain Boolean algebras.
keywords: M: cla, S: ico
Sh:1001
Rosłanowski, A., & Shelah, S. (2019). The last forcing standing with diamonds. Fund. Math., 246(2), 109–159. arXiv: 1406.4217 DOI: 10.4064/fm898-9-2018 MR: 3959246
type: article
abstract: This article continues Rosłanowski and Shelah [RoSh:655], [RoSh:942]. We introduce here yet another property of ({<}\lambda)–strategically complete forcing notions which implies that their \lambda–support iterations do not collapse \lambda^+.
keywords: S: for, (iter)
Sh:1002
Garti, S., & Shelah, S. (2012). The ultrafilter number for singular cardinals. Acta Math. Hungar., 137(4), 296–301. arXiv: 1201.1713 DOI: 10.1007/s10474-012-0245-0 MR: 2992547
type: article
abstract: We prove the consistency of a singular cardinal \lambda with small value of the ultrafilter number \mathfrak{u}_\lambda, and arbitrarily large value of 2^\lambda.
keywords: S: ico, S: for, (inv)
Sh:1003
Baldwin, J. T., Larson, P. B., & Shelah, S. (2015). Almost Galois \omega-stable classes. J. Symb. Log., 80(3), 763–784. DOI: 10.1017/jsl.2015.19 MR: 3395349
type: article
abstract: Theorem. Suppose that an \aleph_0-presentable Abstract Elementary Class (AEC), \mathbf {K}, has the joint embedding and amalgamation properties in \aleph_0 and <2^{\aleph_1} models in \aleph_1. If \mathbf {K} has only countably many models in \aleph_1, then all are small. If, in addition, \mathbf{K} is almost Galois \omega-stable then \mathbf{K} is Galois \omega-stable.
keywords: M: nec, (aec)
Sh:1004
Shelah, S. (2017). A parallel to the null ideal for inaccessible \lambda: Part I. Arch. Math. Logic, 56(3-4), 319–383. arXiv: 1202.5799 DOI: 10.1007/s00153-017-0524-0 MR: 3633799
type: article
abstract: It is well known how to generalize the meagre ideal replacing \aleph_0 by a (regular) cardinal \lambda > \aleph_0 and requiring the ideal to be \lambda^+-c.c and (< \lambda )-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing \aleph_0 by \lambda. So naturally, to call it a generalization we require it to be (< \lambda)-complete and \lambda^+-c.c. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead of forcing) we may look at the Boolean Algebra of \lambda-Borel sets modulo the ideal. Common wisdom have said that there is no such thing, but here surprisingly first we get a positive = existence answer for \lambda a “mild" large cardinal: the weakly compact one. Second, we try to show that this together with the meagre ideal (for \lambda) behaves as in the countable case. In particular, consider the classical Cichoń diagram, which compare several cardinal characterizations of those ideals. Last but not least, we apply this to get consistency results on cardinal invariants for such \lambda’s. We shall deal with other cardinals, and with more properties related forcing notions in a continuation.
keywords: S: for, (inv), (large), (pure(for))
Sh:1005
Shelah, S. (2016). ZF + DC + AX_4. Arch. Math. Logic, 55(1-2), 239–294. arXiv: 1411.7164 DOI: 10.1007/s00153-015-0469-0 MR: 3453586
type: article
abstract: We consider mainly the following version of set theory:“ZF + DC and for every \lambda,\lambda^{\aleph_0} is well ordered", our thesis is that this is a reasonable set theory, e.g. much can be said. In particular, we prove that for a sequence \bar\delta = \langle \delta_s:s \in Y\rangle,\textrm{cf}(\delta_s) large enough compared to Y, we can prove the pcf theorem with minor changes (using true cofinalities not the pseudo one). We then deduce the existence of covering numbers and define and prove existence of truely successor cardinals. From this we show that some diagonalization arguments (more specifically some black boxes and consequence) on Abelian groups. We end but showing some such consequences hold in ZF above.
keywords: S: pcf, S: ods, (ab), (AC)
Sh:1006
Shelah, S. (2013). On incompactness for chromatic number of graphs. Acta Math. Hungar., 139(4), 363–371. arXiv: 1205.0064 DOI: 10.1007/s10474-012-0287-3 MR: 3061483
type: article
abstract: We deal with incompactness. Assume the existence of non-reflecting stationary set of cofinality \kappa. We prove that one can define a graph G whose chromatic number is >\kappa, while the chromatic number of every subgraph G' \subseteq G,|G'| < |G| is \le \kappa. The main case is \kappa = \aleph_0.
keywords: S: ico, (graph)
Sh:1007
Chernikov, A., Kaplan, I., & Shelah, S. (2016). On non-forking spectra. J. Eur. Math. Soc. (JEMS), 18(12), 2821–2848. arXiv: 1205.3101 DOI: 10.4171/JEMS/654 MR: 3574578
type: article
abstract: Non-forking is one of the most important notions in modern model theory capturing the idea of a generic extension of a type (which is a far-reaching generalization of the concept of a generic point of a variety).

To a countable first-order theory we associate its non-forking spectrum — a function of two cardinals \kappa and \lambda giving the supremum of the possible number of types over a model of size \lambda that do not fork over a sub-model of size \kappa. This is a natural generalization of the stability function of a theory.

We make progress towards classifying the non-forking spectra. On the one hand, we show that the possible values a non-forking spectrum may take are quite limited. On the other hand, we develop a general technique for constructing theories with a prescribed non-forking spectrum, thus giving a number of examples. In particular, we answer negatively a question of Adler whether NIP is equivalent to bounded non-forking.

In addition, we answer a question of Keisler regarding the number of cuts a linear order may have. Namely, we show that it is possible that {ded}\kappa<\left({ded}\kappa\right)^{ \omega}.
keywords: M: cla, S: pcf, (unstable)
Sh:1008
Shelah, S. (2013). Non-reflection of the bad set for \check I_{\theta}[\lambda] and pcf. Acta Math. Hungar., 141(1-2), 11–35. arXiv: 1206.2048 DOI: 10.1007/s10474-013-0344-6 MR: 3102967
type: article
abstract: We reconsider here the following related pcf questions and make some advances: (Q1) concerning the ideal \check I_\kappa[\lambda] how much reflection do we have for the bad set S^{bd}_{\lambda,\kappa}\subseteq \{\delta<\lambda: cf(\delta)=\kappa\} assuming it is well defined? (Q2) for an ideal J on \kappa how large are S^{bd}_J[\bar f],S^{ch}_J[\bar f] for \bar f=\langle f_\alpha: \alpha < \lambda\rangle which is <_J-increasing and cofinal in (\prod\limits_{i<\kappa}\lambda_i,<_J)? (Q3) are there somewhat free black boxes?
keywords: S: ico, S: pcf, (normal)
Sh:1009
Malliaris, M., & Shelah, S. (2015). Saturating the random graph with an independent family of small range. In A. Hirvonen, M. Kesala, J. Kontinen, R. Kossak, & A. Villaveces, eds., Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, Berlin, Boston: De Gruyter, pp. 319–337. arXiv: 1208.5585 DOI: 10.1515/9781614516873.319
type: article
abstract: Motivated by Keisler’s order, a far-reaching program of understanding basic model-theoretic structure through the lens of regular ultrapowers, we prove that for a class of regular filters \mathcal{D} on I, |I| = \lambda > \aleph_0, the fact that {\mathcal{P}}(I)/ \mathcal{D} has little freedom (as measured by the fact that any maximal antichain is of size <\lambda, or even countable) does not prevent extending \mathcal{D} to an ultrafilter \mathcal{D}1 on I which saturates ultrapowers of the random graph. “Saturates” means that M^I/\mathcal{D}_1 is \lambda^+-saturated whenever M \models T_{\mathbf{rg}}. This was known to be true for stable theories, and false for non-simple and non-low theories. This result and the techniques introduced in the proof have catalyzed the authors’ subsequent work on Keisler’s order for simple unstable theories. The introduction, which includes a part written for model theorists and a part written for set theorists, discusses our current program and related results.
keywords: M: cla, S: ico, (up)
Sh:1011
Kennedy, J. C., Shelah, S., & Väänänen, J. A. (2015). Regular ultrapowers at regular cardinals. Notre Dame J. Form. Log., 56(3), 417–428. arXiv: 1307.6396 DOI: 10.1215/00294527-3132788 MR: 3373611
type: article
abstract: ADD
keywords: M: odm
Sh:1012
Garti, S., & Shelah, S. (2016). Open and solved problems concerning polarized partition relations. Fund. Math., 234(1), 1–14. arXiv: 1208.6091 DOI: 10.4064/fm763-10-2015 MR: 3509813
type: article
abstract: We list some open problems, concerning the polarized partition relation. We solve a couple of them, by showing that for every singular cardinal \mu there exists a forcing notion \mathbb{P} such that the strong polarized relation {\mu^+ \choose \mu} \rightarrow {\mu^+ \choose \mu}^{1,1}_2 holds in {\rm\bf V}^{\mathbb{P}}.
keywords: S: ico, (pc)
Sh:1013
Gitik, M., & Shelah, S. (2013). Applications of pcf for mild large cardinals to elementary embeddings. Ann. Pure Appl. Logic, 164(9), 855–865. arXiv: 1307.5977 DOI: 10.1016/j.apal.2013.03.002 MR: 3056300
type: article
abstract: The following pcf results are proved: 1. Assume that \kappa>\aleph_0 is a weakly compact cardinal. Let \mu>2^\kappa be a singular cardinal of cofinality \kappa. Then for every regular \lambda<{\rm pp}^+_{\Gamma(\kappa)}(\mu) there is an increasing sequence \langle \lambda_i \mid i<\kappa \rangle of regular cardinals converging to \mu such that \lambda= {\rm tcf}(\prod_{i<\kappa} \lambda_i, <_{J^{bd}_{\kappa}}). 2. Let \mu be a strong limit cardinal and \theta a cardinal above \mu. Suppose that at least one of them has an uncountable cofinality. Then there is \sigma_*<\mu such that for every \chi<\theta the following holds: \theta> {\rm sup}\{ {\rm sup} {\rm pcf}_{\sigma_{*}-{ \rm complete}}(\mathfrak{a}) \mid \mathfrak{a}\subseteq {\rm Reg} \cap (\mu^+,\chi) \text{ and } | \mathfrak{a}|<\mu \}. As an application we show that:

if \kappa is a measurable cardinal and j:V \to M is the elementary embedding by a \kappa–complete ultrafilter over a measurable cardinal \kappa, then for every \tau the following holds:
1. if j(\tau) is a cardinal then j(\tau)=\tau;
2. |j(\tau)|=|j(j(\tau))|;
3. for any \kappa–complete ultrafilter W on \kappa, |j(\tau)|=|j_W(\tau)|.
The first two items provide affirmative answers to questions from [G-Sh] and the third to a question of D. Fremlin.
keywords: S: pcf
Sh:1014
Lücke, P., & Shelah, S. (2014). Free groups and automorphism groups of infinite structures. Forum Math. Sigma, 2, e8, 18. arXiv: 1211.6891 DOI: 10.1017/fms.2014.9 MR: 3264251
type: article
abstract: Let \lambda be a cardinal with \lambda=\lambda^{\aleph_0} and p be either 0 or a prime number. We show that there are fields K_0 and K_1 of cardinality \lambda and characteristic p such that the automorphism group of K_0 is a free group of cardinality 2^\lambda and the automorphism group of K_1 is a free abelian group of cardinality 2^\lambda. This partially answers a question from [MR1736959] and complements results from [MR1934424], [MR2773054] and [MR1720580]. The methods developed in the proof of the above statement also allow us to show that the above cardinal arithmetic assumption is consistently not necessary for the existence of such fields and that it is necessary to use large cardinal assumptions to construct a model of set theory containing a cardinal \lambda of uncountable cofinality with the property that no free group of cardinality greater than \lambda is isomorphic to the automorphism group of a field of cardinality \lambda.
keywords: S: ico, O: alg
Sh:1015
Koszmider, P., & Shelah, S. (2013). Independent families in Boolean algebras with some separation properties. Algebra Universalis, 69(4), 305–312. arXiv: 1209.0177 DOI: 10.1007/s00012-013-0227-2 MR: 3061090
type: article
abstract: We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties which asserted the existence of an uncountable independent family. In fact we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone spaces K_{\mathcal A} of all such Boolean algebras \mathcal A contains a copy of the Čech-Stone compactification of the integers \beta\mathbb{N} and the Banach space C(K_{\mathcal A}) has l_\infty as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.
keywords: O: top, (ba)
Sh:1016
Laskowski, M. C., & Shelah, S. (2015). Borel completeness of some \aleph_0-stable theories. Fund. Math., 229(1), 1–46. arXiv: 1211.0558 DOI: 10.4064/fm229-1-1 MR: 3312114
type: article
abstract: We study \aleph_0-stable theories, and prove that if T either has eni-DOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of \lambda-Borel completeness and prove that such theories are \lambda-Borel complete. Using this, we conclude that an \aleph_0-stable theory satisfies I_{\infty, \aleph_0}(T, \lambda) = 2^\lambda for all cardinals \lambda if and only if T either has eni-DOP or is eni-deep.
keywords: M: cla, S: dst
Sh:1017
Shelah, S. (2014). Ordered black boxes: existence. Geombinatorics, 23(3), 108–126. arXiv: 1302.3426 MR: 3184377
type: article
abstract: We defined ordered black boxes in which for a partial J we try to predict just a bound in J to a function restricted to C_\alpha. The existence results are closely related to pcf, propagating downward. We can start with trivial cases.
keywords: S: ico
Sh:1020
Shelah, S., & Usvyatsov, A. (2019). Minimal stable types in Banach spaces. Adv. Math., 355, 106738, 29. arXiv: 1402.6513 DOI: 10.1016/j.aim.2019.106738 MR: 3994442
type: article
abstract: We prove existence of wide types in a continuous theory expanding a Banach space, and density of minimal wide types among stable types in such a theory. We show that every minimal wide stable type is “generically” isometric to an \ell_2 space. We conclude with a proof of the following formulation of Henson’s Conjecture: every model of an uncountably categorical theory expanding a Banach space is prime over a spreading model, isometric to the standard basis of a Hilbert space.
keywords: M: cla, O: top
Sh:1021
Mildenberger, H., & Shelah, S. (2021). The cofinality of the symmetric group and the cofinality of ultrapowers. Israel J. Math., 242(1), 97–128. DOI: 10.1007/s11856-021-2124-2 MR: 4282078
type: article
abstract: We prove that {\mathfrak{{mcf}}} < {\rm cf} ({\rm{Sym}(\omega)}) and {\mathfrak{{mcf}}} > {\rm cf}({\rm{Sym}(\omega)})= {\mathfrak{{b}}} are both consistent relative to ZFC. This answers a question from [BanakhRepovsZdomskyy] and from [MdSh:967].
keywords: S: for, S: str
Sh:1022
Rosłanowski, A., & Shelah, S. (2014). Around cofin. Colloq. Math., 134(2), 211–225. arXiv: 1304.5683 DOI: 10.4064/cm134-2-5 MR: 3194406
type: article
abstract: We show the consistency of “there is a nice \sigma–ideal I on the reals with add(I)=\omega_1 which cannot be represented as the union of a strictly increasing sequence of length \omega_1 of \sigma-subideals”. This answers a question by Borodulin–Nadzieja and Gła̧b.
keywords: S: for, S: str
Sh:1024
Kuhlmann, F.-V., Kuhlmann, K., & Shelah, S. (2015). Symmetrically complete ordered sets abelian groups and fields. Israel J. Math., 208(1), 261–290. arXiv: 1308.0780 DOI: 10.1007/s11856-015-1199-z MR: 3416920
type: article
abstract: We characterize and construct linealy ordered sets, abelian groups and fields that are symmetrically complete, meaning that the intersection over any chain of closed bounded intervals is nonempty. Such ordered abelian groups and fields are important because generalizations of Banach’s Fixed Point Theorem can be proved for them. We prove that symmetrically complete ordered abelian groups and fields are divisible Hahn products and real closed power series fields, respectively. We show how to extend any given ordered set, abelian group or field to one that is symmetrically complete. A main part of the paper establishes a detailed study of the cofinalities in cuts.
keywords: S: ico, O: alg
Sh:1025
Juhász, I., & Shelah, S. (2015). Strong colorings yield \kappa-bounded spaces with discretely untouchable points. Proc. Amer. Math. Soc., 143(5), 2241–2247. arXiv: 1307.1989 DOI: 10.1090/S0002-9939-2014-12394-X MR: 3314130
type: article
abstract: It is well-known that every non-isolated point in a compact Hausdorff space is the accumulation point of a discrete subset. Answering a question raised by Z. Szentmiklóssy and the first author. We show that this statement fails for countably compact regular spaces, and even for \omega-bounded regular spaces. In fact, there are \kappa-bounded counterexamples for every infinite cardinal \kappa. The proof makes essential use of the so-called strong colorings that were invented by the second author.
keywords: S: ico, O: top, (gt)
Sh:1026
Shelah, S. (2018). The spectrum of ultraproducts of finite cardinals for an ultrafilter. Acta Math. Hungar., 155(2), 201–220. arXiv: 1312.6780 DOI: 10.1007/s10474-018-0847-2 MR: 3831292
type: article
abstract: We complete the characterization of the possible spectrum of regular ultrafilters D on a set I, where the spectrum is the set of infinite cardinals which are ultra-products of finite cardinals modulo D
keywords: S: ico
Sh:1027
Shelah, S. (2019). The colouring existence theorem revisited. Acta Math. Hungar., 159(1), 1–26. arXiv: 1311.1026 DOI: 10.1007/s10474-019-00953-2 MR: 4003692
type: article
abstract: We prove a better colouring theorem for \aleph_3
keywords: S: ico
Sh:1028
Shelah, S. (2020). Quite free complicated Abelian groups, pcf and black boxes. Israel J. Math., 240(1), 1–64. arXiv: 1404.2775 DOI: 10.1007/s11856-020-2051-7 MR: 4193126
type: article
abstract: We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say \aleph_{\omega +1}-free, and on the other hand, are complicated in suitable sense. We choose as our test problem having no non-trivial homomorphism to \mathbb{Z} (known classically for \aleph_1-free, recently for \aleph_n-free). We succeed to prove the existence of even \aleph_{\omega_1 \cdot n}-free ones. This requires building n-dimensional black boxes, which are quite free. This combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardinals, we prove that it is consistent that every \aleph_{\omega_1 \cdot \omega}-free Abelian group has non-trivial homomorphisms to \mathbb{Z}.
keywords: S: ico, S: for, O: alg, O: odo, (ab), (stal), (large), (af)
Sh:1029
Shelah, S. (2016). No universal group in a cardinal. Forum Math., 28(3), 573–585. arXiv: 1311.4997 DOI: 10.1515/forum-2014-0040 MR: 3510831
type: article
abstract: For many classes of models, there are universal members in any cardinal \lambda which “essentially satisfies GCH, i.e. \lambda = 2^{< \lambda}", in particular for the class of a complete first order T (well, if at least if \lambda > |T|). But if the class is “complicated enough", e.g. the class of linear orders, we know that if \lambda is “regular and not so close to satisfying GCH" then there is no universal member. Here we find new sufficient conditions (which we call the olive property), not covered by earlier cases (i.e. fail the so-called \rm SOP_4). The advantage of those conditions is witnessed by proving that the class of groups satisfies one of those conditions.

This version has minor changes , compared to the earlier one.
keywords: M: non, S: ico
Sh:1030
Malliaris, M., & Shelah, S. (2016). Existence of optimal ultrafilters and the fundamental complexity of simple theories. Adv. Math., 290, 614–681. arXiv: 1404.2919 DOI: 10.1016/j.aim.2015.12.009 MR: 3451934
type: article
abstract: We characterize the simple theories in terms of saturation of ultrapowers. This gives a dividing line at simplicity in Keisler’s order and gives a true outside definition of simple theories. Specifically, for any \lambda \geq \mu \geq \theta \geq \sigma such that \lambda = \mu^+, \mu = \mu^{<\theta} and \sigma is uncountable and compact (natural assumptions given our prior work, which allow us to work directly with models), we define a family of regular ultrafilters on \lambda called optimal, prove that such ultrafilters exist and prove that for any \mathcal{D} in this family and any M with countable signature, M^\lambda/\mathcal{D} is \lambda^+-saturated if Th(M) is simple and M^\lambda/ \mathcal{D} is not \lambda^+-saturated if Th(M) is not simple. The proof lays the groundwork for a stratification of simple theories according to the inherent complexity of coloring, and gives rise to a new division of the simple theories: (\lambda, \mu, \theta)-explicitly simple.
keywords: M: cla
Sh:1031
Filipczak, T., Rosłanowski, A., & Shelah, S. On Borel hull operations. Real Anal. Exchange, 40(1), 129–140. arXiv: 1308.3749 http://projecteuclid.org/euclid.rae/1435759199 MR: 3365394
type: article
abstract: We show that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the \sigma–algebra of sets with the property of Baire.
keywords: S: ico, S: str
Sh:1032
Machura, M., Shelah, S., & Tsaban, B. (2016). The linear refinement number and selection theory. Fund. Math., 234(1), 15–40. arXiv: 1404.2239 DOI: 10.4064/fm124-8-2015 MR: 3509814
type: article
abstract: The linear refinement number \mathfrak{lr} is a combinatorial cardinal characteristic of the continuum. This number, which is a relative of the pseudointersection number \mathfrak{p}, showed up in studies of selective covering properties, that in turn were motivated by the tower number \mathfrak{t}.

It was long known that \mathfrak{p}=\min\{\mathfrak{t},\mathfrak{lr}\} and that \mathfrak{lr}\le\mathfrak{d}. We prove that if \mathfrak{lr}=\mathfrak{d} in all models where the continuum is \aleph_2, and that \mathfrak{lr} is not provably equal to any classic combinatorial cardinal characteristic of the continuum.

These results answer several questions from the theory of selection principles.
keywords: S: ico, S: str
Sh:1033
Cherlin, G. L., & Shelah, S. (2016). Universal graphs with a forbidden subgraph: block path solidity. Combinatorica, 36(3), 249–264. arXiv: 1404.5757 DOI: 10.1007/s00493-014-3181-5 MR: 3521114
type: article
abstract: Let C be a finite connected graph for which there is a countable universal C-free graph, and whose tree of blocks is a path. Then the blocks of C are complete. This generalizes a result of Füredi and Komjáth, and fits naturally into a set of conjectures regarding the existence of countable C-free graphs, with C an arbitrary finite connected graph.
keywords: M: odm
Sh:1034
Shelah, S., Veličković, B., & Väänänen, J. A. (2015). Positional strategies in long Ehrenfeucht-Fraïssé games. J. Symb. Log., 80(1), 285–300. arXiv: 1308.0156 DOI: 10.1017/jsl.2014.43 MR: 3320594
type: article
abstract: We prove that it is relatively consistent with {\rm ZF + CH} that there exist two models of cardinality \aleph_2 such that the second player has a winning strategy in the Ehrenfeucht-Fraı̈ssé-game of length \omega_1 but there is no \sigma-closed back-and-forth set for the two models. If {\rm CH} fails, no such pairs of models exist.
keywords: M: smt, S: for
Sh:1035
Chernikov, A., & Shelah, S. (2016). On the number of Dedekind cuts and two-cardinal models of dependent theories. J. Inst. Math. Jussieu, 15(4), 771–784. arXiv: 1308.3099 DOI: 10.1017/S1474748015000018 MR: 3569076
type: article
abstract: For an infinite cardinal \kappa, let {\rm ded}\kappa denote the supremum of the number of Dedekind cuts in linear orders of size \kappa. It is known that \kappa<{\rm ded}\kappa \leq2^{\kappa} for all \kappa and that {\rm ded}\kappa< 2^{\kappa} is consistent for any \kappa of uncountable cofinality. We prove however that 2^{\kappa}\leq{\rm ded}\left( {\rm ded}\left({\rm ded}\left({\rm ded} \kappa\right)\right)\right) always holds. Using this result we calculate the Hanf numbers for the existence of two-cardinal models with arbitrarily large gaps and for the existence of arbitrarily large models omitting a type in the class of countable dependent first-order theories. Specifically, we show that these bounds are as large as in the class of all countable theories.
keywords: M: odm, S: pcf
Sh:1036
Shelah, S. (2022). Forcing axioms for \lambda-complete \mu^+-c.c. MLQ Math. Log. Q., 68(1), 6–26. arXiv: 1310.4042 DOI: 10.1002/malq.201900020 MR: 4413641
type: article
abstract: We note that some form of the condition "p_1, p_2 have a \leq_{\mathbb{Q}}-lub in \mathbb{Q}" is necessary in some forcing axiom for \lambda-complete \mu^+-c.c. forcing notions. We also show some versions are really stronger than others.
keywords: S: for, (iter)
Sh:1037
Baldwin, J. T., Laskowski, M. C., & Shelah, S. (2016). Constructing many atomic models in \aleph_1. J. Symb. Log., 81(3), 1142–1162. arXiv: 1503.00318 DOI: 10.1017/jsl.2015.81 MR: 3569124
type: article
abstract: (none)
keywords: M: nec, M: non
Sh:1038
Shelah, S., & Spinas, O. (2015). Mad spectra. J. Symb. Log., 80(3), 901–916. arXiv: 1402.5616 DOI: 10.1017/jsl.2015.9 MR: 3395354
type: article
abstract: The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on \omega. We treat the problem to characterize those sets \mathcal A which, in some forcing extension of the universe, can be the mad spectrum. We solve this problem to some extent. What remains open is the possible values of min({\mathcal A}) and max({\mathcal A}).
keywords: S: for, S: str
Sh:1039
Greenberg, N., & Shelah, S. (2014). Models of Cohen measurability. Ann. Pure Appl. Logic, 165(10), 1557–1576. arXiv: 1309.3938 DOI: 10.1016/j.apal.2014.05.001 MR: 3226055
type: article
abstract: We show that in contrast with the Cohen version of Solovay’s model, it is consistent for the continuum to be Cohen-measurable and for every function to be continuous on a non-meagre set.
keywords: S: for, S: str
Sh:1040
Garti, S., Magidor, M., & Shelah, S. (2018). On the spectrum of characters of ultrafilters. Notre Dame J. Form. Log., 59(3), 371–379. arXiv: 1601.01409 DOI: 10.1215/00294527-2018-0006 MR: 3832086
type: article
abstract: We show that the character spectrum {\rm Sp}_\chi(\lambda) (for a singular cardinal \lambda of countable cofinality) may include any prescribed set of regular cardinals between \lambda and 2^\lambda.

Nous prouvons que {\rm Sp}_\chi(\lambda) (par un cardinal singulier \lambda avec cofinalitè nombrable) peut comporter tout l’ensemble prescrit de cardinaux reguliers entre \lambda et 2^\lambda.
keywords: S: ico, S: for
Sh:1041
Bagaria, J., & Shelah, S. (2016). On partial orderings having precalibre-\aleph_1 and fragments of Martin’s axiom. Fund. Math., 232(2), 181–197. arXiv: 1502.05500 DOI: 10.4064/fm232-2-6 MR: 3418888
type: article
abstract: We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-\aleph_1, and show that Martin’s axiom restricted to the class of partial orderings that have the property does not imply Martin’s axiom for \sigma-linked partial orderings. This yields a new solution to an old question of the first author about the relative strength of Martin’s axiom for \sigma-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer a question of J. Steprans and S. Watson (1988) by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-\aleph_1 property of a partial ordering while preserving its ccc-ness.
keywords: S: for
Sh:1042
Farah, I., & Shelah, S. (2016). Rigidity of continuous quotients. J. Inst. Math. Jussieu, 15(1), 1–28. arXiv: 1401.6689 DOI: 10.1017/S1474748014000218 MR: 3427592
type: article
abstract: The assertion that the Čech–Stone remainder of the half-line has only trivial automorphisms is independent from ZFC. The consistency of this statement follows from Proper Forcing Axiom and this is the first known example of a connected space with this property. The existence of 2^{\aleph_1} autohomeomorphisms under the Continuum Hypothesis follows from a general model-theoretic fact. We introduce continuous fields of metric models and prove countable saturation of the corresponding reduced products. We also provide an (overdue) proof that the reduced products of metric models corresponding to the Fréchet ideal are countably saturated. This provides an explanation of why the asymptotic sequence C*- algebras and the central sequence C*-algebras are as useful as ultrapowers.
keywords: S: for, S: str
Sh:1043
Shelah, S. (2020). Superstable theories and representation. Sarajevo J. Math., 16(29)(1), 71–82. arXiv: 1412.0421 DOI: 10.5644/sjm MR: 4144091
type: article
abstract: In this paper we give characterizations of the super-stable theories, in terms of an external property called representation. In the sense of the representation property, the mentioned class of first-order theories can be regarded as “not very complicated"
keywords: M: cla
Sh:1044
Fischer, A. J., Goldstern, M., Kellner, J., & Shelah, S. (2017). Creature forcing and five cardinal characteristics in Cichoń’s diagram. Arch. Math. Logic, 56(7-8), 1045–1103. arXiv: 1402.0367 DOI: 10.1007/s00153-017-0553-8 MR: 3696076
type: article
abstract: We use a (countable support) creature construction to show that consistently \mathfrak{d} = \aleph_1 = cov(\mathcal{N}) < non (\mathcal{M}) < non (\mathcal{N}) < cof(\mathcal{N}) < 2^{\aleph_0}.
keywords: S: for, S: str
Sh:1047
Garti, S., & Shelah, S. (2018). Random reals and polarized colorings. Studia Sci. Math. Hungar., 55(2), 203–212. arXiv: 1609.00242 DOI: 10.1556/012.2018.55.2.1393 MR: 3813351
type: article
abstract: We prove that if \mathfrak{r}=\mathfrak{c} then \mathfrak{s}\leq\textrm{cf}(\mathfrak{c}), and conclude that if \mathfrak{r}=\mathfrak{s}=\mathfrak{c} then \mathfrak{s} is a regular cardinal. Nous prouvons que \mathfrak{s}\leq\textrm{cf}(\mathfrak{c}) si \mathfrak{r}=\mathfrak{c}, et conclurons que \mathfrak{s} est un cardinal regulier si \mathfrak{r}=\mathfrak{s}= \mathfrak{c}.
keywords: S: for, S: str
Sh:1048
Shelah, S. (2020). The Hanf number in the strictly stable case. MLQ Math. Log. Q., 66(3), 280–294. arXiv: 1412.0428 DOI: 10.1002/malq.201900021 MR: 4174105
type: article
abstract: Suppose \mathbf {t} = (T, T_1, p) is a triple of two theories in vocabularies \tau \subset \tau_1 of cardinality \lambda and a \tau_1-type p over the empty set: here we fix T and assume it is stable. We show the Hanf number for the property: “there is a model M_1 of T_1 which omits p, but M_1\restriction \tau is saturated" is larger than the Hanf number of L_{\lambda^+, \kappa} but smaller than the Hanf number of L_{(2^\lambda)^+, \kappa} when T is stable with \kappa = \kappa(T).
keywords: M: cla, M: smt
Sh:1050
Malliaris, M., & Shelah, S. (2018). Keisler’s order has infinitely many classes. Israel J. Math., 224(1), 189–230. arXiv: 1503.08341 DOI: 10.1007/s11856-018-1647-7 MR: 3799754
type: article
abstract: We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler’s order. Thus Keisler’s order is infinite and not a well order. Moreover, this chain occurs within the simple unstable theories, considered model-theoretically tame. Keisler’s order is a large scale classification program in model theory, introduced in the 1960s, which compares the complexity of theories. Prior to this paper, it was thought to have finitely many classes, linearly ordered. The model-theoretic complexity we find is witnessed by a very natural class of theories, the n-free k-hypergraphs studied by Hrushovski. Notably, this complexity reflects the difficulty of amalgamation and appears orthogonal to forking.
keywords: M: cla, S: ico
Sh:1051
Malliaris, M., & Shelah, S. (2017). Model-theoretic applications of cofinality spectrum problems. Israel J. Math., 220(2), 947–1014. arXiv: 1503.08338 DOI: 10.1007/s11856-017-1526-7 MR: 3666452
type: article
abstract: We apply the recently developed technology of cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is \lambda-saturated iff it has cofinality \geq \lambda and the underlying order has no (\kappa, \kappa)-gaps for regular \kappa < \lambda. We also answer a question about balanced pairs of models of PA. Second, assuming instances of GCH, we prove that SOP_2 characterizes maximality in the interpretability order \triangleleft^*, settling a prior conjecture and proving that SOP_2 is a real dividing line. Third, we establish the beginnings of a structure theory for NSOP_2, proving that NSOP_2 can be characterized by the existence of few so-called higher formulas. In the course of the paper, we show that \mathfrak{p}_{\mathbf{s}} = \mathfrak{t}_{\mathbf{s}} in any weak cofinality spectrum problem closed under exponentiation (naturally defined). We also prove that the local versions of these cardinals need not coincide, even in cofinality spectrum problems arising from Peano arithmetic. *2015.02.12 The new note from Feb on the delayed lemma make no sense:- the finite satisfiability ensure the formulas are compatible. Look back at the old.. The second theorem which is delayed (union of few pairwise non-contradictory) - a debt, look again
keywords: M: cla, M: odm
Sh:1052
Shelah, S. (2016). Lower bounds on coloring numbers from hardness hypotheses in pcf theory. Proc. Amer. Math. Soc., 144(12), 5371–5383. arXiv: 1503.02423 DOI: 10.1090/proc/13163 MR: 3556279
type: article
abstract: We prove that the statement “for every infinite cardinal \nu, every graph with list chromatic \nu has coloring number at most \beth_\omega(\nu)" proved by Kojman [koj] using the RGCH theorem [sh:460] implies the RGCG theorem via a short forcing argument. By the same method, a better upper bound than \beth_\omega(\nu) in this statement implies stronger forms of the RGCH theorem whose consistency as well as the consistency of their negations are wide open. Thus, the optimality of Kojman’s upper bound is a purely cardinal arithmetic problem, which, as discussed below, may be quite hard to decide.
keywords: S: for, S: pcf
Sh:1053
Shelah, S., & Wohofsky, W. (2016). There are no very meager sets in the model in which both the Borel conjecture and the dual Borel conjecture are true. MLQ Math. Log. Q., 62(4-5), 434–438. DOI: 10.1002/malq.201600002 MR: 3549562
type: article
abstract: We show that the model in [GoKrShWo:969] (for the simultaneous consistency of the Borel Conjecture and the dual Borel Conjecture) actually satisfies a stronger version of the dual Borel Conjecture: there are no uncountable very meager sets.
keywords: S: for, S: str
Sh:1054
Kaplan, I., & Shelah, S. (2016). Forcing a countable structure to belong to the ground model. MLQ Math. Log. Q., 62(6), 530–546. arXiv: 1410.1224 DOI: 10.1002/malq.201400094 MR: 3601093
type: article
abstract: Suppose that P is a forcing notion, L is a language (in \mathbb{V}), \dot{\tau} a P-name such that P forces “\dot{\tau} is a countable L-structure”. In the product P\times P, there are names \dot{\tau_{1}},\dot{\tau_{2}} such that for any generic filter G=G_{1}\times G_{2} over P\times P, \dot{\tau}_{1}\left[G\right]= \dot{\tau}\left[G_{1}\right] and \dot{\tau}_{2}\left[G\right]=\dot{\tau}\left[G_{2}\right]. Zapletal asked whether or not P\times P forces \dot{\tau}_{1} \cong\dot{\tau}_{2} implies that there is some M\in\mathbb{V} such that P forces \dot{\tau}\cong\check{M}. We answer this negatively and discuss related issues.
keywords: M: odm
Sh:1055
Kaplan, I., Lavi, N., & Shelah, S. (2016). The generic pair conjecture for dependent finite diagrams. Israel J. Math., 212(1), 259–287. arXiv: 1410.2516 DOI: 10.1007/s11856-016-1286-9 MR: 3504327
type: article
abstract: We consider a (\mathbf D,\bar\kappa)-homomorphism, \mathbf D a finite diagram. Another case is fixing a compact cardinal \theta, we look at (\bar\kappa,\mathcal{L})-saturated monster \mathfrak{C}. Our intention is to generalize [Sh:900].
keywords: M: nec
Sh:1056
Bartoszyński, T., Larson, P. B., & Shelah, S. (2017). Closed sets which consistently have few translations covering the line. Fund. Math., 237(2), 101–125. DOI: 10.4064/fm191-8-2016 MR: 3615047
type: article
abstract: We characterize those compact subsets K of 2^{\omega} for which one can force the existence of a set X of cardinality less than the continuum such that K+X=2^{\omega}.
keywords: S: for, S: str
Sh:1057
Dow, A. S., & Shelah, S. (2018). Asymmetric tie-points and almost clopen subsets of \mathbb N^*. Comment. Math. Univ. Carolin., 59(4), 451–466. arXiv: 1801.02523 MR: 3914712
type: article
abstract: A tie-point of compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. We review some of the many consistency results that have depended on the construction of tie-points of \mathbb{N}^*. One especially important application, due to Velickovic, was to the existencce of non-trivial involutions on \mathbb{N}^*. A tie-point \mathbb{N}^* has been called symmetric if it is the unique fixed point of an involution. We define the notion of almost clopen set to be the closure of one of the proper relatively clopen subsets of the complement of a tie-point. We explore asymmetries of almost clopen subsets of \mathbb{N}^* in the sense of how may an almost clopen set differ from its natural complementary almost clopen set.
keywords: S: for, O: top
Sh:1058
Raghavan, D., & Shelah, S. (2017). On embedding certain partial orders into the P-points under Rudin-Keisler and Tukey reducibility. Trans. Amer. Math. Soc., 369(6), 4433–4455. arXiv: 1411.0084 DOI: 10.1090/tran/6943 MR: 3624416
type: article
abstract: The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for \sigma-centered posets. In his 1973 paper he showed under this assumption that both {\omega}_{1} and the reals can be embedded. This result was later repeated for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for \sigma-centered posets implies that every partial order of size at most continuum can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility.
keywords: S: for, S: str
Sh:1059
Haber, S., & Shelah, S. (2015). An extension of the Ehrenfeucht-Fraïssé game for first order logics augmented with Lindström quantifiers. In Fields of logic and computation. II, Vol. 9300, Springer, Cham, pp. 226–236. arXiv: 1510.06581 DOI: 10.1007/978-3-319-23534-9_13 MR: 3485648
type: article
abstract: We propose an extension of the Ehrenfeucht-Fraı̈sse game able to deal with logics augmented with Lindström quantifiers. We describe three different games with varying balance between simplicity and ease of use.
keywords: M: odm, O: fin, (fmt)
Sh:1060
Raghavan, D., & Shelah, S. (2017). Two inequalities between cardinal invariants. Fund. Math., 237(2), 187–200. arXiv: 1505.06296 DOI: 10.4064/fm253-7-2016 MR: 3615051
type: article
abstract: We prove two ZFC inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of \omega of aymptotic density 0. We obtain an upper bound on the \ast-covering number, sometimes also called the weak covering number, of this ideal by proving in Section 2 that cov{\ast} (\mathcal{Z}_0) \leq \mathfrak{d}.

In Section 3 we investigate the relationship between the bounding and splitting numbers at regular uncountable cardinals. We prove in sharp contrast to the case when \kappa = \omega, that if \kappa is any regular uncountable cardinal, then \mathfrak{s}_\kappa \leq \mathfrak{b}_\kappa.
keywords: S: str
Sh:1061
Shelah, S. (2015). On failure of 0-1 laws. In Fields of logic and computation. II, Vol. 9300, Springer, Cham, pp. 293–296. arXiv: 2108.03846 DOI: 10.1007/978-3-319-23534-9_18 MR: 3485653
type: article
abstract: Let \alpha \in (0,1)_{\mathbb{R}} be irrational and G_n = G_{n,1/n^\alpha} be the random graph with edge probability 1/n^\alpha; we know that it satisfies the 0-1 law for first order logic. We deal with the failure of the 0-1 law for stronger logics: \mathbb{L}_{\infty,k},k large enough and the inductive logic.
keywords: M: odm, O: fin, (fmt)
Sh:1062
Shelah, S. (2017). Failure of 0-1 law for sparse random graph in strong logics (Sh1062). In Beyond first order model theory, CRC Press, Boca Raton, FL, pp. 77–101. arXiv: 1706.01226 MR: 3729324
type: article
abstract: Let \alpha \in (0,1)_{\mathbb{R}} be irrational and G_n = G_{n,1/n^\alpha} be the random graph with edge probability 1/n^\alpha; we know that it satisfies the 0-1 law for first order logic. We deal with the failure of the 0-1 law for stronger logics: \mathbb{L}_{\infty,k},k large enough and the inductive logic.
keywords: M: odm, O: fin, (fmt)
Sh:1063
Kumar, A., & Shelah, S. (2018). Clubs on quasi measurable cardinals. MLQ Math. Log. Q., 64(1-2), 44–48. DOI: 10.1002/malq.201600003 MR: 3803065
type: article
abstract: We construct a model satisfying \kappa < 2^{\aleph_0} + \clubsuit_{\kappa} +\kappa is quasi measurable. Here, we call \kappa quasi measurable if there is an \aleph_1-saturated \kappa-additive ideal \mathcal{I} over \kappa. We also show that, in this model, forcing with \mathcal{P}(\kappa)/ \mathcal{I} adds one but not \kappa Cohen reals. We introduce a weak club principle and use it to show that, consistently, for some \aleph_{\mathcal{1}}-saturated \kappa-additive ideal \mathcal{I} over \kappa, forcing with P(\kappa) /I adds one but not \kappa random reals.
keywords: S: for, S: str
Sh:1064
Shelah, S. (2021). Atomic saturation of reduced powers. MLQ Math. Log. Q., 67(1), 18–42. arXiv: 1601.04824 DOI: 10.1002/malq.201900006 MR: 4313125
type: article
abstract: 2014.1.24 We continue [1019] looking at infinitary logics but now for filters instead ultrafilters. Earlier this month has written §1 which generalize [Sh:c,VI,2.6], so do not depend on \theta being a compact cardinal, (this apply also to \theta = {\aleph_0} but this is marginal here) In §2 we deal with how to generalize “good ultrafilter” Intentions: (A) we hope to sort out the versions of cp implicit in §2. (B) Even assume T has \theta-cp, can we sort out the maximal T? (C) Recall the problem of building a good filter of \lambda such that the quotient is a complete BA free enough failing chain condition 2014.1.26 [here or F1396] Noted (Friday) that if a model is an ultrapower by a normal ultrafilter on \theta, then looking at all sequences of \theta formulas, we can colour by 2^ \theta colours, such that the colour of a branch determine if the type is realized. THIS is one side on the other SIDE we start with a strong limit \mu of cofinality \theta and a model of cardinality \mu which embed much, then build a directed system of sub-models, indexed by subsets of \lambda each of cardinality at most \theta (<)-directed, bounded below all members of a given stationary subset of \lambda of cofinality \theta.

After publication a full proof was added of what was conclusion 3.10 in the published version and is here 2.10.
keywords: M: cla
Sh:1066
Goldstern, M., Mejı́a, D. A., & Shelah, S. (2016). The left side of Cichoń’s diagram. Proc. Amer. Math. Soc., 144(9), 4025–4042. arXiv: 1504.04192 DOI: 10.1090/proc/13161 MR: 3513558
type: article
abstract: Using a finite support iteration of ccc forcings, we construct a model of \aleph_1<{\textrm{add}}({\mathcal{N}})< \textrm{cov} ({\mathcal{N}})<\mathfrak{b}<\textrm{non} (\mathcal{M})<\textrm{cov}(\mathcal{M}) =\mathfrak{c}.
keywords: S: for, S: str
Sh:1068
Kumar, A., & Shelah, S. (2017). A transversal of full outer measure. Adv. Math., 321, 475–485. DOI: 10.1016/j.aim.2017.10.008 MR: 3715717
type: article
abstract: We prove that for any set of real and equivalence relations on it such that every equivalence class is countable, there is a transversal (= a set of representations of the equivalence relations, i.e. having exactly one member in each equivalence class) with the same outer measure.
keywords: S: str
Sh:1069
Malliaris, M., & Shelah, S. (2017). Open problems on ultrafilters and some connections to the continuum. In Foundations of mathematics, Vol. 690, Amer. Math. Soc., Providence, RI, pp. 145–159. MR: 3656310
type: article
abstract: We discuss a range of open problems at the intersection of set theory, model theory, and general topology, mainly around the construction of ultrafilters. Along the way we prove uniqueness for a weak notion of cut.
keywords: S: ico
Sh:1070
Malliaris, M., & Shelah, S. (2016). Cofinality spectrum problems: the axiomatic approach. Topology Appl., 213, 50–79. DOI: 10.1016/j.topol.2016.08.019 MR: 3563070
type: article
abstract: Let X be a set of definable linear or partial orders in some model. We say that X has uniqueness below some cardinal \mathfrak{t}_* if for any regular \kappa < \mathfrak{t}_*, any two increasing \kappa-indexed sequences in any two orders of X have the same co-initiality. Motivated by recent work, we investigate this phenomenon from several interrelated points of view. We define the lower-cofinality spectrum for a regular ultrafilter \mathcal{D} on \lambda and show that this spectrum may consist of a strict initial segment of cardinals below \lambda and also that it may finitely alternate. We connect these investigations to a question of Dow on the conjecturally nonempty (in ZFC) region of OK but not good ultrafilters, by introducing the study of so-called ‘automorphic ultrafilters’ and proving that the ultrafilters which are automorphic for some, equivalently every, unstable theory are precisely the good ultrafilters. Finally, we axiomatize a general context of “lower cofinality spectrum problems”, a bare-bones framework consisting essentially of a single tree projecting onto two linear orders. We prove existence of a lower cofinality function in this context show that this framework holds of theories which are substantially less complicated than Peano arithmetic, the natural home of cofinality spectrum problems. Along the way we give new analogues of several open problems.
keywords: M: cla
Sh:1071
Shelah, S., & Steprāns, J. (2016). When automorphisms of \mathcal P(\kappa)/[\kappa]^{<\aleph_0} are trivial off a small set. Fund. Math., 235(2), 167–182. DOI: 10.4064/fm222-2-2016 MR: 3549381
type: article
abstract: It is shown that if \kappa > 2^{\aleph_0} and \kappa is less than the first inaccessible cardinal then every automorphism of \mathcal{P} (\kappa)/[\kappa]^{<\aleph_0} is trivial outside of a set of cardinality 2^{\aleph_0}.
keywords: S: ico
Sh:1072
Mohsenipour, S., & Shelah, S. (2018). Set mappings on 4-tuples. Notre Dame J. Form. Log., 59(3), 405–416. arXiv: 1510.02216 DOI: 10.1215/00294527-2018-0002 MR: 3832089
type: article
abstract: In this paper we study set mappings on 4-tuples. We continue a previous work of Komjath and Shelah by getting new finite bounds on the size of free sets. This is obtained by an entirely different forcing construction. Moreover we prove two ZFC results for set mappings on 4-tuples and also as another application of our forcing construction we give a consistency result for set mappings on triples.
keywords: S: ico, S: for
Sh:1074
Kaplan, I., Shelah, S., & Simon, P. (2017). Exact saturation in simple and NIP theories. J. Math. Log., 17(1), 1750001, 18. arXiv: 1510.02741 DOI: 10.1142/S0219061317500015 MR: 3651210
type: article
abstract: A theory T is said to have exact saturation at a singular cardinal \kappa if it has a \kappa-saturated model which is not \kappa^+-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.
keywords: M: cla
Sh:1075
Golshani, M., & Shelah, S. (2016). On cuts in ultraproducts of linear orders I. J. Math. Log., 16(2), 1650008, 34. arXiv: 1510.06278 DOI: 10.1142/S0219061316500082 MR: 3580893
type: article
abstract: For an ultrafilter D on cardinal \kappa, we wonder for which pair (\theta_1, \theta_2) of regular cardinals, we have: for any (\theta_1 + \theta_2)^+-saturated dense linear order J,J^\kappa/D has a cut of cofinality (\theta_1, \theta_2). We deal mainly with the case \theta_1, \theta_2 > 2^\kappa.
keywords: S: ico
Sh:1076
Larson, P. B., & Shelah, S. (2017). Coding with canonical functions. MLQ Math. Log. Q., 63(5), 334–341. DOI: 10.1002/malq.201500060 MR: 3748478
type: article
abstract: A function f from \omega_{1} to the ordinals is called a canonical function for an ordinal \alpha if f represents \alpha in any generic ultrapower induced by forcing with \mathcal{P}(\omega_{1})/\mathrm{NS}_{\omega_{1}}. We introduce here a method for coding sets of ordinals using canonical functions from \omega_{1} to \omega_{1}. Combining this approach with arguments from [Sh:f], we show that for each cardinal \kappa there is a forcing construction preserving cardinalities and cofinalities forcing that every subset of \kappa is in the inner model L(\mathcal{P}(\omega_{1})).
keywords: S: ico, S: for
Sh:1078
Kumar, A., & Shelah, S. (2017). On a question about families of entire functions. Fund. Math., 239(3), 279–288. DOI: 10.4064/fm252-3-2017 MR: 3691208
type: article
abstract: We show that the existence of a continuum sized family mathcal{F} of entire functions such that for each complex number z, the set \{ f(z): f \in \mathcal{F}\} has size less than continuum is undecidable in ZFC plus the negation of CH
keywords: S: for, S: str
Sh:1079
Kumar, A., & Shelah, S. (2017). Avoiding equal distances. Fund. Math., 236(3), 263–267. DOI: 10.4064/fm169-5-2016 MR: 3600761
type: article
abstract: We show that it is consistent that there is a non meager set of reals X each of whose non meager subsets contains equal distances.
keywords: S: for, S: str
Sh:1080
Komjáth, P., & Shelah, S. (2017). Consistently \mathcal P(\omega_1) is the union of less than 2^{\aleph_1} strongly independent families. Israel J. Math., 218(1), 165–173. DOI: 10.1007/s11856-017-1463-5 MR: 3625129
type: article
abstract: It is consistent that \mathcal{P}(\omega_1) is the union of less than 2^{\aleph_1} parts such that if A_0,\dots,A_{n-1},B_0,\dots,B_{m-1} are distinct elements of the same part then |A_0\cap\cdots \cap A_{n-1}\cap (\omega_1-B_0)\cap\cdots \cap(\omega_1-B_{m-1})|=\aleph_1.
keywords: S: for
Sh:1081
Rosłanowski, A., & Shelah, S. (2018). Small-large subgroups of the reals. Math. Slovaca, 68(3), 473–484. arXiv: 1605.02261 DOI: 10.1515/ms-2017-0117 MR: 3805955
type: article
abstract: We are interested in subgroups of the reals that are small in one and large in another sense. We prove that, in ZFC, there exists a non–meager Lebesgue null subgrooup of {\mathbb R}, while it is consistent there there is no non–null meager subgroup of {\mathbb R}.
keywords: S: str
Sh:1082
Kaplan, I., & Shelah, S. (2017). Decidability and classification of the theory of integers with primes. J. Symb. Log., 82(3), 1041–1050. arXiv: 1601.07099 DOI: 10.1017/jsl.2017.16 MR: 3694340
type: article
abstract: We show that under Dickson’s conjecture about the distribution of primes in the natural numbers, the theory Th\left(\mathbb{Z},+,1,0, Pr\right) where Pr is a predicate for the prime numbers and their negations is decidable, unstable and supersimple. This is in contrast with Th\left(\mathbb{Z},+,0,Pr,<\right) which is known to be undecidable by the works of Jockusch, Bateman and Woods.
keywords: M: odm
Sh:1083
Garti, S., Hayut, Y., & Shelah, S. (2017). On the verge of inconsistency: Magidor cardinals and Magidor filters. Israel J. Math., 220(1), 89–102. arXiv: 1601.07745 DOI: 10.1007/s11856-017-1510-2 MR: 3666820
type: article
abstract: We introduce a model-theoretic characterization of Magidor cardinals, from which we infer that Magidor filters are beyond ZFC-inconsistency
keywords: S: for
Sh:1084
Barnea, I., & Shelah, S. (2018). The abelianization of inverse limits of groups. Israel J. Math., 227(1), 455–483. arXiv: 1608.02220 DOI: 10.1007/s11856-018-1741-x MR: 3846331
type: article
abstract: (none)
keywords: S: ico, O: alg
Sh:1085
Cohen, S., & Shelah, S. (2019). Generalizing random real forcing for inaccessible cardinals. Israel J. Math., 234(2), 547–580. arXiv: 1603.08362 DOI: 10.1007/s11856-019-1925-z MR: 4040837
type: article
abstract: The two parallel concepts of “small" sets of the real line are meagre sets and null sets. Those are equivalent to Cohen forcing and Random real forcing for \aleph^{\aleph_0}_0; in spite of this similarity, the Cohen forcing and Random Real forcing have very different shapes. One of these differences is in the fact that the Cohen forcing has an easy natural generalization for \lambda^{\lambda} while \lambda > \aleph_0, corresponding to an extension for the meagre sets, while the Random real forcing didn’t see to have a natural generalization, as Lebesgue measure doesn’t have a generalization for space \lambda^\lambda while \lambda > \aleph_0. In work [1], Shelah found a forcing resembling the properties of Random Real Forcing for \lambda^\lambda while \lambda is a weakly compact cardinal. Here we describe, with additional assumptions, such a forcing for \lambda^\lambda while \lambda is an inaccessible cardinal; this forcing preserves cardinals and cofinalities, however unlike Cohen forcing, does not add undominated reals.
keywords: S: for
Sh:1086
Koszmider, P., Shelah, S., & Świȩtek, M. (2018). There is no bound on sizes of indecomposable Banach spaces. Adv. Math., 323, 745–783. arXiv: 1603.01753 DOI: 10.1016/j.aim.2017.11.002 MR: 3725890
type: article
abstract: Assuming the generalized continuum hypothesis we construct arbitrarily big indecomposable Banach spaces. i.e., such that whenever they are decomposed as X\oplus Y, then one of the closed subspaces X or Y must be finite dimensional. It requires alternative techniques compared to those which were initiated by Gowers and Maurey or Argyros with the coauthors. This is because hereditarily indecompo- sable Banach spaces always embed into \ell_\infty and so their density and cardinality is bounded by the continuum and because dual Banach spaces of densities bigger than continuum are decomposable by a result due to Heinrich and Mankiewicz. The obtained Banach spaces are of the form C(K) for some compact connected Hausdorff space and have few operators in the sense that every linear bounded operator T on C(K) for every f\in C(K) satisfies T(f)=gf+S(f) where g\in C(K) and S is weakly compact or equivalently strictly singular. In particular, the spaces carry the structure of a Banach algebra and in the complex case even the structure of a C^*-algebra.
keywords: O: top
Sh:1087
Golshani, M., & Shelah, S. (2018). On cuts in ultraproducts of linear orders II. J. Symb. Log., 83(1), 29–39. arXiv: 1604.06044 DOI: 10.1017/jsl.2017.87 MR: 3796271
type: article
abstract: We continue our study of the class \mathcal{C}(D), where D is a uniform ultrafilter on a cardinal \kappa and \mathcal{C}(D) is the class of all pairs (\theta_1, \theta_2), where (\theta_1, \theta_2) is the cofinality of a cut in J^\kappa/D and J is some (\theta_1 + \theta_2)^+-saturated dense linear order. We show that if (\theta_1, \theta_2) \in \mathcal{C} (D) and D is \aleph_1-complete or \theta_1 + \theta_2 > 2^\kappa, then \theta_1 = \theta_2.
keywords: S: ico
Sh:1088
Shelah, S., & Steprāns, J. (2021). Universal graphs and functions on \omega_1. Ann. Pure Appl. Logic, 172(8), Paper No. 102986, 43. DOI: 10.1016/j.apal.2021.102986 MR: 4266242
type: article
abstract: It is shown to be consistent with various values of \mathfrak{b} and \mathfrak{d} that there is a universal graph on \omega_1. Moreover, it is also shown that it is consistent that there is a ’ universal graph on \omega_1 - in other words, a universal symmetric function from \omega^2_1 to 2 – but no such function from \omega^2_1 to \omega. The method used relies on iterating well know reals, such as Miller and Laver reals, and alternating this with the PID forcing which adds no new reals.
keywords: S: ico, S: for
Sh:1089
Horowitz, H., & Shelah, S. (2023). A Borel maximal eventually different family. Ann. Pure Appl. Logic, (175). arXiv: 1605.07123 DOI: 10.106/j.apal.2023.103334
type: article
abstract: (none)
keywords: S: ico, S: str
Sh:1090
Horowitz, H., & Shelah, S. (2019). On the non-existence of mad families. Arch. Math. Logic, 58(3-4), 325–338. DOI: 10.1007/s00153-018-0640-5 MR: 3928385
Contains [Sh:E95a], [Sh:E95b]
type: article
abstract: We show that the non-existence of mad families is equiconsistent with ZFC, answering an old question of Mathias. We also consider the above result in the general context of maximal independent sets in Borel graphs, and we construct a Borel graph G such that ZF + DC+ ’there is no maximal independent set in G’ is equiconsistent with ZFC+ ’there exists an inaccessible cardinal’.
keywords: S: for, S: str, S: dst
Sh:1091
Dugas, M. H., Herden, D., & Shelah, S. (2017). An extension of M. C. R. Butler’s theorem on endomorphism rings. In Groups, modules, and model theory—surveys and recent developments, Springer, Cham, pp. 277–284. MR: 3675912
type: article
abstract: (none)
keywords: S: ico, O: alg
Sh:1092
Baldwin, J. T., & Shelah, S. (2022). Hanf numbers for extendibility and related phenomena. Arch. Math. Logic, 61(3-4), 437–464. arXiv: 2111.01704 DOI: 10.1007/s00153-021-00796-1 MR: 4418753
type: article
abstract: In this paper we discuss two theorems whose proofs depend on extensions of the Fraissé method. We prove the Hanf number for the existence of an extendible model (has a proper extension in the class. Here, this means an \infty,\omega-elementary extension) of a (complete) sentence of L_{\omega_1, \omega} is (modulo some mild set theoretic hypotheses that we expect to remove in a later paper) the first measurable cardinal. And we outline the description on an explicit L_{\omega_1, \omega}-sentence \phi_n characterizing \aleph_n for each n. We provide some context for these developments as outlined in the lectures at IPM.
keywords: M: smt
Sh:1093
Horowitz, H., & Shelah, S. (2021). Transcendence bases, well-orderings of the reals and the axiom of choice. Proc. Amer. Math. Soc., 149(2), 851–858. arXiv: 1901.01508 DOI: 10.1090/proc/15242 MR: 4198089
type: article
abstract: We prove that ZF+DC+"there exists a transcendence basis for the reals"+"there is no well-ordering of the reals" is consistent relative to ZFC. This answers a question of Larson and Zapletal.
keywords: S: for, S: str
Sh:1095
Horowitz, H., & Shelah, S. (2023). A Borel maximal cofinitary group. J. Symbolic Logic. arXiv: 1610.01344
type: article
abstract: We construct a Borel maximal cofinitary group
keywords: S: str, S: dst
Sh:1099
Laskowski, M. C., & Shelah, S. (2019). A strong failure of \aleph_0-stability for atomic classes. Arch. Math. Logic, 58(1-2), 99–118. arXiv: 1701.05474 DOI: 10.1007/s00153-018-0623-6 MR: 3902807
type: article
abstract: We study classes of atomic models At_T of a countable, complete first-order theory T . We prove that if At_T is not pcl-small, i.e., there is an atomic model N that realizes uncountably many types over pcl(a) for some finite tuple a from N, then there are 2^aleph1 non-isomorphic atomic models of T, each of size aleph_1.
keywords: M: nec, M: non, S: str, (aec)
Sh:1101
Shelah, S. (2021). Isomorphic limit ultrapowers for infinitary logic. Israel J. Math., 246(1), 21–46. arXiv: 1810.12729 DOI: 10.1007/s11856-021-2226-x MR: 4358271
type: article
abstract: The logic \mathbb{L} ^1_\theta was introduced in [Sh:797]; it is the maximal logic below \mathbb{L} _{\theta,\theta} in which a well ordering is not definable. We investigate it for \theta a compact cardinal. We prove that it satisfies several parallel of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are \mathbb{L} ^1_\theta-equivalent iff for some \omega-sequence of \theta-complete ultrafilters, the iterated ultra-powers by it of those two models are isomorphic.

Also for strong limit \lambda > \theta of cofinality \aleph_0, every complete \mathbb{L} ^1_\theta-theory has a so called a special model of cardinality \lambda, a parallel of saturated. For first order theory T and singular strong limit cardinal \lambda , T has a so called special model of cardinality \lambda. Using “special" in our context is justified by: it is unique (fixing T and \lambda), all reducts of a special model are special too, so we have another proof of interpolation in this case.

This was earlier section 3 of [Sh:1019].
keywords: M: smt, (inf-log), (large)
Sh:1102
Kumar, A., & Shelah, S. (2019). On possible restrictions of the null ideal. J. Math. Log., 19(2), 1950008, 14. DOI: 10.1142/S0219061319500089 MR: 4014888
type: article
abstract: We prove that the null ideal restricted to a non null set of reals could be isomorphic to a variety of sigma ideals. Using this, we show that the following are consistent: (1) There is a non null subset of plane each of whose non null subsets contains three collinear points. (2) There is a partition of a non null set of reals into null sets, each of size \aleph_1, such that every transversal of this partition is null.
keywords: S: ico, S: str, (leb)
Sh:1104
Kumar, A., & Shelah, S. (2019). Saturated null and meager ideal. Trans. Amer. Math. Soc., 371(6), 4475–4491. DOI: 10.1090/tran/7702 MR: 3917229
type: article
abstract: We prove that the meager ideal and the null ideal could both be somewhere \aleph_1-saturated
keywords: S: str
Sh:1105
Larson, P. B., & Shelah, S. (2018). A model of ZFA + PAC with no outer model of ZFAC with the same pure part. Arch. Math. Logic, 57(7-8), 853–859. DOI: 10.1007/s00153-018-0610-y MR: 3850686
type: article
abstract: We produce a model of \mathsf{ZFA} + \mathsf{PAC} such that no outer model of \mathsf{ZFAC} has the same pure sets, answering a question asked privately by Eric Hall.
keywords: S: ods
Sh:1106
Paolini, G., & Shelah, S. (2018). The automorphism group of Hall’s universal group. Proc. Amer. Math. Soc., 146(4), 1439–1445. arXiv: 1703.10540 DOI: 10.1090/proc/13836 MR: 3754331
type: article
abstract: We study the automorphism group of Hall’s universal locally finite group H. We show that in Aut(H) every subgroup of index < 2^\omega lies between the pointwise and the setwise stabilizer of a unique finite subgroup A of H, and use this to prove that Aut(H) is complete. We further show that Inn(H) is the largest locally finite normal subgroup of Aut(H). Finally, we observe that from the work of [312] it follows that for every countable locally finite G there exists G \cong G' \leq H such that every f \in Aut(G') extends to an \hat{f} \in Aut(H) in such a way that f \mapsto \hat{f} embeds Aut(G') into Aut(H). In particular, we solve the three open questions of Hickin on Aut(H) from [3] and give a partial answer to Question VI.5 of Kegel and Wehrfritz from [6].
keywords: S: str, O: alg
Sh:1107
Paolini, G., & Shelah, S. (2020). Automorphism groups of countable stable structures. Fund. Math., 248(3), 301–307. arXiv: 1712.02568 DOI: 10.4064/fm723-4-2019 MR: 4046958
type: article
abstract: For every countable structure M we construct an \aleph_0-stable countable structure N such that Aut(M) and Aut(N) are topologically isomorphic. This shows that it is impossible to detect any form of stability of a countable structure M from the topological properties of the Polish group Aut(M).
keywords: S: str
Sh:1109
Paolini, G., & Shelah, S. (2019). Reconstructing structures with the strong small index property up to bi-definability. Fund. Math., 247(1), 25–35. arXiv: 1703.10498 DOI: 10.4064/fm640-9-2018 MR: 3984277
type: article
abstract: Let \mathbf{K} be the class of countable structures M with the strong small index property and locally finite algebraicity, and \mathbf{K}_* the class of M \in \mathbf{K} such that acl_M(\{ a \}) = \{ a \} for every a \in M. For homogeneous M \in \mathbf{K}, we introduce what we call the expanded group of automorphisms of M, and show that it is second-order definable in Aut(M). We use this to prove that for M, N \in \mathbf{K}_*, Aut(M) and Aut(N) are isomorphic as abstract groups if and only if (Aut(M), M) and (Aut(N), N) are isomorphic as permutation groups. In particular, we deduce that for \aleph_0-categorical structures the combination of strong small index property and no algebraicity implies reconstruction up to bi-definability, in analogy with Rubin’s well-known \forall \exists-interpretation technique of [Rubin] Finally, we show that every finite group can be realized as the outer automorphism group of Aut(M) for some countable \aleph_0-categorical homogeneous structure M with the strong small index property and no algebraicity.
keywords: S: str, O: alg
Sh:1110
Shelah, S., & Spinas, O. (2023). Different cofinalities of tree ideals. Ann. Pure Appl. Logic, 174(8), Paper No. 103290, 18. DOI: 10.1016/j.apal.2023.103290 MR: 4597956
type: article
abstract: We introduce a general framework of generalized tree forcings, GTF for short, that includes the classical tree forcings like Sacks, Silver, Laver or Miller forcing. Using this concept we study the cofinality of the ideal \mathcal{I}(\mathbf {Q}) associated with a GTF \mathbf {Q}. We show that if for two GTF’s \mathbf{Q_0} and \mathbf{Q_1} the consistency of add(\mathcal{I}(\mathbf{Q_0})) < add(\mathcal{I}(\mathbf{Q_1})) holds, then we can obtain the consistency of cof(\mathcal{I}(\mathbf{Q_1})) < cof(\mathcal{I} (\mathbf{Q_0})). We also show that cof(\mathcal{I}(\mathbf{Q})) can consistently be any cardinal of cofinality larger than the continuum.
keywords: S: for, S: str
Sh:1111
Garti, S., & Shelah, S. (2021). Double weakness. Acta Math. Hungar., 163(2), 379–391. arXiv: 2002.03573 DOI: 10.1007/s10474-021-01132-y MR: 4227788
type: article
abstract: We prove that, consistently, there exists a weakly but not strongly inaccessible cardinal \lambda for which the sequence \langle 2^\theta:\theta<\lambda\rangle is not eventually constant and the weak diamond fails at \lambda. We also prove that consistently diamond fails but a parametrized version of weak diamond holds at some strongly inaccessible
keywords: S: ico, S: for, (wd)
Sh:1112
Paolini, G., & Shelah, S. (2017). No uncountable Polish group can be a right-angled Artin group. Axioms Topical Collection “Topological Groups", 6(2), 4. arXiv: 1701.03021
type: article
abstract: We prove that no uncountable Polish group can admit a system of generators whose associated length function satisfies the following conditions: (i) if 0 < k < \omega, then lg(x) \leq lg(x^k); (ii) if lg(y) < k < \omega and x^k = y, then x = e. In particular, the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes results from [Sh:744] and "Polish Group Topologies" by S. Solecki, where this is proved for free and free abelian uncountable groups.
keywords: S: str, O: alg, (grp)
Sh:1114
Shelah, S., & Steprāns, J. (2018). Trivial and non-trivial automorphisms of \mathcal P(\omega_1)/[\omega_1]^{<\aleph_0}. Fund. Math., 243(2), 155–168. DOI: 10.4064/fm402-11-2017 MR: 3846847
type: article
abstract: The following statement is shown to be independent of set theory with the Continuum Hypothesis: There is an automorphism of \mathcal{P}(\omega_1)/[\omega_1]^{<\aleph_0} whose restriction to \mathcal{P} (\alpha) / [\alpha]^{<\aleph_0} is induced by a bijection for every \alpha \in \omega_1, but the automorphism itself is not induced by any bijection on \omega_1.
keywords: S: for, S: str
Sh:1115
Paolini, G., & Shelah, S. (2018). Polish topologies for graph products of cyclic groups. Israel J. Math., 228(1), 305–319. arXiv: 1705.01815 DOI: 10.1007/s11856-018-1765-2 MR: 3874845
type: article
abstract: We give a complete characterization of the graph products of cyclic groups admitting a Polish group topology, and show that they are all realizable as the group of automorphisms of a countable structure. In particular, we characterize the right-angled Coxeter groups (resp. Artin groups) admitting a Polish group topology. This generalizes results from Shelah and Paolini-Shelah.
keywords: S: str, O: alg, (grp)
Sh:1116
Casanovas, E., & Shelah, S. (2019). Universal theories and compactly expandable models. J. Symb. Log., 84(3), 1215–1223. arXiv: 1705.02611 DOI: 10.1017/jsl.2019.16 MR: 4010496
type: article
abstract: Our aim is to solve a quite old question on the difference between expandability and compact expandability. Toward this, we further investigate the logic of countable cofinality
keywords: M: smt, M: odm, (nni), (ua)
Sh:1117
Paolini, G., & Shelah, S. (2017). Group metrics for graph products of cyclic groups. Topology Appl., 232, 281–287. arXiv: 1705.02582 DOI: 10.1016/j.topol.2017.10.016 MR: 3720899
type: article
abstract: We complement the characterization of the graph products of cyclic groups G(\Gamma, {\mathfrak p}) admitting a Polish group topology of [9] with the following result. Let G = G(\Gamma, {\mathfrak p}), then the following are equivalent: [i] there is a metric on \Gamma which induces a separable topology in which E_{\Gamma} is closed; [ii] G(\Gamma, {\mathfrak p}) is embeddable into a Polish group; [iii] G(\Gamma, {\mathfrak p}) is embeddable into a non-Archimedean Polish group. We also construct left-invariant separable group ultrametrics for G = G(\Gamma, {\mathfrak p}) and \Gamma a closed graph on the Baire space, which is of independent interest.
keywords: S: str, O: alg, (grp)
Sh:1118
Kaplan, I., Ramsey, N., & Shelah, S. (2019). Local character of Kim-independence. Proc. Amer. Math. Soc., 147(4), 1719–1732. arXiv: 1707.02902 DOI: 10.1090/proc/14305 MR: 3910436
type: article
abstract: We show that NSOP_1 theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if T is NSOP_1, M \models T, and p is a type over M, then the collection of elementary submodels of size |T| over which p does not Kim-fork is a club of [M]^{|T|} and that this characterizes NSOP_1.
keywords: M: cla, (unstable)
Sh:1119
Shelah, S., & Vasey, S. (2018). Abstract elementary classes stable in \aleph_0. Ann. Pure Appl. Logic, 169(7), 565–587. arXiv: 1702.08281 DOI: 10.1016/j.apal.2018.02.004 MR: 3788738
type: article
abstract: We study abstract elementary classes (AECs) that, in \aleph_0, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). We prove that such classes exhibit super stable-like behavior at \aleph_0. More precisely, there is a superlimit model of cardinality \aleph_0 and the class generated by this superlimit has a type-full good \aleph_0-frame (a local notion of nonforking independence) and a superlimit model of cardinality \aleph_1. This extends the first author’s earlier study of PC_{\aleph_0}-representable AECs and also improves results of Hyttinen-Kesala and Baldwin-Kueker-VanDieren.
keywords: M: nec, (aec)
Sh:1120
Golshani, M., & Shelah, S. (2021). Specializing trees and answer to a question of Williams. J. Math. Log., 21(1), 2050023, 20. arXiv: 1708.02719 DOI: 10.1142/S0219061320500233 MR: 4194557
type: article
abstract: We show that if cf(2^{\aleph_0}) = \aleph_1, then any non-trivial \aleph_1-closed forcing notion of size \le 2^{\aleph_0} is forcing equivalent to Add(\aleph_1, 1), the Cohen forcing for adding a new Cohen subset of \omega_1. We also produce, relative the existence of some large cardinals, a model of ZFC in which 2^{\aleph_0} = \aleph_2 and all \aleph_1-closed forcing notion of size \le 2^{\aleph_0} collapse \aleph_2, and hence are forcing equivalent to Add(\aleph_1, 1). Our results answer a question of Scott Williams from 1978. We also extend a result of Todorcevic and Foreman-Magidor-Shelah by showing that it is consistent that every partial order which adds a new subset of \aleph_2, collapes \aleph_2 or \aleph_3.
keywords: S: for
Sh:1121
Paolini, G., & Shelah, S. (2019). Polish topologies for graph products of groups. J. Lond. Math. Soc. (2), 100(2), 383–403. arXiv: 1711.06155 DOI: 10.1112/jlms.12219 MR: 4017147
type: article
abstract: We give strong necessary conditions on the admissibility of a Polish group topology for an arbitrary graph product of groups G(\Gamma, G_a), and use them to give a characterization modulo a finite set of nodes. As a corollary, we give a complete characterization in case all the factor groups G_a are countable.
keywords: S: pcf
Sh:1122
Goldstern, M., Kellner, J., & Shelah, S. (2019). Cichoń’s maximum. Ann. Of Math. (2), 190(1), 113–143. arXiv: 1708.03691 DOI: 10.4007/annals.2019.190.1.2 MR: 3990602
type: article
abstract: (none)
keywords: S: for, S: str, (inv)
Sh:1123
Shelah, S., & Verner, J. L. (2023). Ramsey partitions of metric spaces. Acta Math. Hungar., 169(2), 524–533. arXiv: 2210.12836 DOI: 10.1007/s10474-023-01318-6 MR: 4594315
type: article
abstract: We investigate the existence of metric spaces which, for any coloring with a fixed number of colors, contain monochromatic isomorphic copies of a fixed starting space K. In the main theorem we construct such a space of size 2^{\aleph_0} for colorings with \aleph_0 colors and any metric space K of size \aleph_0. We also give a slightly weaker theorem for countable ultrametric K where, however, the resulting space has size \aleph_1.
keywords: S: ico
Sh:1124
Malliaris, M., & Shelah, S. (2019). A new look at interpretability and saturation. Ann. Pure Appl. Logic, 170(5), 642–671. arXiv: 1709.04899 DOI: 10.1016/j.apal.2019.01.001 MR: 3926500
type: article
abstract: We investigate the interpretability ordering \trianglelefteq^* using generalized Ehrenfeucht-Mostowski models. This gives a new approach to proving inequalities and investigating the structure of types.
keywords: M: cla
Sh:1126
Shelah, S. (2023). Corrected iteration. Boll. Unione Mat. Ital., 16(3), 521–584. arXiv: 2108.03672 DOI: 10.1007/s40574-022-00338-4 MR: 4627290
type: article
abstract: For \lambda inaccessible, we may consider (< \lambda)-support iteration of some specific (<\lambda)-complete \lambda^+-c.c. forcing notion. But this fails a “preservation by restricting to a sub-sequence of the forcing, we “correct" the iteration to regain it. This is used in another paper in the consistency of cov(meagre) < \mathfrak{d} _\lambda.
keywords: S: for, O: odo, (iter)
Sh:1127
Dow, A. S., & Shelah, S. (2018). On the cofinality of the splitting number. Indag. Math. (N.S.), 29(1), 382–395. arXiv: 1801.02517 DOI: 10.1016/j.indag.2017.01.010 MR: 3739621
type: article
abstract: The splitting number \mathfrak s can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and the second author [Ultrafilters with small generating sets, Israel J. Math., 65, (1989)]
keywords: S: for
Sh:1129
Komjáth, P., Leader, I., Russell, P., Shelah, S., Soukup, D. T., & Vidnyánszky, Z. (2019). Infinite monochromatic sumsets for colourings of the reals. Proc. Amer. Math. Soc., 147(6), 2673–2684. arXiv: 1710.07500 DOI: 10.1090/proc/14431 MR: 3951442
type: article
abstract: N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite coloring of \mathbb{R} so that no infinite sumset X + X is monochromatic. The (rather fascinating) question if the same conclusion holds in ZFC was open until now: we show that under certain set theoretic assumptions for any c: \mathbb{R} \to r with r finite there is an infinite X \subseteq \mathbb{R} so that C\upharpoonright X + X is constant.
keywords: S: ico
Sh:1131
Kellner, J., Latif, A., & Shelah, S. (2019). Another ordering of the ten cardinal characteristics in Cichoń’s diagram. Comment. Math. Univ. Carolin., 60(1), 61–95. arXiv: 1712.00778 DOI: 10.14712/1213-7243.2015.273 MR: 3946665
type: article
abstract: It is consistent that \aleph_1 < \mathrm{add}(\mathrm{Null}) < \mathrm{add}(\mathrm{Meager})= \mathfrak{b} < \mathrm{cov} (\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov} (\mathrm{Meager}) = 2^{\aleph_0}. Assuming four strongly compact cardinals, it is consistent that \aleph_1 < \mathrm{add} (\mathrm{Null}) <\mathrm{add}(\mathrm{Meager})=\mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov}(\mathrm{Meager}) < \mathrm{non}(\mathrm{Null}) < \mathrm{cof}(\mathrm{Meager})= \mathfrak{d} < \mathrm{cof} (\mathrm{Null}) < 2^{\aleph_0}.
keywords: S: for
Sh:1132
Saveliev, D. I., & Shelah, S. (2019). Ultrafilter extensions do not preserve elementary equivalence. MLQ Math. Log. Q., 65(4), 511–516. arXiv: 1712.06198 MR: 4057949
type: article
abstract: We show that there exist models {\mathcal M}_1 and {\mathcal M}_2 such that {\mathcal M}_1 elementarily embeds into {\mathcal M}_2 but their ultrafilter extension \beta({\mathcal M}_1) and \beta({\mathcal M}_2) are not elementarily equivalent.
keywords: M: smt
Sh:1133
Palacı́n, D., & Shelah, S. (2018). On the class of flat stable theories. Ann. Pure Appl. Logic, 169(8), 835–849. arXiv: 1801.01438 DOI: 10.1016/j.apal.2018.04.005 MR: 3802227
type: article
abstract: A new notion of independence relation is given and associated to it the class of flat theories, a subclass of strong stable theories including the superstable ones is introduced. More precisely, after introducing this independence relation, flat theories are defined as an appropriate version of superstability and it is shown that in a flat theory every type has finite weight and therefore flat theories are strong.
keywords: S: ico
Sh:1134
Dow, A. S., & Shelah, S. (2019). Pseudo P-points and splitting number. Arch. Math. Logic, 58(7-8), 1005–1027. arXiv: 1802.04829 DOI: 10.1007/s00153-019-00674-x MR: 4003647
type: article
abstract: We construct a model in which the splitting number is large and every ultrafilter has a small subset with no pseudo-intersection
keywords: S: for, O: top
Sh:1135
Raghavan, D., & Shelah, S. (2019). Two results on cardinal invariants at uncountable cardinals. In Proceedings of the 14th and 15th Asian Logic Conferences, World Sci. Publ., Hackensack, NJ, pp. 129–138. arXiv: 1801.09369 DOI: 10.1142/9789813237551_0006 MR: 3890461
type: article
abstract: We prove two ZFC theorems about cardinal invariants above the continuum which are in sharp contrast to well-known facts about these same invariants at the continuum. It is shown that for an uncoutable regular cardinal \kappa, \mathfrak{b}(\kappa) = \kappa^+ implies \mathfrak{a}(\kappa) = \kappa^+. This improves an earlier result of Blass, Hyttinen and Zhang [3]. It is also shown that if \kappa \ge \beta_\omega is an uncountable regular cardinal, then \mathfrak{d} (\kappa) \le \mathfrak{r}(\kappa). This result partially dualizes an earlier theorem of the authors [6]
keywords: S: ico
Sh:1136
Kumar, A., & Shelah, S. (2021). On some variants of the club principle. Eur. J. Math., 7(1), 1–27. arXiv: 1802.01137 DOI: 10.1007/s40879-020-00425-w MR: 4220028
type: article
abstract: We study some asymptotic variants of the club principle. Along the way, we construct some forcings and use them to separate several of these principles.
keywords: S: for
Sh:1137
Fischer, V., & Shelah, S. (2019). The spectrum of independence. Arch. Math. Logic, 58(7-8), 877–884. DOI: 10.1007/s00153-019-00665-y MR: 4003640
type: article
abstract: We study the set of possible size of maximal independent family, to which we refer as spectrum of independence and denote Spec(mi f). We show that: (1) whenever \kappa_1 < \dots < \kappa_n are finitely many regular uncountable cardinals, it is consistent that \{ \kappa_i\}^n_{i = 1} \subseteq Spec (mif); (2) whenever \kappa has uncountable cofinality, it is consistent that Spec(mif) = \{ \aleph_1, \kappa = \mathfrak{c}\}. Assuming large cardinals, in addition to (1) above, we can provide that (\kappa_i, \kappa_{i +1} ) \cap Spec(mif) = \emptyset for each i, 1 \le i < n.
keywords: S: for
Sh:1138
Rosłanowski, A., & Shelah, S. (2019). Borel sets without perfectly many overlapping translations. Rep. Math. Logic, (54), 3–43. arXiv: 1806.06283 DOI: 10.4467/20842589rm.19.001.10649 MR: 4011916
type: article
abstract: For a cardinal \lambda<\lambda_{\omega_1} we give a ccc forcing notion P such that in V^P there is a \Sigma^0_2 set B\subseteq {}^\omega 2 with a sequence \langle\eta_\alpha: \alpha<\lambda\rangle of distinct elements of {}^\omega 2 such that \big|(\eta_\alpha+B)\cap (\eta_\beta+B)\big|\geq 6 for all \alpha,\beta<\lambda but does not have a perfect set of such \eta’s. The construction closely follows the one from [Sh:522, Section 1].
keywords: S: ico, S: dst
Sh:1139
Bartoszyński, T., & Shelah, S. (2018). A note on small sets of reals. C. R. Math. Acad. Sci. Paris, 356(11-12), 1053–1061. arXiv: 1805.02703 DOI: 10.1016/j.crma.2018.11.003 MR: 3907570
type: article
abstract: We construct an example of a combinatorially large measure zero set
keywords: S: str
Sh:1140
Malliaris, M., & Shelah, S. (2020). An example of a new simple theory. In Trends in set theory, Vol. 752, Amer. Math. Soc., [Providence], RI, pp. 121–151. arXiv: 1804.03254 DOI: 10.1090/conm/752/15133 MR: 4132104
type: article
abstract: We construct a countable simple theory which, in Keisler’s order, is strictly above the random graph (but “barely so”) and also in some sense orthogonal to the building blocks of the recently discovered infinite descending chain. As a result we prove in ZFC that there are incomparable classes in Keisler’s order.
keywords: M: cla
Sh:1141
Shelah, S., & Ulrich, D. (2019). Torsion-free abelian groups are consistently {\mathrm{a}}\Delta^1_2-complete. Fund. Math., 247(3), 275–297. arXiv: 1804.08152 DOI: 10.4064/fm673-12-2018 MR: 4017015
type: article
abstract: Let \textrm{TFAG} be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of ZF^- + \kappa(\omega) exists, then \textrm{TFAG} is a \Delta^1_2-complete; in particular, this is consistent with ZFC. We define the \alpha-ary Schröder- Bernstein property, and show that \textrm{TFAG} fails the \alpha-ary Schröder-Bernstein property for every \alpha < \kappa(\omega). We leave open whether or not \textrm{TFAG} can have the \kappa(\omega)-ary Schröder-Bernstein property; if it did, then it would not be a \Delta^1_2-complete, and hence not Borel complete.
keywords: S: dst, O: alg, (set-mod)
Sh:1142
Corson, S. M., & Shelah, S. (2019). Deeply concatenable subgroups might never be free. J. Math. Soc. Japan, 71(4), 1123–1136. arXiv: 1804.05538 DOI: 10.2969/jmsj/80498049 MR: 4023299
type: article
abstract: We show that certain algebraic structures lack freeness in the absence of the axiom of choice. These include some subgroups of the Baer-Spevcker group \mathbb{Z}^\omega and the Hawaiian earring group. Applications to slenderness, completely metrizable topological groups, length functions and strongly bounded groups are also presented.
keywords: O: alg
Sh:1143
Garti, S., & Shelah, S. (2020). Remarks on generalized ultrafilter, dominating and reaping numbers. Fundamenta Mathematicae, 250(2), 101–115. arXiv: 1806.09286 DOI: 10.4064/fm595-9-2019 MR: 4107530
type: article
abstract: We prove that \textrm{cf}(\mathfrak{u}_\kappa)>\kappa.
keywords: (inv), (up)
Sh:1144
Baumhauer, T., Goldstern, M., & Shelah, S. (2021). The higher Cichoń diagram. Fund. Math., 252(3), 241–314. arXiv: 1806.08583 DOI: 10.4064/fm666-4-2020 MR: 4178868
type: article
abstract: (none)
keywords: S: for
Sh:1145
Horowitz, H., & Shelah, S. (2022). \kappa-madness and definability. MLQ Math. Log. Q., 68(3), 346–351. arXiv: 1805.07048 MR: 4472782
type: article
abstract: Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal \kappa, there are no \Sigma^1_1(\kappa)-\kappa-mad families.
keywords: S: ods
Sh:1146
Mohsenipour, S., & Shelah, S. (2019). On finitary Hindman numbers. Combinatorica, 39(5), 1185–1189. arXiv: 1806.04917 DOI: 10.1007/s00493-019-4002-7 MR: 4039607
type: article
abstract: Spencer asked whether the Paris-Harrington version of Hindman’s theorem has primitive recursive upper bounds. We give a positive answer to this question.
keywords: O: fin
Sh:1148
Paolini, G., & Shelah, S. (2020). On a cardinal invariant related to the Haar measure problem. Israel J. Math., 236(1), 305–316. arXiv: 1809.10442 DOI: 10.1007/s11856-020-1975-2 MR: 4093888
type: article
abstract: (none)
keywords: S: for, (inv)
Sh:1149
Malliaris, M., & Shelah, S. (2022). A separation theorem for simple theories. Trans. Amer. Math. Soc., 375(2), 1171–1205. arXiv: 1810.09604 DOI: 10.1090/tran/8513 MR: 4369245
type: article
abstract: (none)
keywords: (up), (unstable)
Sh:1150
Brendle, J., Halbeisen, L. J., Klausner, L. D., Lischka, M., & Shelah, S. (2023). Halfway new cardinal characteristics. Ann. Pure Appl. Logic, 174(9), Paper No. 103303, 33. arXiv: 1808.02442 DOI: 10.1016/j.apal.2023.103303 MR: 4609469
type: article
abstract: (none)
keywords: S: str
Sh:1152
Horowitz, H., & Shelah, S. (2022). On the definability of mad families of vector spaces. Ann. Pure Appl. Logic. arXiv: 1811.03753 MR: 4354828
type: article
abstract: (none)
keywords: S: str, (pc), (ultraf)
Sh:1153
Kubiś, W., & Shelah, S. (2020). Homogeneous structures with nonuniversal automorphism groups. J. Symb. Log., 85(2), 817–827. arXiv: 1811.09650 DOI: 10.1017/jsl.2020.10 MR: 4206113
type: article
abstract: (none)
keywords: S: ico, S: str, (auto)
Sh:1154
Mildenberger, H., & Shelah, S. (2020). A version of \kappa-Miller forcing. Arch. Math. Logic, 59(7-8), 879–892. arXiv: 1802.07986 DOI: 10.1007/s00153-020-00721-y MR: 4159758
type: article
abstract: (none)
keywords: (pure(for))
Sh:1155
Paolini, G., & Shelah, S. (2020). Some results on Polish groups. Rep. Math. Logic, (55), 61–71. arXiv: 1810.12855 DOI: 10.4467/20842589rm.20.003.12435 MR: 4183417
type: article
abstract: (none)
keywords: S: str, S: dst
Sh:1156
Garti, S., Gitik, M., & Shelah, S. (2020). Cardinal characteristics at \aleph_\omega. Acta Math. Hungar., 160(2), 320–336. arXiv: 1812.05291 DOI: 10.1007/s10474-019-00971-0 MR: 4075404
type: article
abstract: (none)
keywords: S: for, (inv)
Sh:1157
Bergfalk, J., Hrušák, M., & Shelah, S. (2021). Ramsey theory for highly connected monochromatic subgraphs. Acta Math. Hungar., 163(1), 309–322. arXiv: 1812.06386 DOI: 10.1007/s10474-020-01058-x MR: 4217971
type: article
abstract: (none)
keywords: S: ico, (pc), (graph)
Sh:1158
Shelah, S. (2021). Mutual stationarity and singular Jonsson cardinals. Acta Math. Hungar., 163(1), 140–148. DOI: 10.1007/s10474-020-01041-6 MR: 4217962
type: article
abstract: We prove that if the sequence \langle k_n:1 \le n < \omega\rangle contains a so-called gap then the sequence \langle S^{\aleph_n}_{\aleph_{k_n}}:1 \le n < \omega\rangle of stationary sets is not mutually stationary, provided that k_n<n for every n \in \omega. We also prove a sufficient condition for being singular Jonsson cardinals.
keywords: S: ico
Sh:1159
Shelah, S. (2020). On \mathfrak{d}_\mu for \mu singular. Acta Math. Hungar., 161(1), 245–256. DOI: 10.1007/s10474-019-00999-2 MR: 4110369
type: article
abstract: (none)
keywords: S: ico, (inv)
Sh:1160
Raghavan, D., & Shelah, S. (2020). A small ultrafilter number at smaller cardinals. Arch. Math. Logic, 59(3-4), 325–334. arXiv: 1904.02379 DOI: 10.1007/s00153-019-00693-8 MR: 4081063
type: article
abstract: (none)
keywords: (iter), (inv), (ultraf)
Sh:1161
Komjáth, P., & Shelah, S. (2019). Universal graphs omitting finitely many finite graphs. Discrete Math., 342(12), 111596, 4. DOI: 10.1016/j.disc.2019.111596 MR: 3990009
type: article
abstract: (none)
keywords: S: ico, (univ)
Sh:1162
Shelah, S. (2020). No universal in singular. Boll. Unione Mat. Ital., 13(3), 361–368. DOI: 10.1007/s40574-020-00229-6 MR: 4132913
type: article
abstract: (none)
keywords: M: cla, M: non, S: ico, O: odo, (univ)
Sh:1163
Shelah, S. (2021). Colouring of successor of regular, again. Acta Math. Hungar., 165(1), 192–202. arXiv: 1910.02419 DOI: 10.1007/s10474-021-01181-3 MR: 4323594
type: article
abstract: We get a quite maximal version of the colouring property Pr_1 by proving Pr_1(\lambda,\lambda,\lambda,\theta) when \lambda = \partial^+, \partial > \theta are regular cardinals.
keywords: S: ico, (pc)
Sh:1164
Shelah, S. (2023). Universality: new criterion for non-existence. Boll. Unione Mat. Ital., 16(1), 43–64. arXiv: 2108.06727 DOI: 10.1007/s40574-022-00327-7 MR: 4548558
type: article
abstract: We find new “reasons" for a class of models for not having a universal model in a cardinal \lambda. This work, though has consequences in model theory, is really in combinatorial (set theory). We concentrate on a prototypical class which is a simply defined class of models, of combinatorial character - models of T_{\rm ceq} (essentially another representation of T_{\rm feq} which was already considered but the proof with T_{\rm ceq} is more transparent). Models of T_{\rm ceq} consist essentially of an equivalence relation on one set and a family of choice functions for it. This class is not simple (in the model theoretic sense) but seems to be very low among the non-simple (first order complete countable) ones. We give sufficient conditions for the non-existence of a universal model for it in \lambda. This work is continued in [Sh:F2071].
keywords: S: ico, (univ), (unstable)
Sh:1165
Garti, S., Magidor, M., & Shelah, S. (2020). Infinite monochromatic paths and a theorem of Erdös-Hajnal-Rado. Electronic Journal of Combinatorics, 27(2), 11 pages. arXiv: 1907.03254 DOI: 10.37236/8849 MR: 4245063
type: article
abstract: (none)
keywords: S: ico, (pc)
Sh:1166
Goldstern, M., Kellner, J., Mejı́a, D. A., & Shelah, S. (2021). Controlling cardinal characteristics without adding reals. J. Math. Log., 21(3), Paper No. 2150018, 29. arXiv: 2006.09826 DOI: 10.1142/S0219061321500185 MR: 4330526
type: article
abstract: (none)
keywords: S: str, (iter), (inv)
Sh:1167
Malliaris, M., & Shelah, S. (2021). Keisler’s order is not simple (and simple theories may not be either). Adv. Math., 392, Paper No. 108036, 94. arXiv: 1906.10241 DOI: 10.1016/j.aim.2021.108036 MR: 4319768
type: article
abstract: (none)
keywords: (up), (unstable)
Sh:1168
Horowitz, H., & Shelah, S. (2023). On the non-existence of \kappa-mad families. Arch. Math. Logic, 62(7-8), 1033–1039. arXiv: 1906.09538 DOI: 10.1007/s00153-023-00874-6 MR: 4643484
type: article
abstract: (none)
keywords: S: ods
Sh:1169
Corson, S. M., & Shelah, S. (2020). Strongly bounded groups of various cardinalities. Proc. Amer. Math. Soc., 148(12), 5045–5057. arXiv: 1906.10481 DOI: 10.1090/proc/14998 MR: 4163821
type: article
abstract: (none)
keywords: O: alg, (grp)
Sh:1171
Dugas, M. H., Herden, D., & Shelah, S. (2020). \aleph_k-free cogenerators. Rend. Semin. Mat. Univ. Padova, 144, 87–104. arXiv: 1909.00595 DOI: 10.4171/rsmup/58 MR: 4186448
type: article
abstract: (none)
keywords: O: alg, (ab)
Sh:1172
Kaplan, I., Segel, O., & Shelah, S. (2022). Boolean types in dependent theories. J. Symb. Log., 87(4), 1349–1373. arXiv: 2008.03214 DOI: 10.1017/jsl.2022.7 MR: 4510824
type: article
abstract: The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra \mathcal{B} to each formula. We show some basic results regarding the effect of the properties of \mathcal{B} on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author’s result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures — in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.
keywords: M: odm, (unstable)
Sh:1173
Hrušák, M., Ramos-Garcı́a, U. A., Shelah, S., & Mill, J. van. (2021). Countably compact groups without non-trivial convergent sequences. Trans. Amer. Math. Soc., 374(2), 1277–1296. arXiv: 2006.12675 DOI: 10.1090/tran/8222 MR: 4196393
type: article
abstract: (none)
keywords: S: ico, O: top, (pc)
Sh:1175
Shelah, S. (2023). Canonical universal locally finite groups. Rend. Semin. Mat. Univ. Padova, 149, 25–44. arXiv: 2303.03788 DOI: 10.4171/rsmup/117 MR: 4575363
type: article
abstract: (none)
keywords: O: alg
Sh:1177
Goldstern, M., Kellner, J., Mejı́a, D. A., & Shelah, S. (2022). Cichoń’s maximum without large cardinals. J. Eur. Math. Soc. (JEMS), 24(11), 3951–3967. arXiv: 1906.06608 DOI: 10.4171/jems/1178 MR: 4493617
type: article
abstract: (none)
keywords: S: for, (iter), (inv)
Sh:1179
Garti, S., & Shelah, S. (2023). Tiltan and superclub. C. R. Math. Acad. Sci. Paris, 361, 853–861. arXiv: 2305.09490 DOI: 10.5802/crmath.434 MR: 4616154
type: article
abstract: (none)
keywords: S: for, (iter), (wd)
Sh:1180
Hamel, C., Horowitz, H., & Shelah, S. (2020). Turing invariant sets and the perfect set property. MLQ Math. Log. Q., 66(2), 247–250. arXiv: 1912.12558 DOI: 10.1002/malq.202000005 MR: 4130400
type: article
abstract: (none)
keywords: S: str, O: odo
Sh:1181
Shelah, S. (2020). Density of indecomposable locally finite groups. Rend. Semin. Mat. Univ. Padova, 144, 253–270. DOI: 10.4171/rsmup/68 MR: 4186458
type: article
abstract: (none)
keywords: O: alg, (set-mod)
Sh:1184
Shelah, S., & Villaveces, A. (2022). Infinitary logics and A.E.C. Proc. Amer. Math. Soc., 150, 371–380. arXiv: 2010.02145 MR: 4335884
type: article
abstract: (none)
keywords:
Sh:1186
Džamonja, M., & Shelah, S. (2021). On wide Aronszajn trees in the presence of MA. J. Symb. Log., 86(1), 210–223. arXiv: 2002.02396 DOI: 10.1017/jsl.2020.42 MR: 4282706
type: article
abstract: A wide Aronszajn tree is a tree of size and height \omega_1 with no uncountable branches. We prove that under MA(\omega_1) there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and Väänänen from 1994. We also prove that under the same assumption there is no universal Aronszajn tree, improving a result of Todorčević from 2007 who proved the same under the assumption of BPFA for posets of size \aleph_1. Finally, we prove that under MA(\omega_1), every wide Aronszajn tree weakly embeds in an Aronszajn tree.
keywords: S: for
Sh:1189
Poór, M., & Shelah, S. (2021). Characterizing the spectra of cardinalities of branches of Kurepa trees. Pacific J. Math., 311(2), 423–453. arXiv: 2003.01272 DOI: 10.2140/pjm.2021.311.423 MR: 4292936
type: article
abstract: We give a complete characterization of the sets of cardinals that in a suitable forcing extension can be the Kurepa spectrum, that is, the set of cardinalities of branches of Kurepa trees. This answers a question of the first named author.
keywords: S: for
Sh:1190
Komjáth, P., & Shelah, S. (2021). Monocolored topological complete graphs in colorings of uncountable complete graphs. Acta Math. Hungar., 163(1), 71–84. DOI: 10.1007/s10474-020-01125-3 MR: 4217959
type: article
abstract: (none)
keywords:
Sh:1191
Mildenberger, H., & Shelah, S. (2021). Higher Miller forcing may collapse cardinals. J. Symb. Log., 86(4), 1721–1744. DOI: 10.1017/jsl.2021.90 MR: 4362933
type: article
abstract: We show that it is independent whether club-\kappa-Miller forcing preserves \kappa^{++}. We show that under \kappa^{<\kappa} > \kappa, club-\kappa-Miller forcing collapses some cardinal in [\kappa^+,\kappa^{<\kappa}] to \kappa. Answering a question by Brendle, Brooke-Taylor, Friedman and Montoya, we show that the iteration of ultrafilter \kappa-Miller forcing does not have the Laver property.
keywords:
Sh:1192
Kaplan, I., Ramsey, N., & Shelah, S. (2021). Criteria for exact saturation and singular compactness. Ann. Pure Appl. Logic, 172(9), Paper No. 102992, 28. arXiv: 2003.00597 DOI: 10.1016/j.apal.2021.102992 MR: 4264145
type: article
abstract: We introduce the class of unshreddable theories, which contains the simple and NIP theories, and prove that such theories have exactly saturated models in singular cardinals, satisfying certain set-theoretic hypotheses. We also give criteria for a theory to have singular compactness.
keywords:
Sh:1193
Shelah, S., & Soukup, L. (2023). On \kappa-homogeneous, but not \kappa-transitive permutation groups. J. Symb. Log., 88(1), 363–380. arXiv: 2003.02023 DOI: 10.1017/jsl.2021.63 MR: 4550395
type: article
abstract: A permutation group G on a set A is {\kappa}-homogeneous iff for all X,Y\in\bigl[ {A} \bigr]^ {\kappa} with |A\setminus X|=|A\setminus Y|=|A| there is a g\in G with g[X]=Y. G is {\kappa}-transitive iff for any injective function f with dom(f) \cup ran(f)\in \bigl[ {A} \bigr]^ {\le \kappa} and |A\setminus dom(f)|=|A\setminus ran(f)|=|A| there is a g\in G with f\subseteq g.

Giving a partial answer to a question of P. M. Neumann we show that there is an {\omega}-homogeneous but not {\omega}-transitive permutation group on a cardinal {\lambda} provided
1. {\lambda}<{\omega}_{\omega}, or
2. 2^{\omega}<{\lambda}, and {\mu}^{\omega}={\mu}^+ and \Box_{\mu} hold for each {\mu}\le{\lambda} with {\omega}=cf ({\mu})<{{\mu}}, or
3. our model was obtained by adding {\omega}_1 many Cohen generic reals to some ground model.
For {\kappa}>{\omega} we give a method to construct large {\kappa}-homogeneous, but not {\kappa}-transitive permutation groups. Using this method we show that there exists {\kappa}^+-homogeneous, but not {\kappa}^+-transitive permutation groups on {\kappa}^{+n} for each infinite cardinal {\kappa} and natural number n\ge 1 provided V=L.
keywords:
Sh:1194
Shelah, S., & Väänänen, J. A. (2023). Positive logics. Arch. Math. Logic, 62(1-2), 207–223. arXiv: 2008.01145 DOI: 10.1007/s00153-022-00837-3 MR: 4535931
type: article
abstract: (none)
keywords:
Sh:1196
Halevi, Y., Kaplan, I., & Shelah, S. (2022). Infinite stable graphs with large chromatic number. Trans. Amer. Math. Soc., 375(3), 1767–1799. arXiv: 2007.12139 DOI: 10.1090/tran/8570 MR: 4378079
type: article
abstract: (none)
keywords:
Sh:1197
Kaplan, I., Ramsey, N., & Shelah, S. (2023). Exact saturation in pseudo-elementary classes for simple and stable theories. J. Math. Log., 23(2), Paper No. 2250020, 26. arXiv: 2009.08365 DOI: 10.1142/S0219061322500209 MR: 4575271
type: article
abstract: We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analogue of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories.
keywords:
Sh:1198
Golshani, M., & Shelah, S. (2023). On slow minimal reals I. Proc. Amer. Math. Soc., 151(10), 4527–4536. arXiv: 2010.10812 DOI: 10.1090/proc/16397 MR: 4643336
type: article
abstract: Answering a question of Harrington, we show that there exists a proper forcing notion, which adds a minimal real \eta \in \prod_{i<\omega} n^*_i, which is eventually different from any old real in \prod_{i<\omega} n^*_i, where the sequence \langle n^*_i \mid i<\omega \rangle grows slowly.
keywords:
Sh:1199
Goldstern, M., Kellner, J., Mejı́a, D. A., & Shelah, S. (2021). Preservation of splitting families and cardinal characteristics of the continuum. Israel J. Math., 246(1), 73–129. arXiv: 2007.13500 DOI: 10.1007/s11856-021-2237-7 MR: 4358274
type: article
abstract: Using a finite support forcing iteration with many automorphisms, we force different values not only to “all” cardinals in Cichoń’s Diagram, but also to \mathfrak s, \mathfrak r, \mathfrak p, \mathfrak h, and to one of the Knaster numbers, so we get 15 different cardinal characteristics of the continuum.
keywords:
Sh:1201
Mühlherr, B., Paolini, G., & Shelah, S. (2022). First-order aspects of Coxeter groups. J. Algebra, 595, 297–346. arXiv: 2010.13161 DOI: 10.1016/j.jalgebra.2021.12.033 MR: 4359398
type: article
abstract: (none)
keywords:
Sh:1202
Farah, I., & Shelah, S. (2022). Between reduced powers and ultrapowers, II. Trans. Amer. Math. Soc., 375(12), 9007–9034. arXiv: 2011.07352 DOI: 10.1090/tran/8777 MR: 4504659
type: article
abstract: (none)
keywords:
Sh:1203
Brian, W., Dow, A. S., & Shelah, S. (2022). The independence of GCH and a combinatorial principle related to Banach-Mazur games. Arch. Math. Logic, 61(1-2), 1–17. arXiv: 2011.01273 DOI: 10.1007/s00153-021-00770-x MR: 4380012
type: article
abstract: (none)
keywords:
Sh:1207
Kumar, A., & Shelah, S. (2023). Large Turing independent sets. Proc. Amer. Math. Soc., 151(1), 355–367. DOI: 10.1090/proc/16081 MR: 4504631
type: article
abstract: (none)
keywords:
Sh:1209
Shelah, S., & Strüngmann, L. H. (2021). Infinite Combinatorics in Mathematical Biology. BioSystems, 204, 14. DOI: 10.1016/j.biosystems.2021.104392
type: article
abstract: (none)
keywords:
Sh:1211
Halevi, Y., Kaplan, I., & Shelah, S. (2023). Infinite Stable Graphs With Large Chromatic Number II. J. Eur. Math. Soc. (JEMS). arXiv: 2103.13931 DOI: 10.4171/JEMS/1352
type: article
abstract: (none)
keywords:
Sh:1212
Shelah, S., & Steprāns, J. (2024). Some variations on the splitting number. Annals of Pure and Applied Logic, (175). DOI: 10.1016/j.apal.2023.103321
type: article
abstract: (none)
keywords:
Sh:1213
Juhász, I., Shelah, S., Soukup, L., & Szentmiklóssy, Z. (2023). Large strongly anti-Urysohn spaces exist. Topology Appl., 323, Paper No. 108288, 15. arXiv: 2106.00618 DOI: 10.1016/j.topol.2022.108288 MR: 4518085
type: article
abstract: (none)
keywords:
Sh:1215
Golshani, M., & Shelah, S. (2023). The Keisler-Shelah isomorphism theorem and the continuum hypothesis. Fund. Math., 260(1), 59–66. arXiv: 2108.03977 MR: 4516185
type: article
abstract: We show that if for any two elementary equivalent structures \bold M, \bold N of size at most continuum in a countable language, \bold M^{\omega}/ \mathcal U \simeq \bold N^\omega / \mathcal U for some ultrafilter \mathcal U on \omega, then CH holds. We also provide some consistency results about Keisler and Shelah isomorphism theorems in the absence of CH.
keywords:
Sh:1216
Golshani, M., & Shelah, S. (2023). Usuba’s principle UB_\lambda can fail at singular cardinals. J. Symbolic Logic. arXiv: 2107.09339 DOI: 10.1017/jsl.2023.61
type: article
abstract: We answer a question of Usuba by showing that the combinatorial principle UB_\lambda can fail at a singular cardinal. Furthermore, \lambda can be taken to be \aleph_\omega.
keywords:
Sh:1218
Malliaris, M., & Shelah, S. (2024). Some simple theories from a Boolean algebra point of view. APAL, (175). arXiv: 2108.05314 DOI: 10.106/j.apal.2023.103345
type: article
abstract: (none)
keywords:
Sh:1219
Benhamou, T., Garti, S., & Shelah, S. (2023). Kurepa trees and the failure of the Galvin property. Proc. Amer. Math. Soc., 151(3), 1301–1309. arXiv: 2111.11823 DOI: 10.1090/proc/16198 MR: 4531656
type: article
abstract: We prove the consistency of the failure of the Galvin property at some \kappa complete ultrafilter over a measurable cardinal \kappa.
keywords:
Sh:1220
Halbeisen, L. J., Plati, R., Schumacher, S., & Shelah, S. (2023). Four cardinals and their relations in ZF. Ann. Pure Appl. Logic, 174(2), Paper No. 103200, 17. arXiv: 2109.11315 DOI: 10.1016/j.apal.2022.103200 MR: 4498208
type: article
abstract: (none)
keywords:
Sh:1223
Golshani, M., & Shelah, S. (2023). The Keisler-Shelah isomorphism theorem and the continuum hypothesis II. Monatshefte Fur Mathematik, 201, 789–801. arXiv: 2112.15468 MR: 4595008
type: article
abstract: We continue the investigation started in [Sh:1215] about the relation between the Keilser-Shelah isomorphism theorem and the continuum hypothesis. In particular, we show it is consistent that the continuum hypothesis fails and for any given sequence \langle (\mathbb{M}^{1}_n, \mathbb{M}^{2}_n: n < \omega \rangle of models of size at most \aleph_1 in a countable language, if the sequence satisfies a mild extra property, then for every non-principal ultrafilter \mathcal D on \omega, if the ultraproducts \prod\limits_{\mathcal D} \mathbb{M}^{1}_n and \prod\limits_{\mathcal D} \mathbb{M}^{2}_n are elementarily equivalent, then they are isomorphic.
keywords:
Sh:1225
Fischer, V., & Shelah, S. (2022). The spectrum of independence, II. Ann. Pure Appl. Logic, 173(9), Paper No. 103161. DOI: 10.1016/j.apal.2022.103161 MR: 4452678
type: article
abstract: (none)
keywords:
Sh:1228
Dow, A. S., & Shelah, S. (2023). S-spaces and large continuum. Topology Appl., 333, Paper No. 108526, 18. arXiv: 2206.11122 DOI: 10.1016/j.topol.2023.108526 MR: 4579945
type: article
abstract: (none)
keywords:
Sh:1232
Asgharzadeh, M., Golshani, M., & Shelah, S. (2023). Co-Hopfian and boundedly endo-rigid mixed abelian groups. PACIFIC JOURNAL OF MATHEMATICS. arXiv: 2210.17210
type: article
abstract: For a given cardinal \lambda and a torsion abelian group K of cardinality less than \lambda, we present, under some mild conditions (for example \lambda=\lambda^{\aleph_0}), boundedly endo-rigid abelian group G of cardinality \lambda with Tor(G)=K. Essentially, we give a complete characterization of such pairs (K, \lambda). Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existing problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups. In particular, we give useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals \lambda> 2^{\aleph_{0}} for which there is a co-Hopfian abelian group of size \lambda.
keywords:
Sh:E3
Shelah, S. (1996). On some problems in general topology. In Set theory (Boise, ID, 1992–1994), Vol. 192, Amer. Math. Soc., Providence, RI, pp. 91–101. arXiv: 0708.1981 DOI: 10.1090/conm/192/02352 MR: 1367138
type: article
abstract: We prove that Arhangelskii’s problem has a consistent positive answer: if V satisfies CH, then for some \aleph_1-complete \aleph_2-c.c. forcing notion P of cardinality \aleph_2 we have that P forces “CH and there is a Lindelof regular topological space of size \aleph_2 with clopen basis with every point of pseudo-character \aleph_0 (i.e. each singleton is the intersection of countably many open sets)”. Also, we prove the consistency of: CH+ 2^{\aleph_1} > \aleph_2 + "there is no space as above with \aleph_2 points" (starting with a weakly compact cardinal).
keywords: S: for, O: top, (set), (gt)
Sh:E9
Shelah, S. (1996). Remarks on \aleph_1-CWH not CWH first countable spaces. In Set theory (Boise, ID, 1992–1994), Vol. 192, Amer. Math. Soc., Providence, RI, pp. 103–145. arXiv: math/9408202 DOI: 10.1090/conm/192/02353 MR: 1367139
type: article
abstract: (none)
keywords: O: top
Sh:E33
Shelah, S., & Soifer, A. (2003). Axiom of choice and chromatic number of the plane. J. Combin. Theory Ser. A, 103(2), 387–391. DOI: 10.1016/S0097-3165(03)00102-X MR: 1996076
type: article
abstract: We will present here an example of a distance graph on the line R, whose chromatic number depends a great deal upon the axiom of choice. While the setting of our example differs from that of chromatic number of the plane problem, the example illuminates how the value of chromatic number may be affected by inclusion or exclusion of the axiom of choice in the system of axioms for sets.
keywords: S: ods, (set)
Sh:E33a
Shelah, S., & Soifer, A. (2003). Chromatic number of the plane. III. Its future. Geombinatorics, 13(1), 41–46. MR: 1985343
type: article
abstract: (none)
keywords: (set), (graph), (AC)
Sh:E33b
Shelah, S., & Soifer, A. (2004). How the axiom of choice can affect the chromatic number of distance graphs: three examples on the plane. In Proceedings of the Thirty-Fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing, Vol. 166, pp. 5–9. MR: 2121999
type: article
abstract: (none)
keywords: (graph), (AC)
Sh:E51
Shelah, S., & Soifer, A. (2004). Axiom of choice and chromatic number: examples on the plane. J. Combin. Theory Ser. A, 105(2), 359–364. DOI: 10.1016/j.jcta.2004.01.001 MR: 2046089
type: article
abstract: (none)
keywords: S: ods, (set)
Sh:E87
Goldstern, M., Kellner, J., Mejı́a, D. A., & Shelah, S. (2022). Controlling classical cardinal characteristics while collapsing cardinals. Colloq. Math., 170(1), 115–144. arXiv: 1904.02617 DOI: 10.4064/cm8420-2-2022 MR: 4460218
type: article
abstract: (none)
keywords: S: for, (iter), (inv)

Accepted research papers

Research articles (co)authored by S. Shelah accepted in peer reviewed journals.

Sh:832
Greenberg, N., & Shelah, S. Many forcing axioms for all regular uncountable cardinals. Israel J. Math. To appear. arXiv: 2107.05755
type: article
abstract: A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of “for every suitable forcing notion for \lambda” we shall say “for every such family of forcing notions, depending on stationary S\subseteq \lambda, for some such stationary set we have…”. Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.
keywords: S: for, (iter), (unif)
Sh:835
Shelah, S. PCF without choice. Arch. Math. Logic. To appear. arXiv: math/0510229
type: article
abstract: We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of \lambda is well ordered for every \lambda (really local version for a given \lambda). We think that in this frame much of pcf theory, (and combinatorial set theory in general) can be generalized. We prove here, in particular, that there is a proper class of regular cardinals, every large enough successor of singular is not measurable and we can prove cardinal inequalities.

Solving some open problems, we prove that if \mu > \kappa = cf(\mu) > \aleph_{0}, then from a well ordering of P(P(\kappa)) \cup {}^{\kappa >} \mu we can define a well ordering of {}^{\kappa} \mu.
keywords: S: pcf, S: ods, (set), (AC)
Sh:1019
Shelah, S. Model theory for a compact cardinal. Israel J. Math. To appear. arXiv: 1303.5247
type: article
abstract: We would like to develop classification theory for T, a complete theory in \mathbb{L} _{\theta,\theta}(\tau) when \theta is a compact cardinal. We already have bare bones stability theory and it seemed we can go no further. Dealing with ultrapowers (and ultraproducts) naturally we restrict ourselves to “D a \theta-complete ultrafilter on I, probably (I,\theta)-regular". The basic theorems of model theory work and can be generalized (like Łos’ theorem), but can we generalize deeper parts of model theory?

The first section is trying to sort out what occurs to the notion of “stable T" for complete \mathbb{L} _{\theta,\theta}-theories T. We generalize several properties of complete first order T, equivalent to being stable (see the author classification theory book and find out which implications hold and which fail.

In particular, can we generalize stability enough to generalize Chap.VI of the author classification theory book Let us concentrate on saturation in the local sense (types consisting of instances of one formula). We prove that at least we can characterize the T-s (of cardinality \le \theta for simplicity) which are minimal for appropriate cardinal \lambda \ge 2^\kappa + |T| in each of the following two senses. One is generalizing Keisler order \triangleleft, which measures how saturated ultrapowers are. Another generalizes the results on \triangleleft^*. That is, we ask: “Is there an \mathbb{L} _{\theta,\theta}-theory T_1 \supseteq T of cardinality |T| + 2^\theta such that for every model M_1 of T_1 of cardinality > \lambda, the \tau(T)-reduct M of M_1 is \lambda^+-saturated?" Moreover, the two versions of stable used in the characterization are different.
keywords: M: nec, (inf-log)
Sh:1108
Paolini, G., & Shelah, S. The strong small index property for free homogeneous structures. In Research Trends in Contemporary Logic. To appear. arXiv: 1703.10517
type: article
abstract: We show that in countable homogeneous structures with canonical amalgamation and locally finite algebraicity the small index property implies the strong small index property. We use this and the main result of [siniora] to deduce that countable free homogeneous structures in a locally finite irreflexive relational language have the strong small index property. As an application, we exhibit new continuum sized classes of \aleph_0-categorical structures with the strong small index property whose automorphism groups are pairwise non-isomorphic.
keywords: S: str, O: alg
Sh:1125
Golshani, M., & Shelah, S. On C^s_n(\kappa) and the Juhasz-Kunen question. Notre Dame J. Formal Logic. To appear. arXiv: 1709.06249
type: article
abstract: We generalize the combinatorial principles C_n(\kappa), C^s_n(\kappa) and Princ(\kappa) introduced by various authors, and prove some of their properties and connections between them. We also answer a question asked by Juhasz-Kunen about the relation between these principles, by showing that C_n(\kappa) does not imply C_{n+1}(\kappa), for any n>2. We also show the consistency of C(\kappa)+\neg C^s(\kappa).
keywords: S: for, O: top
Sh:1205
Paolini, G., & Shelah, S. Torsion-free abelian groups are Borel complete. Ann. Of Math. To appear. arXiv: 2102.12371
type: article
abstract: (none)
keywords:
Sh:1206
Malliaris, M., & Shelah, S. New simple theories from hypergraph sequences. Model Theory. To appear. arXiv: 2108.05526
type: article
abstract: (none)
keywords:
Sh:1210
Kostana, Z., & Shelah, S. Slicing axiom. Proceedings of the AMS. To appear. arXiv: 2102.11666
type: article
abstract: (none)
keywords:
Sh:1222
Malliaris, M., & Shelah, S. The Turing degrees and Keisler’s order. J. Symbolic Logic. To appear. DOI: 10.1017/jsl.2022.63
type: article
abstract: (none)
keywords:
Sh:1233
Corson, S. M., & Shelah, S. On projections of the tails of a power. Forum Mathematicum. To appear. arXiv: 2211.13749
type: article
abstract: (none)
keywords:

Articles in preparation

Preprints (co)authored by S. Shelah (intended for publication).

Sh:259
Grossberg, R. P., & Shelah, S. On Hanf numbers of the infinitary order property. Preprint. arXiv: math/9809196
type: article
abstract: We study several cardinal, and ordinal–valued functions that are relatives of Hanf numbers. Let \kappa be an infinite cardinal, and let T\subseteq L_{\kappa^+,\omega} be a theory of cardinality \leq\kappa, and let \gamma be an ordinal \geq\kappa^+. For example we look at (1) \mu_{T}^*(\gamma,\kappa):=\min\{\mu^*\forall\phi\in L_{\infty,\omega}, with rk(\phi)<\gamma, if T has the (\phi,\mu^*)-order property then there exists a formula \phi'(x;y)\in L_{\kappa^+,\omega}, such that for every \chi\geq\kappa, T has the (\phi',\chi)-order property\}; and (2) \mu^*(\gamma,\kappa):=\sup\{\mu_{T}^*(\gamma,\kappa)\;|\;T\in L_{\kappa^+,\omega}\}.
keywords: M: nec, (mod)
Sh:311
Shelah, S. A more general iterable condition ensuring \aleph_1 is not collapsed. Preprint. arXiv: math/0404221
type: article
abstract: In a self-contained way, we deal with revised countable support iterated forcing for the reals. We improve theorems on preservation of the property UP, weaker than semi proper, and we hopefully improve the presentation. We continue [Sh:b, Ch.X,XI] (or see [Sh:f, Ch.X,XI]), and Gitik and Shelah [GiSh:191] and [Sh:f, Ch.XIII,XIV] and particularly Ch.XV. Concerning “no new reals” see lately Larson and Shelah [LrSh:746]. In particular, we fulfill some promises from [Sh:f] and give a more streamlined version.
keywords: S: for, (set), (iter)
Sh:322a
Shelah, S., & Usvyatsov, A. Classification over a predicate — the general case I. Preprint. arXiv: 1910.10811
type: article
abstract: We begin systematic development of structure theory for a first order theory stable over a monadic predicate. We show that stability over a predicate implies quantrifier free definability of types over stable sets, introduce an independence notion and explore its properties, prove stable amalgamation results, and show that every type over a model, orthogonal to the predicate, is generically stable.
keywords: M: cla, (mod), (nni)
Sh:322b
Shelah, S., & Usvyatsov, A. Classification over a predicate — the general case II. Preprint.
type: article
abstract: (none)
keywords: M: cla, (mod), (nni)
Sh:421
Asgharzadeh, M., Golshani, M., & Shelah, S. Kaplansky test problems for R-modules in ZFC. Preprint. arXiv: 2106.13068
type: article
abstract: We conclude a long-standing research program in progress since 1954 by giving negative answers to test problems Kaplansky. Among these problems, the first was largely open, but the others were known to be consequences of Jensen’s diamond principle and therefore impossible to answer affirmatively. Let R be a left non-pure semisimple, and let m> 1 be a natural number. For example, we construct an R-module M such that M^n\cong M if and only if m divides n-1, thus solving the first test problem in the negative. As an application, we also construct an R-module M of arbitrary size such that M^{n_1}\cong M^{n_2} if and only if m divides (n_1-n_2), giving a strongly negative answer to the cube problem of whether an R-module M which is isomorphic to M^3 must be isomorphic to its square M^2? We will treat the other two problems similarly. The crux of our method is to construct a ring S and an (R, S)-bimodule with few endomorphisms, for which we rely heavily on techniques from algebra and set theory, in particular the black box.
keywords: M: non, (ab)
Sh:423
Shelah, S. Compactness spectrum. Preprint.
type: article
abstract: (none)
keywords: S: ico, O: alg, (stal)
Sh:538
Shelah, S. Historic iteration with \aleph_\varepsilon-support. Preprint. arXiv: math/9607227
type: article
abstract: One aim of this work is to get a universe in which weak versions of Martin axioms holds for some forcing notions of cardinality \aleph_0,\aleph_1 and \aleph_2 while on \aleph_2 club, the “small" brother of diamond, holds. As a consequence we get the consistency of “there is no Gross space". Another aim is to present a case of “Historic iteration".
keywords: S: for, S: str, (set)
Sh:550
Larson, P. B., & Shelah, S. 0-1 laws. Preprint.
type: article
abstract: We give a framework for dealing with 0-1 laws for first order logic such that expanding by further random structure tends to give us another case of the framework. From another perspective we deal with 0-1 laws when the number of solutions of first order formulas with parameters behaves dichotomically.
keywords: (fmt)
Sh:555
Scheepers, M., & Shelah, S. Embeddings of partial orders into \omega^\omega. Preprint.
type: article
abstract: (none)
keywords: S: for, (set), (app(pcf))
Sh:581
Shelah, S. When 0–1 law hold for G_{n,\bar{p}}, \bar{p} monotonic. Preprint.
type: article
abstract: (none)
keywords: M: odm, O: fin, (fmt)
Sh:637
Shelah, S. 0.1 Laws: Putting together two contexts randomly. Preprint.
type: article
abstract: (none)
keywords: M: odm, O: fin, (fmt)
Sh:656
Golshani, M., & Shelah, S. NNR Revisited. Preprint. arXiv: math/0003115
type: article
abstract: We show if we use countable support iteration of forcing notions not adding reals that satisfy additional conditions, then the limit forcing does not add reals. As a result we prove that we can amalgamate two earlier methods and prove the consistency with ZFC + GCH of two statements gotten separately earlier: Souslin hypothesis and non-club guessing. We also answer a question of Justin Moore by proving the consistency of one further case of “strong failure of club guessing" with GCH.
keywords: S: for, (set), (iter)
Sh:670
Rosłanowski, A., & Shelah, S. Norms on possibilities III: strange subsets of the real line. Preprint.
type: article
abstract: (none)
keywords: S: for, S: str, (set), (creatures)
Sh:707
Cardona, M., & Shelah, S. Long iterations for the continuum. Preprint. arXiv: math/0112238
type: article
abstract: We deal with an iteration theorem of forcing notion with a kind of countable support of nice enough forcing notion which is proper \aleph_2-c.c. forcing notions. We then look at some special cases ({\mathbb Q}_D’s preceded by random forcing).
keywords: S: str, (set), (iter)
Sh:798
Shelah, S., & Väänänen, J. A. The \Delta–closure of L(Q_1) is not finitely generated, assuming CH. Preprint.
type: article
abstract: We prove that, assuming CH, the logic \Delta(L(Q_1)) is not finitely generated. This answers a long-standing open problem
keywords: M: smt
Sh:800
Shelah, S. On complicated models and compact quantifiers. Preprint.
type: article
abstract: What we do can be looked at as:
1. finding and classifying compact second order logic quantifiers on automorphisms of definable models of \psi which are already definable,
2. building a model M such that if we define in M a model N = N_{M, \bar \psi} of \psi, then any automorphism of N is inner (that is, first order definable in M) at least in some respect.
3. This can be looked at as classifying the \psi-s; so for more complicated \psi-s we have fewer such automorphisms.
4. As a test case, we consider the specific examples of “the model completion of the theory of triangle-free graphs."
More elaborately, we look here again at building models M with second order properties. In particular, M such that every isomorphism between two interpretations of a theory t in M is definable in M or at least is “somewhat" definable (e.g. having a dense linear order, saying this holds for a dense family of intervals). For transparency we can concentrate on t-s of finite vocabulary. If we restrict ourselves to finite t-s, this implies that we get a compact logic when we add to first order logic the second order quantifiers on isomorphisms from one interpretation. We already know this in some instances (e.g. t the theory of Boolean Algebras or the theory of ordered fields) but here we try to analyze a general t. Hence, at least for the time being, we try to sort out what we can get by forcing rather than really proving it (in ZFC).

We may consider the question: for a given T if there is an \kappa-iso-rigid-model of T (so \kappa-full), then our constructions give one. For more details, see the introduction to [Sh:384].
keywords: M: non, (mod)
Sh:810
Shelah, S. The height of the automorphism tower of a group. Preprint. arXiv: math/0405116
type: article
abstract: For a group G with trivial center there is a natural embedding of G into its automorphism group, so we can look at the latter as an extension of the group. So an increasing continuous sequence of groups, the automorphism tower, is defined, the height is the ordinal where this becomes fixed, arriving to a complete group. We first show that for any \kappa = \kappa^{\aleph_0} there is a group of cardinality \kappa with height > \kappa^+ (improving the lower bound). Second we show that for many such \kappa there is a group of height > 2^\kappa, so proving that the upper bound essentially cannot be improved.
keywords: S: ods, O: alg, (set), (grp)
Sh:839
Shelah, S. AEC: weight and p-simplicity. Preprint. arXiv: 2305.01970
type: article
abstract: Part I: We would like to generalize imaginary elements, weight of {\rm ortp}(a,M,N),{\mathbf P}-weight, {\mathbf P}-simple types, etc. from [Sh:c, Ch.III,V,§4] to the context of good frames. This requires allowing the vocabulary to have predicates and function symbols of infinite arity, but it seemed that we do not suffer any real loss.

Part II: become [1238] Good frames were suggested in [Sh:h] as the (bare bones) right parallel among a.e.c. to superstable (among elementary classes). Here we consider (\mu,\lambda,\kappa)-frames as candidates for being the right parallel to the class of |T|^+-saturated models of a stable theory (among elementary classes). A loss as compared to the superstable case is that going up by induction on cardinals is problematic (for cardinals of small cofinality). But this arises only when we try to lift. But this context we investigate the dimension.

Part III: become [1239] In the context of Part II, we consider the main gap problem for the parallel of somewhat saturated model; showing we are not worse than in the first order case.
keywords: M: cla, (mod), (sta), (aec)
Sh:842
Shelah, S., & Vasey, S. Categoricity and multidimensional diagrams. Preprint. arXiv: 1805.06291
type: article
abstract: We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high-enough cardinality will have a single model in any high-enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation.
keywords: (mod), (cat), (aec)
Sh:928
Shelah, S., & Usvyatsov, A. Unstable Classes of Metric Structures. Preprint. arXiv: 0810.0734
type: article
abstract: We prove a strong nonstructure theorem for a class of metric structures with an unstable formula. As a consequence, we show that weak categoricity (that is, categoricity up to isomorphisms and not isometries) implies several weak versions of stability. This is the first step in the direction of the investigation of weak categoricity of metric classes.
keywords: M: cla, (mod)
Sh:932
Shelah, S. Maximal failures of sequence locality in a.e.c. Preprint. arXiv: 0903.3614
type: article
abstract: We are interested in examples of a.e.c. with amalgamation having some (extreme) behaviour concerning types. Note we deal with \mathfrak{k} being sequence-local, i.e. local for increasing chains of length a regular cardinal. For any cardinal \theta \ge \aleph_0 we construct an a.e.c. with amalgamation \mathfrak{k} with L.S.T.(\mathfrak{k}) = \theta,|\tau_\mathfrak{K}| = \theta such that \{\kappa:\kappa is a regular cardinal and \mathfrak{K} is not (2^\kappa,\kappa)-sequence-local\} is maximal. In fact we have a direct characterization of this class of cardinals: the regular \kappa such that there is no uniform \kappa^+-complete ultrafilter. We also prove a similar result to “(2^\kappa,\kappa)-compact for types”.
keywords: M: nec, (mod), (aec)
Sh:940
Jarden, A., & Shelah, S. Non forking good frames minus local character. Preprint. arXiv: 1105.3674
type: article
abstract: We prove that if s is an almost good \lambda-frame (i.e. s is a good \lambda-frame except that it satisfies just a weak version of local character), then we can complete the frame s.t. it will satisfy local character too. This theorem has an important application. In [she46] it has proved that (under mild set theoretic assumptions) Categoricity in \lambda,\lambda^+ and intermediate number of models in K_{\lambda^{++}} implies existence of an almost good \lambda-frame. So by our theorem, we can get the local character too. So by categoricity assumptions in \lambda,\lambda^+, \lambda^{++} we can get existence of a good \lambda-frame. Combining this with [sh600], we conclude that the function \lambda \rightarrow I(\lambda,K), which correspond to each cardinal \lambda, the number of models in K of cardinality \lambda, is not arbitrary.
keywords: M: nec, (mod), (aec)
Sh:950
Shelah, S. Dependent dreams: recounting types. Preprint. arXiv: 1202.5795
type: article
abstract: We investigate the class of models of a general dependent theory. We continue [Sh:900] in particular investigating so called “decomposition of types"; thesis is that what holds for stable theory and for Th(\mathbb Q,<) hold for dependent theories. Another way to say this is: we have to look at small enough neighborhood and use reasonably definable types to analyze a type. We note the results understable without reading. First, a parallel to the “stability spectrum", the “recounting of types", that is assume \lambda = \lambda^{< \lambda} is large enough, M a saturated model of T of cardinality \lambda, let \mathbf S_{\text{aut}}(M) be the number of complete types over M up to being conjugate, i.e. we identify p,q when some automorphism of M maps p to q. Whereas for independent T the number is 2^\lambda, for dependent T the number is \le \lambda moreover it is \le |\alpha|^{|T|} when \lambda = \aleph_\alpha. Second, for stable theories “lots of indiscernibility exists" a “too good indiscernible existence theorem" saying, e.g. that if the type tp(d_\beta;\{d_\beta:\beta < \alpha\}) is increasing for \alpha < \kappa = \text{ cf}(\kappa) and \kappa > 2^{|T|} then \langle d_\alpha:\alpha \in S\rangle is indiscernible for some stationary S \subseteq \kappa. Third, for stable T,a model is \kappa-saturated iff it is \aleph_\varepsilon-saturated and every infinite indiscernible set (of elements) of cardinality < \kappa can be increased. We prove here an analog. Fourth, for p \in \mathbf S(M), the number of ultrafilters on the outside definable subsets of M extending p has an absolute bound 2^{|T|}. Restricting ourselves to one \varphi(x,\bar y), the number is finite, with an absolute found (well depending on T and \varphi). Also if M is saturated then p is the average of an indiscernible sequence inside the model. Lastly, the so-called generic pair conjecture was proved in [Sh:900] for \kappa measurable, here it is essentially proved, i.e. for \kappa > |T| + \beth_\omega.
keywords: M: cla, (mod)
Sh:966
Jarden, A., & Shelah, S. Existence of uniqueness triples without stability. Preprint.
type: article
abstract: (none)
keywords: M: nec
Sh:986
Haber, S., & Shelah, S. Random graphs and Lindström quantifiers for natural graph properties. Preprint. arXiv: 1510.06574
type: article
abstract: We study zero-one laws for random graphs. We focus on the following question that was asked by many: Given a graph property P, is there a language of graphs able to express P while obeying the zero-one law? Our results show that there is a (regular) language able to express connectivity and k-colorability for any constant k and still obey the zero-one law. On the other hand we show that in any (semiregular) language strong enough to express Hamiltonicity one can interpret arithmetic and thus the zero-one law fails miserably. This answers a question of Blass and Harary.
keywords: M: odm, O: fin, (fmt)
Sh:1010
Shelah, S., & Usuba, T. \omega_1-Stationary preserving \sigma-Baire posets of size \aleph_1. Preprint.
type: article
abstract:
We prove that the following theories are equiconsistent with {\rm ZFC}:
1. {\rm ZFC}+{\rm CH}+ “there is an \omega _1-stationary preserving \sigma-Baire poset of size \aleph_1 which is not semiproper”.
2. {\rm ZFC}+“Martin’s axiom for semiproper posets of size \aleph_1” + “there is an \omega _1-stationary preserving \sigma-Baire poset of size \aleph_1 which is not semiproper”.
3. {\rm ZFC}+{\rm CH}+“every \omega _1-stationary preserving \sigma-Baire poset of size \aleph_1 is semiproper”.
keywords: S: for
Sh:1028p
Shelah, S. clarifications of proofs for the Journal. Preprint.
type: article
abstract: (none)
keywords: S: ico, S: for, O: alg, (ab), (stal), (af)
Sh:1045
Shelah, S. Quite free Abelian groups with prescribed endomorphism ring. Preprint.
type: article
abstract: In [Sh:1028] we like to build Abelian groups (or R-modules) which on the one hand are quite free, say \aleph_{\omega +1}-free, and on the other hand, are complicated in suitable sense. We choose as our test problem having no non-trivial homomorphism to \mathbb Z (known classically for \aleph_1-free, recently for \aleph_n-free). We get even \aleph_{\omega_1 \cdot n}-free. Other applications were delayed to the present work.

The construction (there and here) requires building n-dimensional black boxes, which are quite free. Here we continue [Sh:1028] (in some ways). In particular, we consider building quite free Abelian groups with a pre-assigned ring of endomorphism.

In §2 we try to elaborate [Sh:1028, §(2B)], on “minimal" {\rm Hom}(G,{}_R R), e.g. for separable p-groups. In §(3C), §(3D) are some old continuation of [Sh:1028, §3], so of unclear status. In §(4A) we control {\rm End}(G,R,+), e.g. =R. In §(4B), done when we have a 1-witness; seems nearly done. In §(4C) we intend to fill the 4-witness case.

Presently (2014.1.13) it seems that §(4A), §(4B) do something, construct from a 1-witness, which §(4C) has to be completed. Also §(2A) do something, not clear how final. In SEPT 2017 lecture on it in Simonfest; have added in the end of sec 1, analysis of the simple case- R is co-torsion free
keywords: S: ico, O: alg, (ab), (stal), (af)
Sh:1046
Kumar, A., & Shelah, S. RVM, RVC revisited: Clubs and Lusin sets. Preprint.
type: article
abstract: A cardinal \kappa is Cohen measurable (RVC) if for some \kappa-additive ideal \mathcal{I} over \kappa, \mathcal{P}(\kappa) / \mathcal{I} is forcing isomorphic to adding \lambda Cohen reals for some \lambda. Such cardinals can be obtained by starting with a measurable cardinal \kappa and adding at least \kappa Cohen reals. We construct various models of RVC with different properties than this model.Our main results are: (1) \kappa = 2^{\omega} is RVC does not decide \clubsuit_S for various stationary S \subseteq \kappa. (2) \kappa \leq \lambda = cf(\lambda) < 2^{\omega} does not decide \clubsuit_S for various stationary S \subseteq \lambda. (3) \kappa = 2^{\omega} is RVC does not decide the existence of a Lusin set of size \kappa.
keywords: S: for, S: str
Sh:1049
Shelah, S. On group well represented as automorphic groups of groups. Preprint.
type: article
abstract: Assuming (less than) \mathbf V = \mathbf L, we characterize group GL such that there are arbitrarily large group H such that \rm{aut}(H) \cong L. In particular it suffies to have one of cardinal >|L|^{\aleph_0}. In addition, if |L| < \aleph_\omega no need for \mathbf V = \mathbf L (full). Similarly for End(G) \cong L so L is semi-group (fill)
keywords: S: ico, O: alg
Sh:1065
Shelah, S., & Ulrich, D. \le_{SP} can have infinitely many classes. Preprint. arXiv: 1804.08523
type: article
abstract: Building off of recent results on Keisler’s order, we show that consistently, \leq_{SP} has infinitely many classes. In particular, we define the property of \leq k-type amalgamation for simple theories, for each 2 \leq k < \omega. If we let T_{n, k} be the theory of the random k-ary, n-clique free random hyper-graph, then T_{n, k} has \leq k-1-type amalgamation but not \leq k-type amalgamation. We show that consistently, if T has \leq k-type amalgamation then T_{k+1, k} \not \leq_{SP} T, thus producing infinitely many \leq_{SP}-classes. The same construction gives a simplified proof of Shelah’s theorem that consistently, the maximal \leq_{SP}-class is exactly the class of unsimple theories. Finally, we show that consistently, if T has <\aleph_0-type amalgamation, then T \leq_{SP} T_{rg}, the theory of the random graph.
keywords: M: cla
Sh:1067
Horowitz, H., & Shelah, S. Saccharinity with ccc. Preprint. arXiv: 1610.02706
type: article
abstract: Consistency of saccharinity for a nice ccc ideal but only for ZF, using inverse limit of finite iterations
keywords: S: for, S: str
Sh:1073
Larson, P. B., & Shelah, S. On the absoluteness of orbital \omega-stability. Preprint.
type: article
abstract: We show that orbital \omega-stability is upwards absolute (but not downwards absolute) for \aleph_0-presented Abstract Elementary Classes satisfying amalgamation and the joint embedding property (each for countable models). We also show that amalgamation does not imply upwards absoluteness of orbital \omega-stability by itself. The corresponding question for the joint embedding property remains open.
keywords: M: smt
Sh:1077
Shelah, S. Random graph: stronger logic but with the zero one law. Preprint. arXiv: 1511.05383
type: article
abstract: We find a logic really stronger than first order for the random graph with edge probability \frac 12 but satisfies the 0-1 law. This means that on the one hand it satisfies the 0-1 law, e.g. for the random graph {\mathcal G}_{n,1/2} and on the other hand there is a formula \varphi(x) such that for no first order \psi(x) do we have: for every random enough {\mathcal G}_{n,1/2} the formulas \varphi(x),\psi(x) equivalent in it.
keywords: M: odm, O: fin, O: odo, (fmt)
Sh:1094
Horowitz, H., & Shelah, S. Solovay’s inaccessible over a weak set theory without choice. Preprint. arXiv: 1609.03078
type: article
abstract: We study the consistency strength of Lebesgue measurability for \Sigma^1_3 sets over Zermelo set theory (Z) in a completely choiceless context. We establish a result analogous to the Solovay-Shelah theorem
keywords: S: for, S: str
Sh:1096
Shelah, S. Strong failure of 0-1 law for LFP and the path logics. Preprint.
type: article
abstract: we continue [1062], in two issues. First we introduce a logic, called path logic which seem natural there and try to prove for it the 0-1 law Second, for the random graph G_{n, n^ \alpha} in [1062] we get only a weak failure of the 01 law, the truth value of the sentence is sometimes close to 1 and sometimes close to 0, but it change slowly. Here we intend to prove that tehere is an interpretaion which for a random enough graph give number theory on n, so we can "define n being odd/even"
keywords: M: odm, O: fin, (fmt)
Sh:1097
Golshani, M., Horowitz, H., & Shelah, S. On the classification of definable ccc forcing notions. Preprint. arXiv: 1610.07553
type: article
abstract: We show that for a Suslin ccc forcing notion \mathbb Q adding a Hechler real, “\text{ZF}+\text{DC}_{\omega_1}+all sets of reals are I_{\mathbb Q,\aleph_0}-measurable” implies the existence of an inner model with a measurable cardinal. We also introduce a wide class of Suslin ccc forcing notions which add a Hechler real, so that the above result applies to them.
keywords: S: for, S: str, S: dst
Sh:1098
Shelah, S. LF groups, aec amalgamation, few automorphisms. Preprint. arXiv: 1901.09747
type: article
abstract: We deal mainly with \mathbf{K} ^{\rm lf}_\lambda, the class of locally finite groups of cardinality \lambda, in particular \mathbf{K} ^{\rm exlf}_\lambda, the class of existentially closed locally finite groups. In §3 we prove that for almost every cardinal \lambda “every locally finite group G of cardinality \lambda can be extended to an existentially closed complete group of cardinality \lambda which moreover is so called (\lambda,\theta)-full; note that §3 which do not rely on §1,§2. (in earlier results G has cardinality < \lambda and also \lambda was restricted).

In §1 we deal with amalgamation bases, for the class of lf (= locally finite) groups, and general suitable classes, we define when it has the (\lambda,\kappa)-amalgamation property which means that “many" models M \in K^{\mathfrak{k} }_\lambda are amalgamation bases and get more than expected. In this case, we deal with a general frame - so called a.e.c., abstract elementary class. In §2 we deal with weak definability of a \in N \backslash M over M, for = existentially closed {\rm lf} group.
keywords: M: non, O: alg, (grp)
Sh:1100
Shelah, S. Creature iteration for inaccessibles. Preprint.
type: article
abstract: Our starting point is [Sh:1004]. As there we concentrate on forcing for inaccessibles and our definition is by induction when we like to get a nice forcing. (A) Mainly we deal with iterations for \lambda inaccessible of creature forcing (so getting appropriate forcing axioms). We concentrate on the case the forcing is strategically (< \lambda)-complete \lambda^+-c.c. (even \lambda-centered) and mainly (i.e. in (A)) on cases leading to \lambda-bounding forcing. In this case we can start with 2^\lambda > \lambda^+ and the forcing preserves various statements. We allow \mathbf U_{\mathfrak x}, e.g. = \lambda^+ to deal, e.g. with the big at universal graphs in \lambda^+ < 2^\lambda, while {\mathfrak d}_\lambda = \lambda^+. The decision of “weakly compact" or demand? is done via the choice of \mathbf j. (B) A different case is in the same framework but naturally assuming 2^\lambda = \lambda^+. The forcing satisfies only the \lambda^{++}-c.c. and is \kappa-proper so do not collapse \lambda^+. We may make 2^\lambda arbitrarily large or weaken the demands on forcing and get 2^\lambda = \lambda^{++}. ’ We only later do something concerning this. (C) Changing the frame somewhat, we allow adding unbounded \lambda-reals (i.e. \eta \in {}^\lambda \lambda) without adding \lambda-Cohens. For this we need to assume \lambda is measurable and use a fix normal ultrafilter \mathbb{E} on it.

(D) For some purposes we need stronger changes in the framework: allowing H’s in the \mathbf i’s. This includes (f,g)-bounding.
keywords: S: for
Sh:1103
Horowitz, H., & Shelah, S. Mad families and non-meager filters. Preprint. arXiv: 1701.02806
type: article
abstract: We prove the consistency of ZF + DC + "there are no mad families" + "there exists a non-meager filter on \omega" relative to ZFC, answering a question of Neeman and Norwood. We also introduce a weaker version of madness, and we strengthen the result from [HwSh:1090] by showing that no such families exist in our model.
keywords: S: str, (AC)
Sh:1113
Horowitz, H., Karagila, A., & Shelah, S. Madness and regularity properties. Preprint. arXiv: 1704.08327
type: article
abstract: Starting from an inaccessible cardinal, we construct a model of ZF + DC where there exists a mad family and all sets of reals are \mathbb{Q}-measurable for \omega^\omega-bounding sufficiently absolute forcing notions \mathbb{Q}.
keywords: S: for, S: str, (AC)
Sh:1126a
Shelah, S. up to 1 Sept 2021 version of 1126 (simple bold m only. Preprint.
type: article
abstract: This is the pre 2021-09-01 version of [1126] with E'_ \mathbf{m} a two place relation
keywords: S: for, (iter)
Sh:1128
Garti, S., & Shelah, S. Length and Depth. Preprint. arXiv: 1804.05304
type: article
abstract: Addressing a question of Monk, we prove that the inequality \prod_{i<\kappa}{\rm Length}({\bf B}_i)/D < {\rm Length} (\prod_{i<\kappa}{\bf B}_i/D) can be established in ZFC, and a large gap is consistent with ZFC.
keywords: S: ico, S: for
Sh:1130
Larson, P. B., & Shelah, S. An Extendable structure with a rigid elementary extension. Preprint. arXiv: 1711.07333
type: article
abstract: (none)
keywords: M: nec
Sh:1147
Baldwin, J. T., & Shelah, S. Maximal models up to the first measurable in ZFC. Preprint. arXiv: 2111.01709
type: article
abstract: (none)
keywords: (diam)
Sh:1170
Rosłanowski, A., & Shelah, S. Borel sets without perfectly many overlapping translations II. In. Preprint. arXiv: 1909.00937
type: article
abstract: For a countable ordinal \varepsilon we construct \Sigma^0_2 subset of the Cantor space for which we one may force \aleph_\varepsilon translations with intersections of size \geq 2\iota, but with no perfect set of such translations in any ccc extension. These sets have uncountably many translations with intersections of size \geq 2\iota in ZFC.
keywords: S: str, S: dst
Sh:1174
Shelah, S. More forcing for no ultrafilters. Preprint.
type: article
abstract: (none)
keywords: S: for, (iter), (ultraf)
Sh:1176
Shelah, S. Partition theorems for expanded trees. Preprint. arXiv: 2108.13955
type: article
abstract: We look for partition theorems for large subtrees for suitable uncountable trees and colourings. We concentrate on sub-trees of {}^{\kappa \geq}2 expanded by a well ordering of each level. Unlike earlier works, we do not ask the embedding to preserve the height of the tree but the equality of levels is preserved. We get consistency results without large cardinals. The intention is to apply it to model theoretic problems.
keywords: S: for, (pc), (pure(for))
Sh:1178
Larson, P. B., & Shelah, S. Universally measurable sets may all be \boldsymbol{\Delta}^{1}_{2}. Preprint. arXiv: 2005.10399
type: article
abstract: We produce a forcing extension of the constructible universe \bf L in which every sufficiently regular subset of any Polish space is a continuous image of a coanalytic set. In particular, we show that consistently every universally measurable set is \Delta^{1}_{2}, partially answering question CG from David Fremlin’s problem list [FQL].
keywords: S: for, S: str, S: dst
Sh:1182
Golshani, M., & Shelah, S. Iterated Ramsey bounds for the Hales-Jewett numbers; withdrawn. Preprint. arXiv: 1912.08643
type: article
abstract: Consider the Hales-Jewett theorem. The k-dimensional version of it tells us that the combinatorial space \mathcal{U}_{M, \Lambda} = \{ \eta \mid \eta: M \to \Lambda \} has, under suitable assumptions, monochromatic k-dimensional subspaces, where by a k-dimensional subspace we mean there exist a partition \langle N_0, N_1, \cdots, N_k \rangle of M such that N_1, \cdots, N_k \neq \emptyset (but we allow N_0 to be empty) and some \rho_0: N_0 \to \Lambda, such that the subspace consists of those \rho \in \mathcal{U}_{M, \Lambda} such that for 0<l<k+1, \rho \restriction N_l is constant and \rho \restriction N_0= \rho_0.

It seems natural to think it is better to have each N_{l}, 0<l<k+1 a singleton. However it is then impossible to always find monochromatic k-dimensional subspaces (for example color \eta by 0 if |\eta^{-1}\{\alpha \}| is an even number and by 1 otherwise). But modulo restricting the sign of each |\eta^{-1}\{\alpha \}|, we prove the parallel theorem– whose proof is not related to the Hales-Jewett theorem. We then connect the two numbers by showing that the Hales-Jewett numbers are not too much above the present ones. This gives an alternative proof of the Hales-Jewett theorem.
keywords: (fc)
Sh:1183
Baldwin, J. T., Laskowski, M. C., & Shelah, S. An analog of U-rank for atomic classes. Preprint.
type: article
abstract: (none)
keywords: (diam)
Sh:1185
Poór, M., & Shelah, S. Universal graphs between a strong limit singular and its power. Preprint. arXiv: 2201.00741
type: article
abstract: (none)
keywords:
Sh:1187
Rosłanowski, A., & Shelah, S. Borel sets without perfectly many overlapping translations, III. Preprint. arXiv: 2009.03471
type: article
abstract: We expand the results of Rosłanowski and Shelah [RoSh:1138,RoSh:1170] to all Abelian Polish groups (H,+). We show that under the Martin Axiom, if \aleph_\alpha<{\mathfrak c}, \alpha<\omega_1 and 4\leq\iota<\omega, then there exists a \Sigma^0_2 set B\subseteq H which has \aleph_\alpha many pairwise \iota–nondisjoint translations but not a perfect set of such translations.
keywords: S: str
Sh:1188
Greenberg, N., Richter, L., Shelah, S., & Turetsky, D. More on bases of uncountable free Abelian groups. Preprint.
type: article
abstract: We extend results found by Greenberg, Turetsky, and Westrick in [GTW] and investigate effective properties of bases of uncountable free abelian groups. Assuming V=L, we show that if \kappa is a regular uncountable cardinal and X is a \Delta_1^1(L_\kappa) subset of \kappa, then there is a \kappa-computable free abelian group whose bases cannot be effectively computed by X. Unlike in [GTW], we give a direct construction.
keywords: M: odm, O: alg, (ab)
Sh:1195
Barnea, I., & Shelah, S. Inverse Limits of left adjoint functors on pointed sets. Preprint. arXiv: 2006.13705
type: article
abstract: (none)
keywords:
Sh:1200
Shelah, S. Existence of universal models. Preprint.
type: article
abstract: (none)
keywords:
Sh:1204
Horowitz, H., & Shelah, S. Abstract Corrected iterations. Preprint. arXiv: 2302.08581
type: article
abstract: (none)
keywords:
Sh:1208
Kumar, A., & Shelah, S. Weak projections of the null ideal. Preprint.
type: article
abstract: (none)
keywords:
Sh:1214
Paolini, G., & Shelah, S. On the existence of uncountable Hopfian and co-Hopfian abelian groups. Preprint. arXiv: 2107.11290
type: article
abstract: (none)
keywords:
Sh:1217
Asgharzadeh, M., Golshani, M., & Shelah, S. Graphs represented by Ext. Preprint. arXiv: 2110.11143
type: article
abstract: This paper opens and discusses the question originally due to Daniel Herden, who asked for which graph (\mu,R) we can find a family \{\mathbb G_\alpha: \alpha < \mu\} of abelian groups such that for each \alpha,\beta\in\mu

Ext(\mathbb G_\alpha, \mathbb G_\beta) = 0 iff (\alpha,\beta) \in R.

In this regard, we present four results. First, we give a connection to Quillen’s small object argument which helps Ext vanishes and uses to present useful criteria to the question. Suppose \lambda = \lambda^{\aleph_0} and \mu = 2^\lambda. We apply Jensen’s diamond principle along with the criteria to present \lambda-free abelian groups representing bipartite graphs. Third, we use a version of the black box to construct in ZFC, a family of \aleph_1-free abelian groups representing bipartite graphs. Finally, applying forcing techniques, we present a consistent positive answer for general graphs.
keywords:
Sh:1221
Malliaris, M., & Shelah, S. Shearing in some simple rank one theories. Preprint. arXiv: 2109.12642
type: article
abstract: (none)
keywords:
Sh:1224
Kellner, J., Latif, A., & Shelah, S. On automorphisms of \mathcal P(\lambda)/[\lambda]^{<\lambda}. Preprint. arXiv: 2206.02228
type: article
abstract: (none)
keywords:
Sh:1226
Poór, M., & Shelah, S. On the weak Borel chromatic number and cardinal invariants of the continuum. Preprint. arXiv: 2302.10141
type: article
abstract: (none)
keywords:
Sh:1227
Golshani, M., & Shelah, S. Adding highly undefinable sets over L. Preprint. arXiv: 2311.02322
type: article
abstract: We sow that there exists a generic extension of the Gödel’s constructible universe in which diamond holds and there exists a subset Y \subseteq \omega_1 such that for stationary many \delta < \omega_1, the set Y \cap \delta is not definable in the structure (L_{F(\delta)}, \in), where F(\delta) > \delta is the least ordinal such that L_{F(\delta)}\models“\delta is countable”.
keywords:
Sh:1229
Golshani, M., & Shelah, S. The measuring principle and the continuum hypothesis. Preprint. arXiv: 2207.08048
type: article
abstract: We show that one can force the Measuring principle without adding any new reals. We also show that it is consistent with the large continuum. These results answer two famous questions of Justin Moore.
keywords:
Sh:1230
Dobrinen, N., & Shelah, S. The Halpern–Läuchli Theorem at singular cardinals and failures of weak versions. Preprint. arXiv: 2209.11226
type: article
abstract: (none)
keywords:
Sh:1231
Mildenberger, H., & Shelah, S. Tiltan (=club) revisited. Preprint.
type: article
abstract: (none)
keywords:
Sh:1234
Poór, M., & Shelah, S. Universal graphs at the successors of small singulars. Preprint.
type: article
abstract: (none)
keywords:
Sh:1235
Kumar, A., & Shelah, S. Turing independence and Baire category. Preprint.
type: article
abstract: (none)
keywords:
Sh:1236
Garti, S., & Shelah, S. Superclub, splitting, separating statements. Preprint. arXiv: 2302.00904
type: article
abstract: (none)
keywords:
Sh:1237
Paolini, G., & Shelah, S. Anti-classification results for rigidity conditions in Abelian and nilpotent groups. Preprint. arXiv: 2303.03778
type: article
abstract: (none)
keywords:
Sh:1238
Shelah, S. AEC for strictly stable. Preprint. arXiv: 2305.02020
type: article
abstract: Good frames were suggested in [Sh:h] as the (bare-bones) parallel, in the context of AECs, to superstable (among elementary classes). Here we consider (\mu,\lambda,\kappa)-frames as candidates for being (in the context of AECs) the correct parallel to the class of |T|^+-saturated models of a strictly stable theory (among elementary classes). One thing we lose compared to the superstable case is that going up by induction on cardinals is problematic (for stages of small cofinality). But this arises only when we try to lift such classes to higher cardinals. Also, we may use, as a replacement, the existence of prime models over unions of increasing chains. For this context we investigate the dimension.
keywords:
Sh:1239
Shelah, S. AEC for strictly stable II. Preprint.
type: article
abstract: (none)
keywords:
Sh:1240
Rosłanowski, A., & Shelah, S. Borel sets without perfectly many overlapping translations IV. Preprint. arXiv: 2302.12964
type: article
abstract: (none)
keywords:
Sh:1241
Shelah, S., & Steprāns, J. Higher Dimensional Universal functions from lower dimensional ones. Preprint.
type: article
abstract: (none)
keywords:
Sh:1242
Hrušák, M., Shelah, S., & Zhang, J. More Ramsey theory for highly connected monochromatic subgraphs. Preprint. arXiv: 2305.00882
type: article
abstract: An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey Theorem on uncountable cardinals asserting that if we color edges of the complete graph we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušák and Shelah are answered by showing that assuming the consistency of suitable large cardinals the following are relatively consistent with \mathsf{ZFC}: \kappa\to_{hc} (\kappa)^2_\omega for every regular cardinal \kappa\geq \omega_2 and \neg\mathsf{CH}+ \aleph_2 \to_{hc} (\aleph_1)^2_\omega. Building on a work of Lambie-Hanson, we also show that \aleph_2 \to_{hc} [\aleph_2]^2_{\omega,2} is consistent with \neg\mathsf{CH}. To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.
keywords:
Sh:1243
Halbeisen, L. J., Plati, R., & Shelah, S. Implications of Ramsey Choice Principles in ZF. Preprint. arXiv: 2306.00743
type: article
abstract: (none)
keywords:
Sh:1244
Baldwin, J. T., Laskowski, M. C., & Shelah, S. When does \aleph_1-categoricity imply \omega-stability? Preprint. arXiv: 2308.13942
type: article
abstract: (none)
keywords:
Sh:1245
Asgharzadeh, M., Golshani, M., & Shelah, S. Naturality and Definability III. Preprint. arXiv: 2309.02090
type: article
abstract: In this paper, we deal with the notions of naturality from category theory and definablity from model theory and their interactions. In this regard, we present three results. First, we show, under some mild conditions, that naturality implies definablity. Second, by using reverse Easton iteration of Cohen forcing notions, we construct a transitive model of ZFC in which every uniformisable construction is weakly natural. Finally, we show that if F is a natural construction on a class \mathcal K of structures which is represented by some formula, then it is uniformly definable without any extra parameters. Our results answer some questions by Hodges and Shelah.
keywords:
Sh:1246
Asgharzadeh, M., Golshani, M., & Shelah, S. Expressive power of infinitary logic and absolute co-Hopfianity. Preprint. arXiv: 2309.16997
type: article
abstract: Recently, Paolini and Shelah have constructed absolutely Hopfian torsion-free abelian groups of any given size. In contrast, we show that this is not necessarily the case for absolutely co-Hopfian groups. We apply the infinitary logic to show a dichotomy behavior of co-Hopfian groups, by showing that the first beautiful cardinal is the turning point for this property. Namely, we prove that there are no absolute co-Hopfian abelian groups above the first beautiful cardinal. An extension of this result to the category of modules over a commutative ring is given.
keywords:
Sh:1247
Kumar, A., & Shelah, S. Remarks on some cardinal invariants and partition relations. Preprint.
type: article
abstract: (none)
keywords:
Sh:1248
Paolini, G., & Shelah, S. TORSION-FREE ABELIAN GROUPS ARE FAITHFULLY BOREL COMPLETE AND PURE EMBEDDABILITY IS A COMPLETE ANALYTIC QUASI-ORDER. Preprint.
type: article
abstract: (none)
keywords: O: alg, O: odo, (ab)
Sh:1249
Garti, S., Hayut, Y., & Shelah, S. On a problem of Erdös and Hajnal. Preprint.
type: article
abstract: (none)
keywords: M: cla, S: ico, (graph), (unstable)
Sh:1250
Haber, S., Hershko, T., Mirabi, M., & Shelah, S. First order logic with equicardinality in random graphs. Preprint.
type: article
abstract: (none)
keywords: (fmt)
Sh:1251
Kellner, J., & Shelah, S. Nowhere trivial automorphisms of P(\lambda)/[\lambda]^{<\lambda}, for \lambda inaccessible. Preprint.
type: article
abstract: (none)
keywords:
Sh:1252
Kostana, Z., Rinot, A., & Shelah, S. Diamond on Kurepa trees. Preprint. arXiv: 2404.02715
type: article
abstract: We introduce a new weak variation of diamond that is meant to only guess the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by proving it is compatible with Martin’s axiom. We then prove that this principle is nontrivial by showing it may consistently fail.
keywords:
Sh:E102
Kolman, O., & Shelah, S. Categoricity and amalgamation for AEC and \kappa measurable. Preprint. arXiv: math/9602216
type: article
abstract: In the original version of this paper, we assume a theory T that the logic \mathbb L _{\kappa, \aleph_{0}} is categorical in a cardinal \lambda > \kappa, and \kappa is a measurable cardinal. There we prove that the class of model of T of cardinality <\lambda (but \geq |T|+\kappa) has the amalgamation property; this is a step toward understanding the character of such classes of models.

In this revised version we replaced the class of models of T by \mathfrak k, an AEC (abstract elementary class) which has LS-number {<} \, \kappa, or at least which behave nicely for ultrapowers by D, a normal ultra-filter on \kappa.

Presently sub-section §1A deals with T \subseteq \mathbb L_{\kappa^{+}, \aleph_{0}} (and so does a large part of the introduction and little in the rest of §1), but otherwise, all is done in in the context of AEC.
keywords: (aec)

Non-research articles

Published survey articles, opinion pieces, interviews, etc.

Sh:1151
Shelah, S. (2021). Divide and conquer: dividing lines and universality. Theoria, 87(2), 259–348. DOI: 10.1111/theo.12289 MR: 4329456
type: opinion
abstract: (none)
keywords: M: cla, (univ), (unstable)
Sh:E16
Shelah, S. (1993). The future of set theory. In Set theory of the reals (Ramat Gan, 1991), Vol. 6, Bar-Ilan Univ., Ramat Gan, pp. 1–12. arXiv: math/0211397 MR: 1234276
type: opinion
abstract: (none)
keywords: S: ods, O: odo
Sh:E23
Shelah, S. (2003). Logical dreams. Bull. Amer. Math. Soc. (N.S.), 40(2), 203–228. arXiv: math/0211398 DOI: 10.1090/S0273-0979-03-00981-9 MR: 1962296
type: opinion
abstract: We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic
keywords: O: odo
Sh:E25
Shelah, S. (2002). You can enter Cantor’s paradise! In Paul Erdős and his mathematics, II (Budapest, 1999), Vol. 11, János Bolyai Math. Soc., Budapest, pp. 555–564. arXiv: math/0102056 MR: 1954743
type: opinion
abstract: (none)
keywords: S: pcf, O: odo, (set)
Sh:E35
Shelah, S. (2008). A bothersome question. Internat. Math. Nachrichten, 208, 27–30.
type: opinion
abstract: (none)
keywords: S: pcf, O: odo, (set)
Sh:E72
Shelah, S. (2013). Dependent classes E72. In European Congress of Mathematics, Eur. Math. Soc., Zürich, pp. 137–157. MR: 3469119
type: opinion
abstract: (none)
keywords: M: cla, O: odo, (mod)
Sh:E73
Shelah, S. (2014). Reflecting on logical dreams. In Interpreting Gödel, Cambridge Univ. Press, Cambridge, pp. 242–255. MR: 3468189
type: opinion
abstract: (none)
keywords: O: odo
Sh:E74
Malliaris, M., & Shelah, S. (2013). General topology meets model theory, on \mathfrak p and \mathfrak t. Proc. Natl. Acad. Sci. USA, 110(33), 13300–13305. DOI: 10.1073/pnas.1306114110 MR: 3105597
type: opinion
abstract: (none)
keywords: S: str, O: odo

Non-peer-reviewed research articles

Abstracts and research articles (co)authored by S. Shelah, published without peer review

Sh:217
Sageev, G., & Shelah, S. (1986). There are Noetherian domain in every cardinality with free additive groups. Abstracts Amer. Math. Soc., 7, 369. 86T-03-268, 269 arXiv: 0705.4132
type: nonreviewed
abstract: The result is as stated in the title. A full version appear in [E109] in the author list.
keywords: O: alg, (ab), (stal)
Sh:E4
Shelah, S. (1984). An \aleph _2 Souslin tree from a strange hypothesis. Abstracts Amer. Math. Soc., 160, 198. 84T-03
type: nonreviewed
abstract: (none)
keywords: S: ico
Sh:E5
Marcus, L., Redmond, T., & Shelah, S. (1985). Completeness of State Deltas, Aerospace Corporation. Tech. Rep. ATR-85(8354)-5
type: nonreviewed
abstract: (none)
keywords: M: smt, O: fin
Sh:E17
Shelah, S. (1971). Two cardinal and power like models: compactness and large group of automorphisms. Notices Amer. Math. Soc., 18(2), 425. 71 T-El5
type: nonreviewed
abstract: (none)
keywords: M: smt, M: odm
Sh:E30
Shelah, S. (1980). Going to Canossa. Abstracts Amer. Math. Soc., 1, 630. 80T-E85
type: nonreviewed
abstract: (none)
keywords: S: for, S: str, S: dst, (iter)

Corrections

Corrections, clarifications and explanations of other publications.

Sh:25
Shelah, S. (1973). Errata to: First order theory of permutation groups. Israel Journal of Mathematics, 15, 437–441.
Correction of [Sh:24]
type: correction
abstract: (none)
keywords: M: smt
Sh:154a
Shelah, S., & Stanley, L. J. (1986). Corrigendum to: “Generalized Martin’s axiom and Souslin’s hypothesis for higher cardinals” [Israel J. Math. 43 (1982), no. 3, 225–236]. Israel J. Math., 53(3), 304–314. DOI: 10.1007/BF02786563 MR: 852482
corrigendum to [Sh:154]
type: correction
abstract: (none)
keywords: S: for, (set)
Sh:240a
Foreman, M. D., Magidor, M., & Shelah, S. (1989). Correction to: “Martin’s maximum, saturated ideals, and nonregular ultrafilters. I” [Ann. of Math. (2) 127 (1988), no. 1, 1–47]. Ann. Of Math. (2), 129(3), 651. DOI: 10.2307/1971520 MR: 997316
correction to [Sh:240]
type: correction
abstract: (none)
keywords: S: for, (normal)
Sh:326a
Shelah, S. (1992). Erratum to: Vive la différence. I. Nonisomorphism of ultrapowers of countable models. In Set theory of the continuum, Vol. 26, Springer, New York, p. 419. MR: 1233826
Correction of [Sh:326]
type: correction
abstract: (none)
keywords: (iter), (set-mod), (up)
Sh:406a
Fremlin, D. H., & Shelah, S. Postscript to Shelah & Fremlin [Sh:406]. Preprint.
strengthens a theorem of [Sh:406]
type: correction
abstract: (none)
keywords: S: for, S: str, (set), (leb), (creatures)
Sh:446a
Shelah, S. (2020). Retraction of “Baire property and axiom of choice”. Israel J. Math., 240(1), 443. DOI: 10.1007/s11856-020-2083-z MR: 4193139
Retraction of [Sh:446]
type: correction
abstract: (none)
keywords:
Sh:465a
Shelah, S., & Steprāns, J. (1994). Erratum: “Maximal chains in {}^\omega\omega and ultrapowers of the integers” [Arch. Math. Logic 32 (1993), no. 5, 305–319]. Arch. Math. Logic, 33(2), 167–168. arXiv: math/9308202 DOI: 10.1007/BF01352936 MR: 1271434
erratum to [Sh:465]
type: correction
abstract: This note is intended as a supplement and clarification to the proof of Theorem 3.3 of [ShSr:465]; namely, it is consistent that {\mathfrak b}=\aleph_1 yet for every ultrafilter U on \omega there is a \leq^* chain \{f_\xi:\xi\in\omega_2\} such that \{f_\xi/U:\xi\in\omega_2\} is cofinal in \omega/U.
keywords: S: for, S: str
Sh:533a
Blass, A. R., Gurevich, Y., & Shelah, S. (2001). Addendum to: “Choiceless polynomial time” [Ann. Pure Appl. Logic 100 (1999), no. 1-3, 141–187;MR1711992 (2001a:68036)]. Ann. Pure Appl. Logic, 112(1), 117. DOI: 10.1016/S0168-0072(01)00086-0 MR: 1854233
Correction of [Sh:533]
type: correction
abstract: (none)
keywords: O: fin, (fmt)
Sh:559a
Eklof, P. C., & Shelah, S. New non-free Whitehead groups (corrected version). Preprint. arXiv: math/9711221
corrected version of [Sh:559]
type: correction
abstract: We show that it is consistent that there is a strongly \aleph_1-free \aleph_1-coseparable group of cardinality \aleph_1 which is not \aleph_1-separable.
keywords: S: for, O: alg, (ab), (iter), (stal), (wh), (unif)
Sh:700a
Shelah, S. Are \mathfrak a and \mathfrak d your cup of tea? Revisited. Preprint. arXiv: 2108.03666
Revised version of [Sh:700]
type: correction
abstract: This was non-essentially revised in late 2020. First point is noting that the proof of Theorem 4.3 in [Sh:700], which says that the proof giving the consistency \mathfrak{b} = \mathfrak{d} = \mathfrak{u} < \mathfrak{a} also gives \mathfrak{s} = \mathfrak{d}. The proof uses a measurable cardinal and a c.c.c. forcing so it gives large \mathfrak{d} and assumes a large cardinal.

Second point is adding to the results of §2,§3 which say that (in §3 with no large cardinals) we can force {\aleph_1} < \mathfrak{b} = \mathfrak{d} < \mathfrak{a}. We like to have {\aleph_1} < \mathfrak{s} \le \mathfrak{b} = \mathfrak{d} < \mathfrak{a}. For this we allow in §2,§3 the sets K_t to be uncountable; this requires non-essential changes. In particular, we replace usually {\aleph_0}, {\aleph_1} by \sigma , \partial. Naturally we can deal with \mathfrak{i} and similar invariants.

Third we proofread the work again. To get \mathfrak{s} we could have retained the countability of the member of the I_t-s but the parameters would change with A \in I_t, well for a cofinal set of them; but the present seems simpler.

We intend to continue in [Sh:F2009].
keywords: S: for, (iter), (inv)
Sh:927a
Baldwin, J. T., Kolesnikov, A. S., & Shelah, S. Correction for “The Amalgamation Spectrum”. Preprint.
Correction of [Sh:927]
type: correction
abstract: (none)
keywords: (diam)
Sh:990a
Shelah, S., & Steprāns, J. Non-trivial automorphisms of \mathcal P(\mathbb N)/[\mathbb N]^{<\aleph_0} from variants of small dominating number (corrected). Preprint.
Corrected version of [Sh:990]
type: correction
abstract: (none)
keywords: S: ico, (auto)
Sh:E11
Shelah, S. Also quite large {\frak b}\subseteq {\textrm{pcf}}({\frak a}) behave nicely. Preprint. arXiv: math/9906018
Correction of [Sh:371]
type: correction
abstract: The present note is an answer to complainds of E.Weitz on [Sh:371]. We present a corrected version of a part of chapter VIII of Cardinal Arithmetic
keywords: S: pcf, (set)
Sh:E12
Shelah, S. Analytical Guide and Updates to [Sh:g]. Preprint. arXiv: math/9906022
Correction of [Sh:g]
type: correction
abstract:
Part A: A revised version of the guide in [Sh:g], with corrections and expanded to include later works.

Part B: Corrections to [Sh:g].

Part C: Contains some revised proof and improved theorems.

Part D: Contains a list of relevant references.

Recent (July 2022) additions
- §14 = 12a.2 on: no choice
- On inner models 12.29, see [Sh:805].
- On Black Boxes and abelian groups 14.44 - 14.53, see [Sh:750] and [Sh:898]
- On somewhat free scales 2.16, see [Sh:1008],
- On n-dimensional Black Boxes, quite free abelian groups such that Hom(G,\mathbb{Z}) = \{0\}, 14.56, see [Sh:1028],
- Survey on the existence of universal models; in particular, abelian groups 14.32, see [Sh:1151].
keywords: S: pcf, (set)
Sh:E19a
Džamonja, M., & Shelah, S. (2000). Erratum: “\clubsuit does not imply the existence of a Suslin tree” [Israel J. Math. 113 (1999), 163–204]. Israel J. Math., 119, 379. MR: 1802661
Retraction of [Sh:E19]
type: correction
abstract: (none)
keywords: (iter)
Sh:E22
Göbel, R., & Shelah, S. (2001). An addendum and corrigendum to: “Almost free splitters” [Colloq. Math. vol. 81 no. 2, 193–221; MR1715347 (2000m:20092)]. Colloq. Math., 88(1), 155–158. arXiv: math/0009063 DOI: 10.4064/cm88-1-11 MR: 1814921
corrects an error in [Sh:682]
type: correction
abstract: (none)
keywords: O: alg, (ab), (stal)
Sh:E28
Shelah, S. Details on [Sh:74]. Preprint.
Details on [Sh:74]
type: correction
abstract: (none)
keywords: M: smt, S: ico, (mod)
Sh:E54
Shelah, S. Comments to Universal Classes. Preprint.
Comments on [Sh:h]
type: correction
abstract: (none)
keywords: M: nec, (mod)

Retracted publications

Papers that contain serious errors and have therefore been withdrawn.

Sh:446
Judah, H. I., & Shelah, S. (1993). Retracted: Baire property and axiom of choice. Israel J. Math., 84(3), 435–450. arXiv: math/9211213 DOI: 10.1007/BF02760952 MR: 1244679
See [Sh:446a]
type: withdrawn
abstract: Was withdrawn. Old abstract: We show that

(1) If ZF is consistent then the following theory is consistent “ZF + DC(\omega_{1}) + Every set of reals has Baire property” and

(2) If ZF is consistent then the following theory is consistent “ZFC + ‘every projective set of reals has Baire property’ + ‘any union of \omega_{1} meager sets is meager’ ”.
keywords:
Sh:E19
Džamonja, M., & Shelah, S. (1999). Retracted: \clubsuit does not imply the existence of a Suslin tree. Israel J. Math., 113, 163–204. arXiv: math/9612226 DOI: 10.1007/BF02780176 MR: 1729446
See [Sh:E19a]
type: withdrawn
abstract: We prove that \clubsuit does not imply the existence of a Suslin tree, so answering a question of I. Juhász.
keywords: S: for, (set)
Sh:E91
Ben-David, S., & Shelah, S. (1996). Retracted: The two-cardinals transfer property and resurrection of supercompactness. Proc. Amer. Math. Soc., 124(9), 2827–2837. NB: There is a flaw in the proof of the main theorem DOI: 10.1090/S0002-9939-96-03327-8 MR: 1326996
type: withdrawn
abstract: We show that the transfer property (\aleph_1,\aleph_0) \rightarrow(\lambda^+, \lambda) for singular \lambda, does not imply (even) the existence of a non-reflecting stationary subset of \lambda^+. The result assumes the consistency of ZFC with the existence of inifinitely many supercompact cardinals. We employ a technique of “resurrection of supercompactness”. Our forcing extension destroys the supercompactness of some cardinals, to show that in the extended model they still carry some of their compactness properties (such as reflection of stationary sets), we show that their supercompactness can be resurrected via a tame forcing extension.
keywords: (set-mod), (large)

Unpublished notes

Remarks, lecture notes, etc., not intended for publication.

Sh:360a
Shelah, S. The primal framework. Part C: Premature Minimality. Preprint.
additions to [Sh:360]
type: note
abstract: (none)
keywords: M: nec, (mod)
Sh:532
Shelah, S. Borel rectangles. Preprint.
type: note
abstract: We prove the consistency of the existence of co-\aleph_1-Souslin equivalence relation on {}^\omega 2 with any pregiven \aleph_\alpha class, \alpha < \omega_1 but not a perfect set of pairwise non-equivalent. We deal also with co-\kappa-Souslin relations, equivalence relations, exact characterizations and \Pi^1_2-equivalence relations and rectangles.

To 666: problem on equalities of x’s; deal with co-\kappa-Souslin deal with the k-notation and the \alpha-notation.

2012.8.16 In Poland - July- has written a new try , in order to get a Borel relation with aleph_alpha rectangle but no more, for any countable ordinal alpha. What was written was only the forcing. Yesterday, proofread it, expand - still has to write kappa-Delta pair proof, and thoughts about more than \aleph_y w_1 But the rank in [522, §4] seem not OK, will try to revise
keywords: S: for, S: dst, (mod), (inf-log)
Sh:804
Matet, P., & Shelah, S. Positive partition relations for P_\kappa(\lambda). Preprint. arXiv: math/0407440
type: note
abstract: Let \kappa a regular uncountable cardinal and \lambda a cardinal >\kappa, and suppose \lambda^{<\kappa} is less than the covering number for category {\rm cov}({\mathcal M}_{\kappa, \kappa}). Then

(a) I_{\kappa,\lambda}^+\mathop{\longrightarrow}\limits^\kappa (I_{\kappa,\lambda}^+,\omega+1)^2,

(b) I_{\kappa,\lambda}^+\mathop{\longrightarrow}\limits^\kappa [I_{\kappa,\lambda}^+]_{\kappa^+}^2 if \kappa is a limit cardinal, and

(c) I_{\kappa,\lambda}^+ \mathop{\longrightarrow}\limits^\kappa (I_{\kappa,\lambda}^+)^2 if \kappa is weakly compact.
keywords: S: ico, (set), (normal)
Sh:969a
Goldstern, M., Kellner, J., Shelah, S., & Wohofsky, W. An overview of the proof in Borel Conjecture and Dual Borel Conjecture. Preprint. arXiv: 1112.4424
Explanation to [Sh:969]
type: note
abstract: In this note, we give a rather informal overview (including two diagrams) of the proof given in our paper "Borel Conjecture and Dual Borel Conjecture" [Sh:969] (let us call it the "main paper"). This overview was originally a section in the main paper (the section following the introduction), but the referee found it not so illuminating, so we removed it from the main paper.

Since we think the introduction may be helpful to some readers (as opposed to the referee, who found the diagrams "mystifying") we preserve it in this form. The overview is supposed to complement the paper, and not to be read independently. The emphasis of this note is on giving the reader some vague understanding, at the expense of correctness of the claims.
keywords: S: str, (iter)
Sh:1018
Shelah, S. Compactness of chromatic number II. Preprint. arXiv: 1302.3431
type: note
abstract: At present true but not clear how more interesting than [1006]; rethink about this; see F1296
keywords: S: ico
Sh:1219a
Garti, S., & Shelah, S. Aristotelian Poetry. Preprint. arXiv: 2109.04682
type: note
abstract: (none)
keywords:
Sh:E6
Bartoszyński, T., & Shelah, S. Borel conjecture and 2^{\aleph_0}>\aleph_2. included in Bartoszynski Judah “Set theory. On the structure of the real line” (1995) in 8.3.B Preprint.
type: note
abstract: (none)
keywords: S: for, S: str, (set), (leb)
Sh:E13
Goldstern, M., & Shelah, S. A cardinal invariant related to homogeneous families. Preprint. arXiv: math/9707201
type: note
abstract: Define \kappa < {\frak{ex}} iff every independent family of size \kappa can be extended to a homogeneous family. We show {\frak {ex}} = cov(meager).
keywords: S: str, (inv)
Sh:E20
Shelah, S. A continuation of [Sh:691]. Preprint. arXiv: math/9912165
[Sh:691]
type: note
abstract: (none)
keywords: S: for, (set)
Sh:E29
Shelah, S. 3 lectures on pcf. Preprint.
type: note
abstract: (none)
keywords: S: pcf, (set)
Sh:E34
Shelah, S. On model completion of T_{\mathrm{aut}}. Preprint. arXiv: math/0404180
type: note
abstract: We characterize stable T for which the model completion of T_\sigma=T_{\rm aut} is stable (i.e., every completion is). Then we prove that “some completion is stable” is different and characterize it. We also prove that any such model completion satisfies NSOP_3. Finally we show that there is an unstable (complete first order) T with T_{\rm aut} having model completion.
keywords: M: odm, (mod)
Sh:E36
Shelah, S. Good Frames. Preprint.
type: note
abstract: (none)
keywords: M: nec, (mod)
Sh:E39
Kanovei, V., Reeken, M., & Shelah, S. Fully saturated extensions of standard universe. Preprint.
type: note
abstract: (none)
keywords: S: ods, (set)
Sh:E40
Shelah, S. A collection of abstracts of Shelah’s Papers. Preprint. arXiv: 2209.01617
type: note
abstract: These are abstracts of most of the papers up to publication no. [Sh:143] (and [Sh:217]), mostly compiled in 1980/81 with R. Grossberg. Also more details than in the originals were added in [Sh:5], [Sh:8] [Sh:217], and [C2], [C3] were added (the Cxx are representations of the author’s works).
keywords: O: odo
Sh:E43
Shelah, S. Revised GCH. Preprint.
type: note
abstract: (none)
keywords: S: pcf, O: odo, (set)
Sh:E47
Shelah, S., & Väänänen, J. A. On the Method of Identities. Preprint.
type: note
abstract: (none)
keywords: M: smt, S: ico
Sh:E50
Firstenberg, E., & Shelah, S. Perpendicular Indiscernible Sequences in Real Closed Fields. Preprint. arXiv: 1208.1302
type: note
abstract: (none)
keywords: M: cla, O: alg, (mod)
Sh:E52
Shelah, S. Consistency of “the ideal of null restricted to some A is \kappa–complete not \kappa^+–complete, \kappa weakly inaccessible and {\mathrm{cov}}({\mathrm{meagre}})=\aleph_1”. Preprint. arXiv: math/0504201
type: note
abstract: In this note we give an answer to the following question of Grinblat (Moti Gitik asked about it):

Is it consistent that for some set X, {\rm cov}({\rm NULL}\restriction X)=\lambda is a weakly inaccessible cardinal (so X not null of course) while {\rm cov}(\rm meagre) is small, say it is \aleph_1.
keywords: S: for, S: str, (set)
Sh:E56
Shelah, S. Density is at most the spread of the square. Preprint. arXiv: 0708.1984
type: note
abstract: (none)
keywords: O: top
Sh:E64
Gruenhut, E., & Shelah, S. Abstract matrix-tree. Preprint.
type: note
abstract: (none)
keywords: S: ico, (set)
Sh:E65
Cohen, M., & Shelah, S. Ranks for strongly dependent theories. Preprint. arXiv: 1303.3441
type: note
abstract: (none)
keywords: M: cla, (mod)
Sh:E66
Shelah, S. Selected Papers of Abraham Robinson. Preprint.
type: note
abstract: (none)
keywords: M: odm, O: odo, (mod)
Sh:E67
Shelah, S. Forcing is Great. Preprint.
type: note
abstract: (none)
keywords: S: for, O: odo, (mod)
Sh:E68
Shelah, S. Inner product space with no ortho-normal basis without choice. Preprint. arXiv: 1009.1441
type: note
abstract: (none)
keywords: O: alg, (stal)
Sh:E69
Shelah, S. PCF: The Advanced PCF Theorems. Preprint. arXiv: 1512.07063
type: note
abstract: (none)
keywords: S: pcf, (set)
Sh:E70
Shelah, S. (2012). On Model Theory (from: Plenary speakers answer two questions). Wiad. Mat., 48(2), 59–65. arXiv: 1208.1301 https://wydawnictwa.ptm.org.pl/index.php/wiadomosci-matematyczne/article/view/321/326
type: note
abstract: (none)
keywords: M: odm, O: odo, (set)
Sh:E71
Shelah, S. ECM presentation: Classifying classes of structures in model theory. Preprint.
type: note
abstract: These are based on slides of the plenary talk of the ECM. We shall try to explain a new and surprising result that strongly indicates that there is more to be discovered about so-called dependent theories; and we introduce some basic definitions, results and themes of model theory required to explain it. In particular we present first order theories and their classes of models, so-called elementary classes. Among them we look for dividing lines, that is we try to classify them. It is not a priori clear but it turn out that there are good, interesting dividing lines, ones for which which there is much to be said on both sides of the divide. In this frame we explain about two notable ones: stable classes and dependent ones.
keywords: O: odo
Sh:E75
Shelah, S. Categoricity of Classes of Models. Preprint.
type: note
abstract: (none)
keywords: M: nec, (mod)
Sh:E76
Shelah, S. On reaping number having countable cofinality. Preprint. arXiv: 1401.4649
type: note
abstract: (none)
keywords: S: str, (set)
Sh:E78
Shelah, S. From spring 1979 collection of preprints. Preprint.
type: note
abstract: (none)
keywords: O: odo
Sh:E79
Shelah, S. There may exist a unique Ramsey ultrafilter. Preprint.
type: note
abstract: PUT IN arXiv! def 0.5=L5.4(3). omit the union sign
keywords: S: for, S: str, (set)
Sh:E80
Shelah, S. Countably closed in ccc extension. Preprint. http://mathoverflow.net/questions/193522/#199287
type: note
abstract: (none)
keywords: S: for, (set)
Sh:E81
Shelah, S. Bigness properties for \kappa-trees and linear order. Preprint.
type: note
abstract: (none)
keywords: M: non, (mod)
Sh:E82
Shelah, S. Bounding forcing with chain conditions for uncountable cardinals. Preprint.
type: note
abstract: We discuss variants of the forcing notion from [Sh:1004]
keywords: S: for, (set)
Sh:E83
Dow, A. S., & Shelah, S. (2023). On the bounding, splitting, and distributivity number. Comment. Math. Univ. Carolin., 64(3), 331–351. arXiv: 2202.00372
type: note
abstract: (none)
keywords: S: for, S: str
Sh:E88
Shelah, S. (2021). Applying set theory. Axioms. DOI: 10.3390/axioms10040329
type: note
abstract: (none)
keywords: S: ico, S: for, S: str
Sh:E89
Larson, P. B., & Shelah, S. The number of models of a fixed Scott rank, for a counterexample to the analytic Vaught conjecture. Preprint. arXiv: 1903.09753
type: note
abstract: (none)
keywords: (diam)
Sh:E90
Shelah, S. (2020). Struggling with the Size of Infinity — The Paul Bernays lectures 2020, ETH Zürich. supplementary material for the talks, which can be found at https://video.ethz.ch/speakers/bernays/2020.html
type: note
abstract: (none)
keywords: O: odo
Sh:E93
Shelah, S. (1988). Classifying general classes, 1 videocassette (NTSC; 1/2 inch; VHS) (60 min.); sd., col; American Mathematical Society, Providence, RI. A plenary address presented at the International Congress of Mathematicians held in Berkeley, California, August 1986, Introduced by Ronald L. Graham MR: 1055086
type: note
abstract: (none)
keywords: M: nec
Sh:E94
Shelah, S. Power set modulo small, the singular of uncountable cofinality. Preprint.
type: note
abstract: (none)
keywords: S: ico
Sh:E98
Malliaris, M., & Shelah, S. (2021). Notes on the stable regularity lemma. Bull. Symb. Log., 27(4), 415–425. arXiv: 2012.09794 DOI: 10.1017/bsl.2021.69 MR: 4386783
type: note
abstract: (none)
keywords: (fc)
Sh:E99
Shelah, S. (1973). On the monadic (second order) theory of order. Notices A.M.S., 19, A–22.
type: note
abstract: (none)
keywords: (mon)
Sh:E100
Shelah, S. (1973). On the monadic theory of order II. Notices A.M.S., 19, A–282.
type: note
abstract: (none)
keywords: M: smt, (mon)
Sh:E101
Shelah, S. Colouring sucessor of regular, more on [1163]. Preprint.
type: note
abstract: (none)
keywords: S: ico, (pc)
Sh:E103
Shelah, S. Theories with minimal universality spectrum. Preprint.
type: note
abstract: (none)
keywords: S: ico, (univ)
Sh:E104
Shelah, S. Non P-point preserved by many. Preprint.
type: note
abstract: While paper 980 prepare the ground for much more we shall show here the following

Assuming CH , there is a non-P-point ultra-filter on the natural numbers which is preserved by Sacks forcing and many more.

This answer a question communicated to me by Mark Poor
keywords: S: for, (iter)
Sh:E105
Sageev, G., & Shelah, S. There are Noetherian domains in every cardinality with free additive groups. Preprint.
type: note
abstract: (none)
keywords: O: alg
Sh:E106
Shelah, S. Lecture on: Categoricity of atomic classes in small cardinals in ZFC. Preprint.
[Sh:F2195]
type: note
abstract: An atomic class K is the class of atomic first order models of a countable first order theory (assuming there are sucj models. Under the weak GCH it had been proved that if such class is categorical in every \aleph _n then is is categorical in every cardinal and is so called excellent, and results when we assume categoricity for \aleph _1, \dots, \aleph _n. The lecture is on a ZFC result in this direction for n=1. More specifically if K is categorical in \aleph _1 and has a model of cardinality > 2^{\aleph_0} then is is {\aleph_0}-stable, which implies having stable amalgamation, and is the first case of excellence

This a work in preparation by Baldwin, J. T., Laskowski, M. C. and Shelah, S.
keywords: M: nec
Sh:E107
Shelah, S. Some results in set theory. Preprint.
type: note
abstract: Has appeared in Abstracts American Mathematical Society 1982 page 522
keywords: S: ods
Sh:E108
Shelah, S. Stable frames and weights. Preprint. arXiv: 2304.04467
type: note
abstract: This was [839] from 2002 till winter 2023 when by editor request it was divided to current [839, [1248], [1249] and further corrected.. This is the complete version as of 2015 in AMS TEX
keywords: M: nec, (aec)
Sh:E109
Sageev, G., & Shelah, S. (1986). There are Noether Noetherian. domain in every cardinality with free additive groups. Abstracts Amer. Math. Soc., 7, 369. 86T-03-268, 269. Preprint.
type: note
abstract: (none)
keywords: O: alg
Sh:E110
Shelah, S. Notes on ESTS lecture Logical dreams 2023. Preprint.
type: note
abstract: Those informal notes were supposed to help the author lecture in the Colloquium of the ESTS in Sept 2023. You can find the talk here: https://tubedu.org/w/rs4UzTEFwqXTWCxdSsLfyi
keywords: O: odo
Sh:E111
Shelah, S. Strong Covering Lemma and Ch in \mathbf{V} [r]. Preprint.
type: note
abstract: This had appeared as a chapter in the "Cardianl arithmetic" boook an earlier version in the "Proper forcing" book (first addition not in the second one)
keywords:

Texts in other authors' publications

Appendices (with new mathematics), and forewords.

Sh:395
Shelah, S. (1990). Appendix to: Small uncountable cardinals and topology by Jerry E. Vaughan, North-Holland, Amsterdam, pp. 217–218. MR: 1078647
type: otherauthors
abstract: (none)
keywords: S: ico, S: str, (set), (inv)
Sh:E21
Shelah, S. (2002). On a Question of Grinblat (Appendix to: Algebras of sets and combinatorics, by L. Grinblat), American Mathematical Society, Providence, RI, pp. 247–250. arXiv: math/9912163 MR: 1923171
type: otherauthors
abstract: (none)
keywords: S: str
Sh:E84
Shelah, S. (2017). Foreword to: Beyond first order model theory. (J. Iovino, Ed.), CRC Press, Boca Raton, FL, pp. xi–xii. MR: 3726900
type: otherauthors
abstract: (none)
keywords: O: odo
Sh:E92
Shelah, S. (1998). Foreword to: The incompleteness phenomenon by Martin Goldstern and Haim Judah, A K Peters, Ltd., Natick, MA, p. viii. MR: 1690312
type: otherauthors
abstract: (none)
keywords: O: odo

Parts of books

Preprints that are now (sometimes in changed form) part of a book (or another article).

Sh:88a
Shelah, S. (1985). Appendix. In Classification of nonelementary classes. II. Abstract elementary classes., pp. 483–495.
appendix of [Sh:88]
type: partof
abstract: question 2021-07-08 has it appear in the proceedings? problem in citation in 1164
keywords: M: nec, (aec)
Sh:88r
Shelah, S. (2009). Abstract elementary classes near \aleph_1. In Classification theory for abstract elementary classes, Vol. 18, College Publications, London, p. vi+813. arXiv: 0705.4137
Ch. I of [Sh:h]
type: partof
abstract: We prove in ZFC, no \psi\in L_{\omega_1,\omega}[\mathbf Q] have unique model of uncountable cardinality, this confirms the Baldwin conjecture. But we analyze this in more general terms. We introduce and investigate a.e.c. and also versions of limit models, and prove some basic properties like representation by PC class, for any a.e.c. For PC_{\aleph_0}-representable a.e.c. we investigate the conclusion of having not too many non-isomorphic models in \aleph_1 and \aleph_2, but have to assume 2^{\aleph_0} < 2^{\aleph_1} and even 2^{\aleph_1} < 2^{\aleph_2}.
keywords: M: nec, M: smt, (mod), (cat), (wd), (aec)
Sh:171
Shelah, S. (1986). Classifying generalized quantifiers. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 1–46. DOI: 10.1007/BFb0098504 MR: 850052
Part of [Sh:d]
type: partof
abstract: We address the question of classifying generalized quantifier ranging on a family of relations of fixed arity on a fixed set U. We consider two notions of order (and equivalence) by interpretability. In essence, every such family is equivalent to one consisting of equivalence relations but we have to add a well ordering of length \lambda. In the weaker equivalence, there is a gap in cardinality between the size of equivalence relations which can be interpreted and the one needed to interpret, which consistently may occurs. Of course, the problem of classifying families of equivalence relations is clear. The proof proceed by cases, summed up in section 7, where we concentrate on the less fine equivalence relation. [Note pages 24,25 where interchanged]

NOTE: if U is countable, the gap in cardinals disappear even for the weaker (i.e. finer) equivalence every such family is equivalent to a family of equivalence relations (if you want just to follow a proof of this then read):
0.6: definition
1.5 page 5: how to represent equivalence relations, hence later we can deal with a family with one isomorphism type
2.2 page 7: finish the analysis for interpreting cases of monadic quantifier
3.5 page 15: finish the analysis of interpreting one to one functions, concerning uniformity see the remark at the bottom of page 15
4.5 page 16: definition of \lambda_2 (R)
4.17 page 23: reduction on a relation of cardinality \le \lambda_2 (R)
Def 5.1 page 26: defining \lambda_3
Definition 5.11A(?) defining K^{word} page 3(?)
Fact 5.2 page 26(?) \lambda_2 \in \{\lambda_3,\;lambda_2^+\}
5.11,5.12, 5.14 pages 31,32,33: analyses the case \lambda_2\neq\lambda_3,
rest on the case of equality page 35 line -16; defining \chi page 35 line -2

6.11 page 40: finishing this case and from now on \chi\ge cardinality(R) when U has cardinality \aleph_0
Explanation of the from of the paper:
The original aim of this article was to prove that for every K (a family of relations on U on a fixed arity) its quantifier is equivalent to one for KU, a family of equivalence relations (all such classes are assumed to be closed under isomorphism). Sections 1-6 were written for this and realize it to large extent. Clearly it suffice to deal with I with one isomorphism type as long as the interpretations are uniform. But two essential difficulties arise (1) the quantifier \exists^{word}_{\lambda} (a well ordering of length \lambda), provably is not biinterpretable with \exists_K for any family K of equivalence classes. This is not so serious: just add another case. (2) It is consistent that there are cardinals \chi,\lambda satisfying \chi\le\lambda \le 2^\chi such that R is e.g. a family of \chi sunsets of \lambda, again we cannot reduce this to equivalence relations in general (see section 8)
Only under the assumptions V=L and considering more liberal notion of bi-interpretability (\equiv_{exp} rather than \equiv_{int} the desired result is gotten. However when the gap degenerates for any reason we get the original hope (note \exists^{word}_\omega is second order quantification).
keywords: M: smt, (mod)
Sh:197
Shelah, S. (1986). Monadic logic: Hanf numbers. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 203–223. DOI: 10.1007/BFb0098511 MR: 850059
Part of [Sh:d]
type: partof
abstract: (none)
keywords: M: smt, (mod), (inf-log), (mon)
Sh:212
Shelah, S. (1986). The existence of coding sets. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 188–202. DOI: 10.1007/BFb0098510 MR: 850058
Part of [Sh:d]
type: partof
abstract: (none)
keywords: S: ico
Sh:228
Shelah, S. (1986). On the \mathrm{no}(M) for M of singular power. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 120–134. DOI: 10.1007/BFb0098507 MR: 850055
Part of [Sh:d]
type: partof
abstract: (none)
keywords: M: smt, (mod)
Sh:229
Shelah, S. (1986). Existence of endo-rigid Boolean algebras. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 91–119. arXiv: math/9201238 DOI: 10.1007/BFb0098506 MR: 850054
Part of [Sh:d]
type: partof
abstract: In [Sh:89] we, answering a question of Monk, have explicated the notion of “a Boolean algebra with no endomorphisms except the ones induced by ultrafilters on it” (see §2 here) and proved the existence of one with character density \aleph_0, assuming first \diamondsuit_{\aleph_1} and then only CH. The idea was that if h is an endomorphism of B, not among the “trivial” ones, then there are pairwise disjoint D_n\in B with h(d_n)\not\subset d_n. Then we can, for some S\subset\omega, add an element x such that d\leq x for n\in S, x\cap d_n=0 for n\not\in S while forbidding a solution for \{y\cap h(d_n):n\in S\}\cup\{y\cap h(d_n)=0:n\not\in S\}. Further analysis showed that the point is that we are omitting positive quantifier free types. Continuing this Monk succeeded to prove in ZFC, the existence of such Boolean algebras of cardinality 2^{\aleph_0}.

We prove (in ZFC) the existence of such B of density character \lambda and cardinality \lambda^{\aleph_0} whenever \lambda>\aleph_0. We can conclude answers to some questions from Monk’s list. We use a combinatorial method from [Sh:45],[Sh:172], that is represented in Section 1.
keywords: M: non, (ba)
Sh:232
Shelah, S. (1986). Nonstandard uniserial module over a uniserial domain exists. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 135–150. DOI: 10.1007/BFb0098508 MR: 850056
Part of [Sh:d]
type: partof
abstract: (none)
keywords: O: alg, (ab), (stal)
Sh:233
Shelah, S. (1986). Remarks on the numbers of ideals of Boolean algebra and open sets of a topology. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 151–187. DOI: 10.1007/BFb0098509 MR: 850057
Part of [Sh:d]
type: partof
abstract: (none)
keywords: S: ico, (ba), (gt), (inv(ba))
Sh:234
Shelah, S. (1986). Classification over a predicate. II. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 47–90. DOI: 10.1007/BFb0098505 MR: 850053
Part of [Sh:d]
type: partof
abstract: (none)
keywords: M: cla, (mod)
Sh:237a
Shelah, S. (1986). On normal ideals and Boolean algebras. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 247–259. DOI: 10.1007/BFb0098513 MR: 850061
Part of [Sh:d]
type: partof
abstract: (none)
keywords: S: ico, (ba), (normal)
Sh:237b
Shelah, S. (1986). A note on \kappa-freeness of abelian groups. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 260–268. DOI: 10.1007/BFb0098514 MR: 850062
Part of [Sh:d]
type: partof
abstract: (none)
keywords: O: alg, (ab), (stal)
Sh:237c
Shelah, S. (1986). On countable theories with models—homogeneous models only. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 269–271. DOI: 10.1007/BFb0098515 MR: 850063
Part of [Sh:d]
type: partof
abstract: (none)
keywords: M: odm
Sh:237d
Shelah, S. (1986). On decomposable sentences for finite models. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 272–275. DOI: 10.1007/BFb0098516 MR: 850064
Part of [Sh:d]
type: partof
abstract: (none)
keywords: M: odm, O: fin
Sh:237e
Shelah, S. (1986). Remarks on squares. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 276–279. DOI: 10.1007/BFb0098517 MR: 850065
Part of [Sh:d]
type: partof
abstract: (none)
keywords: S: ico
Sh:247
Shelah, S. (1986). More on stationary coding. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 224–246. DOI: 10.1007/BFb0098512 MR: 850060
Part of [Sh:d]
type: partof
abstract: (none)
keywords: M: cla, (set)
Sh:282a
Shelah, S. (1994). Colorings. In D. M. Gabbay, A. Macintyre, & D. Scott, eds., Cardinal Arithmetic, Vol. 29, Oxford University Press.
Apdx. 1 of [Sh:g]
type: partof
abstract: (none)
keywords: S: pcf
Sh:300a
Shelah, S. (2009). Universal Classes: Stability theory for a model. In Classification Theory for Abstract Elementary Classes II.
Ch. V of [Sh:i]
type: partof
abstract: We deal with a universal class of models K, (i.e. a structure \in K iff any finitely generated substructure \in K). We prove that either in K there are long orders (hence many complicated models) or K, under suitable order \le_{\mathfrak s} is an a.e.c. with some stability theory built in. For this we deal with the existence of indiscernible sets and (introduce and prove existence) of convergence sets. Moreover, improve the results on the existence of indiscernible sets such that for some first order theories, we get strong existence results for set of elements, whereas possibly for some sets of n-tuples this fails. In later sub-chapters we continue going up in a spiral - getting either non-structure or showing closed affinity to stable, but the dividing lines are in general missing for first order classes.
keywords: M: nec
Sh:300b
Shelah, S. (2009). Universal Classes: Axiomatic Framework [Sh:h]. In Classification Theory for Abstract Elementary Classes II.
Ch. V (B) of [Sh:i]
type: partof
abstract: (none)
keywords: M: nec
Sh:300c
Shelah, S. (2009). Universal Classes: A frame is not smooth or not \chi-based. In Classification Theory for Abstract Elementary Classes II.
Ch. V (C) of [Sh:i]
type: partof
abstract: (none)
keywords: M: nec
Sh:300d
Shelah, S. (2009). Universal Classes: Non-Forking and Prime Modes. In Classification Theory for Abstract Elementary Classes II.
Ch. V (D) of [Sh:i]
type: partof
abstract: (none)
keywords: M: nec
Sh:300e
Shelah, S. (2009). Universal Classes: Types of finite sequences. In Classification Theory for Abstract Elementary Classes II.
Ch. V (E) of [Sh:i]
type: partof
abstract: (none)
keywords: M: nec
Sh:300f
Shelah, S. (2009). Universal Classes: The heart of the matter. In Classification Theory for Abstract Elementary Classes II.
Ch. V (F) of [Sh:i]
type: partof
abstract: (none)
keywords: M: nec
Sh:300g
Shelah, S. (2009). Universal Classes: Changing the framework. In Classification Theory for Abstract Elementary Classes II.
Ch. V (G) of [Sh:i]
type: partof
abstract: (none)
keywords: M: nec
Sh:300x
Shelah, S. (2009). Bibliography. In Classification Theory for Abstract Elementary Classes.
Bibliography for [Sh:h]
type: partof
abstract: (none)
keywords: M: nec
Sh:300z
Shelah, S. (2009). Annotated Contents. In Classification Theory for Abstract Elementary Classes [Sh:h].
Annotated Contents for [Sh:i]
type: partof
abstract: (none)
keywords: M: nec
Sh:309
Shelah, S. (2022). Black boxes. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 65, 69–130. arXiv: 0812.0656 MR: 4636538
Ch. IV of The Non-Structure Theory" book [Sh:e]
type: partof
abstract: We shall deal comprehensively with Black Boxes, the intention being that provably in ZFC we have a sequence of guesses of extra structure on small subsets, the guesses are pairwise with quite little interaction, are far but together are "dense". We first deal with the simplest case, were the existence comes from winning a game by just writing down the opponent’s moves. We show how it help when instead orders we have trees with boundedly many levels, having freedom in the last. After this we quite systematically look at existence of black boxes, and make connection to non-saturation of natural ideals and diamonds on them.
keywords: M: non, S: ico
Sh:331
Shelah, S. A complicated family of members of trees with \omega +1 levels. Preprint. arXiv: 1404.2414
Ch. VI of The Non-Structure Theory" book [Sh:e]
type: partof
abstract: (none)
keywords: M: smt, (mod)
Sh:333
Shelah, S. (1994). Bounds on Power of singulars: Induction. In Cardinal Arithmetic, Vol. 29, Oxford University Press.
Ch. VI of [Sh:g]
type: partof
abstract: (none)
keywords: S: pcf, (set), (normal)
Sh:345a
Shelah, S. (1994). Basic: Cofinalities of small reduced products. In Cardinal Arithmetic, Vol. 29, Oxford University Press.
Ch. I of [Sh:g]
type: partof
abstract: (none)
keywords: S: pcf
Sh:345b
Shelah, S. (1994). Entangled Orders and Narrow Boolean Algebras. In Cardinal Arithmetic, Vol. 29, Oxford University Press.
Apdx. 2 of [Sh:g]
type: partof
abstract: (none)
keywords: S: pcf, (ba)
Sh:355
Shelah, S. (1994). \aleph _{\omega +1} has a Jonsson Algebra. In Cardinal Arithmetic, Vol. 29, Oxford University Press.
Ch. II of [Sh:g]
type: partof
abstract: (none)
keywords: S: pcf, (set)
Sh:363
Shelah, S. On spectrum of \kappa-resplendent models. Preprint. arXiv: 1105.3774
Ch. V of [Sh:e]
type: partof
abstract: We prove that some natural “outside" property of counting models up to isomorphism is equivalent (for a first order class) to being stable.

For a model, being resplendent is a strengthening of being \kappa-saturated. Restricting ourselves to the case \kappa > |T| for transparency, to say a model M is \kappa-resplendent means:

when we expand M by < \kappa individual constants \langle c_i : i < \alpha \rangle, if (M, c_i)_{ < \alpha } has an elementary extension expandable to be a model of T' where {\rm Th}((M, c_i)_{i < \alpha} ) \subseteq T', |T'| < \kappa then already (M, c_i)_{i < \alpha} can be expanded to a model of T' .

Trivially, any saturated model of cardinality \lambda is \lambda-resplendent. We ask: how many \kappa-resplendent models of a (first order complete) theory T of cardinality \lambda are there? We restrict ourselves to cardinals \lambda = \lambda^\kappa + 2^{|T|} and ignore the case \lambda = \lambda^{<\kappa} + |T| < \lambda^\kappa. Then we get a complete and satisfying answer: this depends only on T being stable or unstable. In this case proving that for stable T we get few, is not hard; in fact, every resplendent model of T is saturated hence it is determined by its cardinality up to isomorphism. The inverse is more problematic because naturally we have to use Skolem functions with any \alpha < \kappa places. Normally we use relevant partition theorems (Ramsey theorem or Erdős-Rado theorem), but in our case the relevant partitions theorems fail so we have to be careful.
keywords: M: cla, M: non, (mod), (nni), (sta)
Sh:365
Shelah, S. (1994). There are Jonsson algebras in many inaccessible cardinals. In Cardinal Arithmetic, Vol. 29, Oxford University Press.
Ch. III of [Sh:g]
type: partof
abstract: (none)
keywords: S: pcf
Sh:371
Shelah, S. (1994). Advanced: cofinalities of small reduced products. In Cardinal Arithmetic, Vol. 29, Oxford University Press.
Ch. VIII of [Sh:g]
See [Sh:E11]
type: partof
abstract: (none)
keywords: S: pcf
Sh:380
Shelah, S. (1994). Jonsson Algebras in an inaccessible \lambda not \lambda-Mahlo. In Cardinal Arithmetic, Vol. 29, Oxford University Press.
Ch. IV of [Sh:g]
type: partof
abstract: (none)
keywords: S: ico, S: pcf, (set)
Sh:384
Shelah, S. Compact logics in ZFC: Constructing complete embeddings of atomless Boolean rings. Preprint.
Ch. X of “The Non-Structure Theory" book [Sh:e]
type: partof
abstract: A debt.high , together with 482. Now [F1649] for casanovas call for proving: there for enough cardinals yk, preferably uk=yk^yk, we have more then compactness and LST, but together n cardianlity yk
keywords: M: non, (mod)
Sh:386
Shelah, S. (1994). Bounding pp(\mu ) when cf(\mu ) > \mu > \aleph _0 using ranks and normal ideals. In Cardinal Arithmetic, Vol. 29, Oxford University Press.
Ch. V of [Sh:g]
type: partof
abstract: (none)
keywords: S: pcf, (set), (normal)
Sh:400
Shelah, S. (1994). Cardinal Arithmetic. In Cardinal Arithmetic, Vol. 29, Oxford University Press.
Ch. IX of [Sh:g]
type: partof
abstract: (none)
keywords: S: pcf, (set)
Sh:482
Shelah, S. Compactness of the Quantifier on “Complete embedding of BA’s”. Preprint. arXiv: 1601.03596
Ch. XI of "The Non-Structure Theory" book [Sh:e]
type: partof
abstract: 1. add thanks to isf-bsf + old

2. copy abstract- how was it a5Xiv-ed without abstract
keywords: M: smt, M: non
Sh:511
Shelah, S. Building complicated index models and Boolean algebras. Preprint. arXiv: 2401.15644
Ch. VII of [Sh:e]
type: partof
abstract: We build models using an indiscernible model sub-structures of {}^{\kappa \ge}\lambda and related more complicated structures. We use this to build various Boolean algebras.
keywords: M: non, (set), (pure(for))
Sh:600
Shelah, S. (2009). Categoricity in abstract elementary classes: going up inductively. In Classification Theory for Abstract Elementary Classes. arXiv: math/0011215
Ch. II of [Sh:h]
type: partof
abstract: We deal with beginning stability theory for “reasonable" non-elementary classes without any remnants of compactness like dealing with models above Hanf number or by the class being definable by \mathbb L_{\omega_1,\omega}. We introduce and investigate good \lambda-frame, show that they can be found under reasonable assumptions and prove we can advance from \lambda to \lambda^+ when non-structure fail. That is, assume 2^{\lambda^{+n}} < 2^{\lambda^{+n+1}} for n < \omega. So if an a.e.c. is cateogorical in \lambda,\lambda^+ and has intermediate number of models in \lambda^{++} and 2^\lambda < 2^{\lambda^+} < 2^{\lambda^{++}}, LS(\mathfrak{K}) \le \lambda). Then there is a good \lambda-frame \mathfrak{s} and if \mathfrak{s} fails non-structure in \lambda^{++} then \mathfrak{s} has a successor \mathfrak{s}^+, a good \lambda^+-frame hence K^\mathfrak{s}_{\lambda^{+3}} \ne \emptyset, and we can continue.
keywords: M: nec, (mod), (nni), (cat)
Sh:705
Shelah, S. (2009). Toward classification theory of good \lambda frames and abstract elementary classes. In Classification Theory for Abstract Elementary Classes. arXiv: math/0404272
Ch. III of [Sh:h]
type: partof
abstract: Our main aim is to investigate a good \lambda-frame \mathfrak{s} which is as in the end of [600], i.e. \mathfrak{s} is n-successful for every n (i.e. we can define a good \lambda^{+n}-frame \mathfrak{s}^{+n} such that \mathfrak{s}^{+0} =\mathfrak{s},\mathfrak{s}^{+(n+1)} = (\mathfrak{s}^{+n})^+). We would like to prove then K^\mathfrak{s} has model in every cardinal > \lambda, and it is categorical in one of them iff it is categorical in every one of them. For this we shall show that K_{\mathfrak{s}^{+n}}’s are similar to superstable elementary classse with prime existence. (Actually also K^\mathfrak{s}_{\ge \lambda^{+ \omega}}, but the full proof are delayed).
keywords: M: nec, (mod), (aec)
Sh:734
Shelah, S. (2009). Categoricity and solvability of A.E.C., quite highly. In Classification Theory for Abstract Elementary Classes. arXiv: 0808.3023
Ch. IV of [Sh:h]
type: partof
abstract: We investigate in ZFC what can be the family of large enough cardinals \mu in which an a.e.c. \mathfrak{K} is categorical or even just solvable. We show that for not few cardinals \lambda < \mu there is a superlimit model in \mathfrak{K}_\lambda. Moreover, our main result is that we can find a good \lambda-frame \mathfrak{s} categorical in \lambda such that \mathfrak{K}_\mathfrak{s} \subseteq \mathfrak{K}_\lambda. We then show how to use 705 to get categoricity in every large enough cardinality if \mathfrak{K} has cases of \mu-amalgamation for enough \mu and 2^\mu < 2^{\mu^{+1}} < \ldots < 2^{\mu^{+n}} \ldots for enough \mu.
keywords: M: nec, (mod), (cat)
Sh:838
Shelah, S. (2009). Non-structure in \lambda^{++} using instances of WGCH. In Classification theory for abstract elementary classes II. arXiv: 0808.3020
Ch. VII of [Sh:i]
type: partof
abstract: Here we try to redo, improve and continue the non-structure parts in some works on a.e.c., which uses weak diamond, in \lambda^+ and \lambda^{++} getting better and more results and do what is necessary for the book on a.e.c. So we rework and improve non-structure proofs from [Sh:87b, §6], [Sh:88r] (or [Sh:88]), [Sh:E46], (or [Sh:576], [Sh:603]) and fulfill promises from [Sh:88r], [Sh:600], [Sh:705]. Comparing with [Sh:576] we make the context closer to the examples, hence hopefully improve transparency, though losing some generality. Toward this we work also on the positive theory, i.e. structure side of “low frameworks" like almost good \lambda-frames.
keywords: M: nec, M: non, S: ico, (mod), (nni), (aec)
Sh:E8
Shelah, S. A note on \kappa-freeness. Now in [Sh:d] pp. 260–268 Preprint. arXiv: math/0404207
Part of [Sh:d]
type: partof
abstract: (none)
keywords: O: alg, (stal)
Sh:E46
Shelah, S. (2009). Categoricity of an abstract elementary class in two successive cardinals, revisited. In Classification Theory for Abstract Elementary Classes II.
Ch. 6 of [Sh:i]
type: partof
abstract: We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models, or existence of large cardinals). We prove (assuming a weak version of GCH around \lambda) that if {\frak K} is categorical in \lambda,\lambda^+, LS({\frak K}) \le \lambda and has intermediate number of models in \lambda^{++}, then {\frak K} has a model in \lambda^{+++}.
keywords: M: nec, M: non
Sh:E53
Shelah, S. Introduction and Annotated Contents. Preprint. arXiv: 0903.3428
introduction of [Sh:h]
type: partof
abstract: (none)
keywords: M: nec, (mod)
Sh:E58
Shelah, S. Existence of endo-rigid Boolean Algebras. Preprint. arXiv: 1105.3777
Ch. I of [Sh:e]
type: partof
abstract: (none)
keywords: O: alg, (stal)
Sh:E59
Shelah, S. General non-structure theory and constructing from linear orders; to appear in Beyond first order model theory II. Preprint. arXiv: 1011.3576
Ch. III of The Non-Structure Theory" book [Sh:e]
type: partof
abstract: The theme of the first two sections, is to prepare the framework of how from a “complicated” family of so called index models I \in K_1 we build many and/or complicated structures in a class K_2. The index models are characteristically linear orders, trees with \kappa+1 levels (possibly with linear order on the set of successors of a member) and linearly ordered graphs; for this we formulate relevant complicatedness properties (called bigness).

In the third section we show stronger results concerning linear orders. If for each linear order I of cardinality \lambda > \aleph_0 we can attach a model M_I \in K_\lambda in which the linear order can be embedded such that for enough cuts of I, their being omitted is reflected in M_I, then there are 2^\lambda non-isomorphic cases. We also do the work for some applications.
keywords: M: non, (mod)
Sh:E60
Shelah, S. Constructions with instances of GCH: applying. Preprint.
Ch. VIII of [Sh:e]
type: partof
abstract: (none)
keywords: M: non, (mod)
Sh:E61
Shelah, S. Constructions with instances of GCH: proving. Preprint.
part of Ch. IX of [Sh:e]
type: partof
abstract: (none)
keywords: M: non, (mod)
Sh:E62
Shelah, S. Combinatorial background for Non-structure. Preprint. arXiv: 1512.04767
Appendix of [Sh:e]
type: partof
abstract: (none)
keywords: M: non, S: ico
Sh:E63
Shelah, S. Quite Complete Real Closed Fields revisited. Preprint.
part of Ch. 9 of [Sh:e]
type: partof
abstract: (none)
keywords: M: odm
Sh:E95a
Horowitz, H., & Shelah, S. Can you take Toernquist’s inaccessible away? Preprint. arXiv: 1605.02419
Has been incorporated (as one of two parts) into [Sh:1090]
type: partof
abstract: (none)
keywords: S: for, S: dst
Sh:E95b
Horowitz, H., & Shelah, S. Maximal independent sets in Borel graphs and large cardinals. Preprint. arXiv: 1606.04765
Has been incorporated (as one of two parts) into [Sh:1090]
type: partof
abstract: (none)
keywords: S: for, S: ods, (graph), (AC)

Preliminary versions, reprints

Published papers that are preliminary versions or a reprint of a journal publication

Sh:54a
Shelah, S. (1978). The lazy model theorist’s guide to stability. In Six days of model theory, ed. P. Henrard, Paul Castella, Switzerland 1661 Albeuve, pp. 9–76.
Reprint of [Sh:54]
type: reprint
abstract: (none)
keywords: M: cla, M: nec, (mod)
Sh:244
Gurevich, Y., & Shelah, S. (1985). Fixed-point extensions of first-order logic. In 26th Annual Symposium on Foundations of Computer Science (sfcs 1985), IEEE Computer Science Society Press, pp. 346–353. DOI: 10.1109/SFCF.1985.27
Conference proceedings version of [Sh:244a]
type: reprint
abstract: (none)
keywords: M: odm, O: fin
Sh:E85
Fuchino, S., & Shelah, S. (2001). Models of real-valued measurability. Sūrikaisekikenkyūsho Kōkyūroku, (1202), 38–60. Axiomatic set theory (Japanese) (Kyoto, 2000) MR: 1855549
Preliminary version of [Sh:763]
type: reprint
abstract: (none)
keywords: S: str
Sh:E86
Shelah, S., & Shioya, M. (2001). Nonreflecting stationary sets in \mathcal{P}_\kappa\lambda. Sūrikaisekikenkyūsho Kōkyūroku, (1202), 61–65. Axiomatic set theory (Japanese) (Kyoto, 2000) MR: 1855550
Preliminary version of [Sh:764]
type: reprint
abstract: (none)
keywords: S: ico, (ref)