# Publications

#### Publications (co)authored by S. Shelah

You can select other lists here.

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.

### Table of contents:

- Books (9 items)
- Published research articles (1066 items)
- Accepted articles (8 items)
- Articles in preparation (106 items)
- Non-research articles (8 items)
- Non-peer-reviewed research articles (5 items)
- Corrections (14 items)
- Retracted publications (4 items)
- Unpublished notes (42 items)
- Texts in other authors' publications (3 items)
- Parts of other publications (57 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

type: book

(none)

keywords: - Sh:b

Shelah, S. (1982).*Proper forcing*, Vol. 940, Springer-Verlag, Berlin-New York, p. xxix+496. MR: 675955

type: book

(none)

keywords: - Sh:c

Shelah, S. (1990).*Classification theory and the number of nonisomorphic models*, Second, Vol. 92, North-Holland Publishing Co., Amsterdam, p. xxxiv+705. MR: 1083551

type: book

(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

(none)

keywords: - Sh:e

Shelah, S.*Non-structure theory*, Oxford University Press. To appear.

Contains [Sh:309], [Sh:331], [Sh:363], [Sh:384], [Sh:421], [Sh:482], [Sh:511], [Sh:E58], [Sh:E59], [Sh:E60], [Sh:E61], [Sh:E62], [Sh:E63]

type: book

(none)

keywords: - Sh:f

Shelah, S. (1998).*Proper and improper forcing*, Second, Springer-Verlag, Berlin, p. xlviii+1020. DOI: 10.1007/978-3-662-12831-2 MR: 1623206

See [Sh:253], [Sh:263]

type: book

(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

(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

(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

(none)

keywords:

### Published research articles

Peer reviewed research articles (co)authored by S. Shelah that are published.

- Sh:1

Shelah, S. (1969).*Stable theories*. Israel J. Math.,**7**, 187–202. DOI: 10.1007/BF02787611 MR: 0253889

type: article

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

(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

(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

(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

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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(none)

keywords: S: ico - Sh:42

Shelah, S. (1975).*The monadic theory of order*. Ann. Of Math. (2),**102**(3), 379–419. DOI: 10.2307/1971037 MR: 0491120

type: article

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(none)

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

(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

(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

(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

(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

(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

(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

(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

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

(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

(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

(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

(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

(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

(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

(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

(none)

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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

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

(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

(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

(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

(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

(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

(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

(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

(none)

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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

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

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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

We continue here [Sh276] but we do not relay on it. The motivation was a conjecture of Galvin stating that 2^{\omega}\geq \omega_2 + \omega_2\to [\omega_1]^{n}_{h(n)} 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. 2020-10-27 Two years ago have a handwritten page elaborating the proof of 2.6(1); see file 288.1810006

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

(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

(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

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

(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

(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

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

(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

(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

(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

(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

(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

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

(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

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

(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

(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

(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

(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

(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

(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

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

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.

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

(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

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

(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

(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

(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

(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

(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

(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

(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

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

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

(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

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

(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

(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

(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

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

(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

(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

(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

(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

(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

(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

(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

(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

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

(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

(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

(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

(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

(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

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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

(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

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

(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

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

(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

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

(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

(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

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

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

(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

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

(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

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

(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

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

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

(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

(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

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

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

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

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

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

(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

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

(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

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

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

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

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

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

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

(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

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

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

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

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

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

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

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

(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

(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

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

(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

(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

(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

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

(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

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

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

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

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

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

(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

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

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

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

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

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

(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

(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

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

(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

The main result is the followingTheorem: 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

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

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

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

(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

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

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

(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

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

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

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

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

(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

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

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

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

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

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

(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

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

(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

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

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

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

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

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

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

(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

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 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 isThe 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\Rightarrow\lambda^{[\theta]}=\lambda and

(b) there is a family {\mathcal 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 {\mathcal 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

We prove (ZF+DC) e.g. : if \mu=|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

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

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

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

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.*On uniformly antisymmetric functions*. Real Anal. Exchange,**19**(1), 218–225. arXiv: math/9308222 MR: 1268847

type: article

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

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

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

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

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

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

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

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

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

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

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 cl_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}\backslash\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

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

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

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

(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

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

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

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

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

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

§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

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 proveTheorem (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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(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

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

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

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

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

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

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

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 cardinality of ultraproduct, the depth of ultraproducts of Boolean Algebras. Also we give cardinal invariant for each \lambda with pcf restriction and investigate further T_D(f).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels 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). For this we consider candidates (countable models to which the definition applies), and the older Souslin proper. 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 define c.c.c. ideals, then they preserve the positiveness of any old positive set. We also prove that (among such forcing notions) the only one commuting with Cohen is Cohen itself.

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

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

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

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

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

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

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: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

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

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

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

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

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

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

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

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

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: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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

type: article

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

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

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

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: https://doi.org/10.1090/dimacs/058/09 MR: 1903854

type: article

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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).*Middle diamond*. Arch. Math. Logic,**44**(5), 527–560. arXiv: math/0212249 DOI: 10.1007/s00153-004-0239-x MR: 2210145

type: article

Under some cardinal arithmetic assumptions, we prove that many stationary subsets of \lambda of a right cofinality has the “middle diamond”. In particular, for many regular \kappa for every large enough regular \lambda we have the middle diamond on \{\delta<\lambda:{\rm cf}(\delta)=\kappa\}. This is a strong negation of uniformization.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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^+ (moreover, for \mu^+, any scale for \mu^+ has a bad stationary set of cofinality \aleph_1). This answers a question of Foreman and Todorcevic 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 andIf 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: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

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

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

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

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

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

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

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

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

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

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

(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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 MR: 3666349

type: article

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

type: article

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 \rm con(\mathfrak{d}_\lambda>cov_\lambda(meagre))*. Trans. Amer. Math. Soc.,**373**(8), 5351–5369. arXiv: 0904.0817 DOI: 10.1090/tran/7948 MR: 4127879

type: article

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

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

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

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

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

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

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

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

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

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

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

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) - S