%% Saharon Shelah's bibliography
%% (public version of listb.bib)
% generated on: 10 Sep 2017, 20:47 GMT
% Disclaimer: This bibliography has its
% own idiosyncrasies. For example, all titles
% are enclosed in double braces, thus ensuring that capitalization
% is maintained. Also: even unpublished papers will be
% called "article".
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@article{Sh:1,
author = {Shelah, Saharon},
ams-subject = {(02.50)},
journal = {Israel Journal of Mathematics},
review = {MR 40-7102},
pages = {187--202},
title = {{Stable theories}},
volume = {7},
year = {1969},
},
@article{Sh:2,
author = {Shelah, Saharon},
ams-subject = {(05.04)},
journal = {Journal of Combinatorial Theory},
review = {MR 39-2652},
pages = {298--300},
title = {{Note on a min-max problem of Leo Moser}},
volume = {6},
year = {1969},
},
@article{Sh:3,
author = {Shelah, Saharon},
ams-subject = {(02.50)},
journal = {Annals of Mathematical Logic},
review = {MR 44-2593},
pages = {69--118},
title = {{Finite diagrams stable in power}},
volume = {2},
year = {1970},
},
@article{Sh:4,
author = {Shelah, Saharon},
ams-subject = {(02.50)},
journal = {Journal of Symbolic Logic},
review = {MR 44-52},
pages = {73--82},
title = {{On theories $T$ categorical in $|T|$}},
volume = {35},
year = {1970},
},
@article{Sh:5,
author = {Shelah, Saharon},
ams-subject = {(02.35)},
journal = {Israel Journal of Mathematics},
review = {MR 41-6674},
pages = {75--79},
title = {{On languages with non-homogeneous strings of quantifiers}},
volume = {8},
year = {1970},
},
@article{Sh:6,
author = {Shelah, Saharon},
ams-subject = {(02.50)},
journal = {Pacific Journal of Mathematics},
review = {MR 42-2932},
pages = {541--545},
title = {{A note on Hanf numbers}},
volume = {34},
year = {1970},
},
@article{Sh:7,
author = {Shelah, Saharon},
ams-subject = {(02H13)},
journal = {Journal of Symbolic Logic},
review = {MR 48:3735},
pages = {83--84},
title = {{On the cardinality of ultraproduct of finite sets}},
volume = {35},
year = {1970},
},
@article{Sh:8,
author = {Shelah, Saharon},
ams-subject = {(02H05)},
journal = {Israel Journal of Mathematics},
review = {MR 46:1581},
pages = {193--198},
title = {{Two cardinal compactness}},
volume = {9},
year = {1971},
},
@article{Sh:9,
author = {Shelah, Saharon},
ams-subject = {(02.52)},
journal = {Annals of Mathematical Logic},
review = {MR 44-56},
pages = {441--447},
title = {{Remark to ``local definability theory'' of Reyes}},
volume = {2},
year = {1970},
},
@article{Sh:10,
author = {Shelah, Saharon},
ams-subject = {(02H05)},
journal = {Annals of Mathematical Logic},
review = {MR 47:6475},
pages = {271--362},
title = {{Stability, the f.c.p., and superstability; model theoretic
properties of formulas in first order theory}},
volume = {3},
year = {1971},
},
@article{Sh:11,
author = {Shelah, Saharon},
ams-subject = {(02.50)},
journal = {Pacific Journal of Mathematics},
review = {MR 44-2594},
pages = {811--818},
title = {{On the number of non-almost isomorphic models of $T$ in a
power}},
volume = {36},
year = {1971},
},
@article{Sh:12,
author = {Shelah, Saharon},
ams-subject = {(02.50)},
journal = {Israel Journal of Mathematics},
review = {MR 43-4652},
pages = {473--487},
title = {{The number of non-isomorphic models of an unstable first-order
theory}},
volume = {9},
year = {1971},
},
@article{Sh:13,
author = {Shelah, Saharon},
ams-subject = {(02H99)},
journal = {Israel Journal of Mathematics},
review = {MR 45:6608},
pages = {224--233},
title = {{Every two elementarily equivalent models have isomorphic
ultrapowers}},
volume = {10},
year = {1971},
},
@article{Sh:14,
author = {Shelah, Saharon},
ams-subject = {(02H99)},
journal = {Annals of Mathematical Logic},
review = {MR 45:3187},
pages = {75--114},
title = {{Saturation of ultrapowers and Keisler's order}},
volume = {4},
year = {1972},
},
@article{Sh:15,
author = {Shelah, Saharon},
ams-subject = {(02H05)},
journal = {Journal of Symbolic Logic},
review = {MR 47:4787},
pages = {107--113},
title = {{Uniqueness and characterization of prime models over sets for
totally transcendental first-order theories}},
volume = {37},
year = {1972},
},
@article{Sh:16,
author = {Shelah, Saharon},
ams-subject = {(02H10)},
journal = {Pacific Journal of Mathematics},
review = {MR 46:7018},
pages = {247--261},
title = {{A combinatorial problem; stability and order for models and
theories in infinitary languages}},
volume = {41},
year = {1972},
},
@article{Sh:17,
author = {Shelah, Saharon},
ams-subject = {(02H13)},
journal = {Israel Journal of Mathematics},
review = {MR 46:3292},
pages = {23--31},
title = {{For what filters is every reduced product saturated?}},
volume = {12},
year = {1972},
},
@article{Sh:18,
author = {Shelah, Saharon},
ams-subject = {(02H13)},
journal = {Journal of Symbolic Logic},
review = {MR 56:5272},
pages = {247--267},
title = {{On models with power-like orderings}},
volume = {37},
year = {1972},
},
@article{ErSh:19,
author = {Erdos, Paul and Shelah, Saharon},
trueauthor = {Erd\H{o}s, Paul and Shelah, Saharon},
ams-subject = {(04A20)},
journal = {Israel Journal of Mathematics},
review = {MR 47:8312},
pages = {207--214},
title = {{Separability properties of almost-disjoint families of sets}},
volume = {12},
year = {1972},
},
@article{ScSh:20,
author = {Schmerl, James H. and Shelah, Saharon},
ams-subject = {(02H05)},
journal = {Journal of Symbolic Logic},
review = {MR 47:6474},
pages = {531--537},
title = {{On power-like models for hyperinaccessible cardinals}},
volume = {37},
year = {1972},
},
@incollection{ErSh:21,
author = {Erdos, Paul and Shelah, Saharon},
trueauthor = {Erd\H{o}s, Paul and Shelah, Saharon},
booktitle = {Graph theory and applications (Proc. Conf., Western
Michigan Univ., Kalamazoo, Mich., 1972; dedicated to the memory of J.
W. T. Youngs)},
ams-subject = {(05A15)},
review = {MR 49:2415},
pages = {75--79},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{On problems of Moser and Hanson}},
volume = {303},
year = {1972},
},
@article{Sh:22,
author = {Shelah, Saharon},
ams-subject = {(02H10)},
journal = {Proceedings of the American Mathematical Society},
review = {MR 45:3188},
pages = {509--514},
title = {{A note on model complete models and generic models}},
volume = {34},
year = {1972},
},
@article{GlSh:23,
author = {Galvin, Fred and Shelah, Saharon},
ams-subject = {(04A20)},
journal = {Journal of Combinatorial Theory. Ser. A},
review = {MR 48:8240},
pages = {167--174},
title = {{Some Counterexamples in the Partition Calculus}},
volume = {15},
year = {1973},
},
@article{Sh:24,
author = {Shelah, Saharon},
ams-subject = {(02H15)},
journal = {Israel Journal of Mathematics},
review = {MR 54:4972},
pages = {149--162},
title = {{First order theory of permutation groups}},
volume = {14},
year = {1973},
},
@article{Sh:25,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
pages = {437--441},
title = {{Errata to: First order theory of permutation groups}},
volume = {15},
year = {1973},
},
@article{Sh:26,
author = {Shelah, Saharon},
ams-subject = {(04A20)},
journal = {Israel Journal of Mathematics},
review = {MR 48:5864},
pages = {262--277},
title = {{Notes on combinatorial set theory}},
volume = {14},
year = {1973},
},
@article{MoSh:27,
author = {Moran, Gadi and Shelah, Saharon},
ams-subject = {(90D05)},
journal = {Israel Journal of Mathematics},
review = {MR 47:10084},
pages = {442--449},
title = {{Size direction games over the real line. III}},
volume = {14},
year = {1973},
},
@article{Sh:28,
author = {Shelah, Saharon},
ams-subject = {(02B15)},
journal = {Israel Journal of Mathematics},
review = {MR 49:20},
pages = {282--300},
title = {{There are just four second-order quantifiers}},
volume = {15},
year = {1973},
},
@article{Sh:29,
author = {Shelah, Saharon},
ams-subject = {(05A05)},
journal = {Journal of Combinatorial Theory. Ser. A},
review = {MR 48:10824},
pages = {199--208},
title = {{A substitute for Hall's theorem for families with infinite
sets}},
volume = {16},
year = {1974},
},
@incollection{MzSh:30,
author = {McKenzie, Ralph and Shelah, Saharon},
booktitle = {Proceedings of the Tarski Symposium (Univ. California,
Berkeley, Calif., 1971)},
ams-subject = {(02H15)},
review = {MR 50:12711},
pages = {53--74},
publisher = {Amer. Math. Soc., Providence, R.I},
series = {Proc. Sympos. Pure Math.},
title = {{The cardinals of simple models for universal theories}},
volume = {XXV},
year = {1974},
},
@incollection{Sh:31,
author = {Shelah, Saharon},
booktitle = {Proceedings of the Tarski Symposium (Univ. of California,
Berkeley, Calif., 1971)},
ams-subject = {(02G20)},
review = {MR 51:10074},
pages = {187--203},
publisher = {Amer. Math. Soc., Providence, R.I},
series = {Proc. Sympos. Pure Math.},
title = {{Categoricity of uncountable theories}},
volume = {XXV},
year = {1974},
},
@incollection{EHSh:32,
author = {Erdos, Paul and Hajnal, Andras and Shelah, Saharon},
trueauthor = {Erd\H{o}s, Paul and Hajnal, Andras and Shelah, Saharon},
booktitle = {Topics in topology (Proc. Colloq., Keszthely, 1972)},
ams-subject = {(05C15)},
review = {MR 50:9662},
pages = {243--255},
publisher = {North-Holland, Amsterdam},
series = {Colloq. Math. Soc. Janos Bolyai},
title = {{On some general properties of chromatic numbers}},
volume = {8},
year = {1974},
},
@article{Sh:33,
author = {Shelah, Saharon},
ams-subject = {(02H05)},
journal = {Pacific Journal of Mathematics},
review = {MR 51:132},
pages = {163--168},
title = {{The Hanf number of omitting complete types}},
volume = {50},
year = {1974},
},
@article{Sh:34,
author = {Shelah, Saharon},
ams-subject = {(02B25)},
journal = {Journal of Symbolic Logic},
review = {MR 51:5263},
pages = {399--404},
title = {{Weak definability in infinitary languages}},
volume = {38},
year = {1973},
},
@article{MlSh:35,
author = {Milner, Eric C. and Shelah, Saharon},
ams-subject = {(04A20)},
journal = {Canadian Journal of Mathematics. Journal Canadien de
Mathematiques},
review = {MR 51:10107},
pages = {948--961},
title = {{Sufficiency conditions for the existence of transversals}},
volume = {26},
year = {1974},
},
@article{Sh:36,
author = {Shelah, Saharon},
ams-subject = {(54A25)},
journal = {General Topology and Applications},
review = {MR 58:2674},
pages = {251--259},
title = {{Remarks on cardinal invariants in topology}},
volume = {7},
year = {1977},
},
@article{Sh:37,
author = {Shelah, Saharon},
ams-subject = {(02H05)},
journal = {Proceedings of the American Mathematical Society},
review = {MR 50:9573},
pages = {207--213},
title = {{A two-cardinal theorem}},
volume = {48},
year = {1975},
},
@incollection{Sh:38,
author = {Shelah, Saharon},
booktitle = {Infinite and finite sets (Colloq., Keszthely, 1973;
dedicated to P. Erd\H{o}s on his 60th birthday)},
ams-subject = {(05C35)},
review = {MR 51:7944},
pages = {1241--1256},
publisher = {North-Holland, Amsterdam},
series = {Colloq. Math. Soc. Janos Bolyai},
title = {{Graphs with prescribed asymmetry and minimal number of
edges}},
volume = {10 (III)},
year = {1975},
},
@article{Sh:39,
author = {Shelah, Saharon},
ams-subject = {(02H15)},
journal = {Israel Journal of Mathematics},
review = {MR 49:8856},
pages = {314--328},
title = {{Differentially closed fields}},
volume = {16},
year = {1973},
},
@incollection{Sh:40,
author = {Shelah, Saharon},
booktitle = {Infinite and finite sets (Colloq., Keszthely, 1973;
dedicated to P. Erd\H{o}s on his 60th birthday)},
ams-subject = {(04A20)},
review = {MR 53:10584},
pages = {1257--1276},
publisher = {North-Holland, Amsterdam},
series = {Colloq. Math. Soc. Janos Bolyai},
title = {{Notes on partition calculus}},
volume = {10 (III)},
year = {1975},
},
@incollection{MlSh:41,
author = {Milner, Eric C. and Shelah, Saharon},
booktitle = {Infinite and finite sets (Colloq., Keszthely, 1973;
dedicated to P. Erd\H{o}s on his 60th birthday)},
ams-subject = {(04A20)},
review = {MR 51:12534},
pages = {1115--1126},
publisher = {North Holland, Amsterdam},
series = {Colloq. Math. Soc. Janos Bolyai},
title = {{Some theorems on transversals}},
volume = {10 (III)},
year = {1975},
},
@article{Sh:42,
author = {Shelah, Saharon},
ams-subject = {(02G05)},
journal = {Annals of Mathematics},
review = {MR 58:10390},
pages = {379--419},
title = {{The monadic theory of order}},
volume = {102},
year = {1975},
},
@article{Sh:43,
author = {Shelah, Saharon},
ams-subject = {(02H05)},
journal = {Transactions of the American Mathematical Society},
review = {MR 51:12510},
pages = {342--364},
title = {{Generalized quantifiers and compact logic}},
volume = {204},
year = {1975},
},
@article{Sh:44,
author = {Shelah, Saharon},
ams-subject = {(02K05)},
journal = {Israel Journal of Mathematics},
review = {MR 50:9582},
pages = {243--256},
title = {{Infinite abelian groups, Whitehead problem and some
constructions}},
volume = {18},
year = {1974},
},
@incollection{Sh:45,
author = {Shelah, Saharon},
booktitle = {Model theory and algebra (A memorial tribute to Abraham
Robinson)},
ams-subject = {(20K10)},
review = {MR 54:425},
pages = {384--402},
publisher = {Springer, Berlin},
series = {Lecture Notes in Math.},
title = {{Existence of rigid-like families of abelian $p$-groups}},
volume = {498},
year = {1975},
},
@article{Sh:46,
author = {Shelah, Saharon},
ams-subject = {(04A20)},
journal = {Israel Journal of Mathematics},
review = {MR 55:109},
pages = {1--12},
title = {{Colouring without triangles and partition relation}},
volume = {20},
year = {1975},
},
@article{MShS:47,
author = {Makowsky, Johann A. and Shelah, Saharon and Stavi, Jonathan},
ams-subject = {(02B20)},
journal = {Annals of Mathematical Logic},
review = {MR 56:15362},
pages = {155--192},
title = {{$\Delta$-logics and generalized quantifiers}},
volume = {10},
year = {1976},
},
@article{Sh:48,
author = {Shelah, Saharon},
ams-subject = {(02H10)},
journal = {Israel Journal of Mathematics},
review = {MR 52:83},
pages = {127--148},
title = {{Categoricity in $\aleph _{1}$ of sentences in $L_{\omega
_{1},\omega}(Q)$}},
volume = {20},
year = {1975},
},
@article{Sh:49,
author = {Shelah, Saharon},
ams-subject = {(02H05)},
journal = {Proceedings of the American Mathematical Society},
review = {MR 55:7764},
pages = {134--136},
title = {{A two-cardinal theorem and a combinatorial theorem}},
volume = {62},
year = {1977},
},
@article{Sh:50,
author = {Shelah, Saharon},
ams-subject = {(04A20)},
journal = {Journal of Combinatorial Theory. Ser. A},
review = {MR 53:12958},
pages = {110--114},
title = {{Decomposing uncountable squares to countably many chains}},
volume = {21},
year = {1976},
},
@incollection{Sh:51,
author = {Shelah, Saharon},
booktitle = {Proceedings of the International Congress of
Mathematicians (Vancouver, B. C., 1974)},
ams-subject = {(02H05)},
review = {MR 54:10008},
pages = {259--263},
publisher = {Canad. Math. Congress, Montreal, Que},
title = {{Why there are many nonisomorphic models for unsuperstable
theories}},
volume = {1},
year = {1974},
},
@article{Sh:52,
author = {Shelah, Saharon},
ams-subject = {(02H13)},
journal = {Israel Journal of Mathematics},
review = {MR 52:10410},
pages = {319--349},
title = {{A compactness theorem for singular cardinals, free algebras,
Whitehead problem and transversals}},
volume = {21},
year = {1975},
},
@article{LtSh:53,
author = {Litman, A. and Shelah, Saharon},
ams-subject = {(02H05)},
journal = {Israel Journal of Mathematics},
review = {MR 57:9522},
pages = {331--338},
title = {{Models with few isomorphic expansions}},
volume = {28},
year = {1977},
},
@article{Sh:54,
author = {Shelah, Saharon},
ams-subject = {(02H05)},
journal = {Logique et Analyse},
review = {MR 58:27447},
nt = {Comptes Rendus de la Semaine d'Etude en Theorie des Modeles (Inst.
Math., Univ. Catholique Louvain, Louvain-la-Neuve, 1975).},
pages = {241--308},
title = {{The lazy model-theoretician's guide to stability}},
volume = {18},
year = {1975},
},
@incollection{Sh:54a,
author = {Shelah, Saharon},
booktitle = {Six days of model theory},
fromwhere = {IL},
pages = {9-76},
publisher = {ed. P. Henrard, Paul Castella, Switzerland 1661 Albeuve},
series = {Proceedings of a conference in Louvain-le-Neuve, March 1975},
title = {{The lazy model theorist's guide to stability}},
year = {1978},
},
@article{McSh:55,
author = {Macintyre, Angus and Shelah, Saharon},
ams-subject = {(02H15)},
journal = {Journal of Algebra},
review = {MR 55:12511},
pages = {168--175},
title = {{Uncountable universal locally finite groups}},
volume = {43},
year = {1976},
},
@article{Sh:56,
author = {Shelah, Saharon},
ams-subject = {(02H05)},
journal = {Israel Journal of Mathematics},
review = {MR 58:5173},
note = {A special volume, Proceedings of the Symposium in memory of A.
Robinson, Yale, 1975},
pages = {273--286},
title = {{Refuting Ehrenfeucht conjecture on rigid models}},
volume = {25},
year = {1976},
},
@article{AmSh:57,
author = {Amit, R. and Shelah, Saharon},
ams-subject = {(02G20)},
journal = {Israel Journal of Mathematics},
review = {MR 58:5162},
pages = {200--208},
title = {{The complete finitely axiomatized theories of order are
dense}},
volume = {23},
year = {1976},
},
@article{Sh:58,
author = {Shelah, Saharon},
ams-subject = {(02G05)},
journal = {Israel Journal of Mathematics},
review = {MR 58:21562},
pages = {32--44},
title = {{Decidability of a portion of the predicate calculus}},
volume = {28},
year = {1977},
},
@article{HiSh:59,
author = {Hiller, Howard L. and Shelah, Saharon},
ams-subject = {(02K05)},
journal = {Israel Journal of Mathematics},
review = {MR 56:2820},
pages = {313--319},
title = {{Singular cohomology in $L$}},
volume = {26},
year = {1977},
},
@article{HLSh:60,
author = {Hodges, Wilfrid and Lachlan, Alistair H. and Shelah, Saharon},
ams-subject = {(04A20)},
journal = {Bulletin of the London Mathematical Society},
review = {MR 57:16085},
pages = {212--215},
title = {{Possible orderings of an indiscernible sequence}},
volume = {9},
year = {1977},
},
@article{Sh:61,
author = {Shelah, Saharon},
ams-subject = {(02K99)},
journal = {Ann. Sci. Univ. Clermont},
review = {MR 58:21622},
note = {Proceedings of Symposium in Clermont-Ferand, July 1975},
pages = {1--29},
title = {{Interpreting set theory in the endomorphism semi-group of a
free algebra or in a category}},
volume = {13},
year = {1976},
},
@article{MwSh:62,
author = {Makowsky, Johann A. and Shelah, Saharon},
ams-subject = {(03C80)},
journal = {Transactions of the American Mathematical Society},
review = {MR 81b:03041},
pages = {215--239},
title = {{The theorems of Beth and Craig in abstract model theory. I.
The abstract setting}},
volume = {256},
year = {1979},
},
@article{SeSh:63,
author = {Stern, Jacques and Shelah, Saharon},
ams-subject = {(03C65)},
fromwhere = {IL},
journal = {Transactions of the American Mathematical Society},
review = {MR 80a:03047},
pages = {147--171},
title = {{The Hanf number of the first order theory of Banach spaces}},
volume = {244},
year = {1978},
},
@article{Sh:64,
author = {Shelah, Saharon},
ams-subject = {(02K05)},
journal = {Israel Journal of Mathematics},
review = {MR 57:9538},
pages = {193--204},
title = {{Whitehead groups may be not free, even assuming CH. I}},
volume = {28},
year = {1977},
},
@article{DvSh:65,
author = {Devlin, Keith J. and Shelah, Saharon},
ams-subject = {(02K05)},
journal = {Israel Journal of Mathematics},
review = {MR 57:9537},
pages = {239--247},
title = {{A weak version of $\diamondsuit $ which follows from
$2^{\aleph _{0}}<2^{\aleph _{1}}$}},
volume = {29},
year = {1978},
},
@article{Sh:66,
author = {Shelah, Saharon},
ams-subject = {(03C15)},
journal = {The Journal of Symbolic Logic},
review = {MR 80b:03037},
pages = {550--562},
title = {{End extensions and numbers of countable models}},
volume = {43},
year = {1978},
abstract = {The answer to the question from page 562 (the end). is
negative; have known a solution but not sure if have Not record
it. \endgraf 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},
},
@article{Sh:67,
author = {Shelah, Saharon},
ams-subject = {(02H13)},
journal = {Journal of Symbolic Logic},
review = {MR 58:10414},
pages = {475--480},
title = {{On the number of minimal models}},
volume = {43},
year = {1978},
},
@article{Sh:68,
author = {Shelah, Saharon},
ams-subject = {(02H05)},
journal = {Israel Journal of Mathematics},
review = {MR 58:21572},
pages = {57--64},
title = {{Jonsson algebras in successor cardinals}},
volume = {30},
year = {1978},
},
@incollection{Sh:69,
author = {Shelah, Saharon},
booktitle = {Word problems, II (Conf. on Decision Problems in Algebra,
Oxford, 1976)},
ams-subject = {(20F06)},
review = {MR 81j:20047},
pages = {373--394},
publisher = {North-Holland, Amsterdam-New York},
series = {Studies in Logic and Foundations of Mathematics},
title = {{On a problem of Kurosh, Jonsson groups, and applications}},
volume = {95},
year = {1980},
},
@article{GuSh:70,
author = {Gurevich, Yuri and Shelah, Saharon},
ams-subject = {(03C85)},
journal = {The Journal of Symbolic Logic},
review = {MR 81a:03038b},
pages = {491--502},
title = {{Modest theory of short chains. II}},
volume = {44},
year = {1979},
},
@article{Sh:71,
author = {Shelah, Saharon},
ams-subject = {(03E10)},
journal = {The Journal of Symbolic Logic},
review = {MR 82c:03070},
pages = {56--66},
title = {{A note on cardinal exponentiation}},
volume = {45},
year = {1980},
},
@article{Sh:72,
author = {Shelah, Saharon},
ams-subject = {(03C85)},
journal = {Annals of Mathematical Logic},
review = {MR 80b:03047a},
pages = {57--72},
title = {{Models with second-order properties. I. Boolean algebras with
no definable automorphisms}},
volume = {14},
year = {1978},
},
@article{Sh:73,
author = {Shelah, Saharon},
ams-subject = {(03C85)},
journal = {Annals of Mathematical Logic},
review = {MR 80b:03047b},
pages = {73--87},
title = {{Models with second-order properties. II. Trees with no
undefined branches}},
volume = {14},
year = {1978},
},
@article{Sh:74,
author = {Shelah, Saharon},
ams-subject = {(03C85)},
journal = {Annals of Mathematical Logic},
review = {MR 80b:03047c},
pages = {223--226},
title = {{Appendix to: ``Models with second-order properties. II.
Trees with no undefined branches'' (Annals of Mathematical Logic
14(1978), no. 1, 73--87)}},
volume = {14},
year = {1978},
},
@article{Sh:75,
author = {Shelah, Saharon},
ams-subject = {(46B99)},
journal = {Israel Journal of Mathematics},
review = {MR 80b:46033},
pages = {181--191},
title = {{A Banach space with few operators}},
volume = {30},
year = {1978},
},
@article{Sh:76,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
journal = {The Journal of Symbolic Logic},
review = {MR 81k:03050},
pages = {505--509},
title = {{Independence of strong partition relation for small cardinals,
and the free-subset problem}},
volume = {45},
year = {1980},
},
@article{Sh:77,
author = {Shelah, Saharon},
ams-subject = {(03C60)},
journal = {Bulletin de la Societe Mathematique de Grece. Nouvelle
Serie},
review = {MR 80j:03047},
note = {A special volume dedicated to the memory of Papakyriakopoulos},
pages = {17--27},
title = {{Existentially-closed groups in $\aleph _{1}$ with special
properties}},
volume = {18},
year = {1977},
},
@article{Sh:78,
author = {Shelah, Saharon},
ams-subject = {(03C50)},
journal = {The Journal of Symbolic Logic},
review = {MR 80k:03031},
pages = {319--324},
title = {{Hanf number of omitting type for simple first-order
theories}},
volume = {44},
year = {1979},
},
@article{Sh:79,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
journal = {The Journal of Symbolic Logic},
review = {MR 80m:03066},
pages = {215--220},
title = {{On uniqueness of prime models}},
volume = {44},
year = {1979},
},
@article{Sh:80,
author = {Shelah, Saharon},
ams-subject = {(02K05)},
journal = {Israel Journal of Mathematics},
review = {MR 58:21606},
pages = {297--306},
title = {{A weak generalization of MA to higher cardinals}},
volume = {30},
year = {1978},
},
@article{ADSh:81,
author = {Avraham (Abraham), Uri and Devlin, Keith J. and Shelah,
Saharon},
ams-subject = {(02K05)},
journal = {Israel Journal of Mathematics},
review = {MR 58:21602},
pages = {19--33},
title = {{The consistency with CH of some consequences of Martin's axiom
plus $2^{\aleph _{0}}>\aleph _{1}$}},
volume = {31},
year = {1978},
},
@article{Sh:82,
author = {Shelah, Saharon},
ams-subject = {(03C80)},
journal = {Archiv fur Mathematische Logik und Grundlagenforschung},
review = {MR 83a:03031},
pages = {1--11},
title = {{Models with second order properties. III. Omitting types for
$L(Q)$}},
volume = {21},
year = {1981},
},
@article{GgSh:83,
author = {Giorgetta, Donato and Shelah, Saharon},
ams-subject = {(03C60)},
fromwhere = {D,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 86e:03035},
note = {Proceedings of the 1980/1 Jerusalem Model Theory year.},
pages = {123--148},
title = {{Existentially closed structures in the power of the
continuum}},
volume = {26},
year = {1984},
},
@article{RuSh:84,
author = {Rubin, Matatyahu and Shelah, Saharon},
ams-subject = {(03C80)},
journal = {The Journal of Symbolic Logic},
review = {MR 81h:03078},
pages = {265--283},
title = {{On the elementary equivalence of automorphism groups of
Boolean algebras; downward Skolem-Lowenheim theorems and compactness of
related quantifiers}},
volume = {45},
year = {1980},
},
@article{DvSh:85,
author = {Devlin, Keith J. and Shelah, Saharon},
ams-subject = {(54E30)},
journal = {Canadian Journal of Mathematics. Journal Canadien de
Mathematiques},
review = {MR 81d:54022},
pages = {241--251},
title = {{A note on the normal Moore space conjecture}},
volume = {31},
year = {1979},
},
@article{DvSh:86,
author = {Devlin, Keith J. and Shelah, Saharon},
ams-subject = {(54D15)},
journal = {Proceedings of the London Mathematical Society. Third
Series},
review = {MR 80m:54031},
pages = {237--252},
title = {{Souslin properties and tree topologies}},
volume = {39},
year = {1979},
},
@article{Sh:87,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
title = {{See [Sh:87a] and [Sh:87b]}},
},
@article{Sh:87a,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 85m:03024a},
pages = {212--240},
title = {{Classification theory for nonelementary classes, I. The number
of uncountable models of $\psi \in L_{\omega _{1},\omega }$. Part A}},
volume = {46},
year = {1983},
},
@article{Sh:87b,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 85m:03024b},
pages = {241--273},
title = {{Classification theory for nonelementary classes, I. The number
of uncountable models of $\psi \in L_{\omega _{1},\omega }$. Part B}},
volume = {46},
year = {1983},
},
@incollection{Sh:88,
author = {Shelah, Saharon},
booktitle = {Classification theory (Chicago, IL, 1985)},
ams-subject = {(03C75)},
fromwhere = {IL},
review = {MR 91h:03046},
note = {Proceedings of the USA--Israel Conference on
Classification Theory, Chicago, December 1985; ed. Baldwin, J.T.},
pages = {419--497},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{Classification of nonelementary classes. II. Abstract
elementary classes}},
volume = {1292},
year = {1987},
},
@incollection{Sh:88a,
author = {Shelah, Saharon},
booktitle = {Classification theory (Chicago, IL, 1985)},
fromwhere = {IL},
note = {Proceedings of the USA--Israel Conference on
Classification Theory, Chicago, December 1985; ed. Baldwin, J.T.},
pages = {483--495},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{Appendix: on stationary sets (in ``Classification
of nonelementary classes. II. Abstract elementary classes'')}},
volume = {1292},
year = {1987},
},
@inbook{Sh:88r,
author = {Shelah, Saharon},
booktitle = {Classification theory for abstract elementary classes},
fromwhere = {IL},
note = {Chapter I. 0705.4137. arxiv:0705.4137 },
title = {{Abstract elementary classes near $\aleph_1$}},
abstract = {We prove in ZFC, no $\psi\in L_{\omega_1,\omega}[\mathbf
Q]$ have unique model of uncountable cardinality, this confirms
the Baldwin conjecture. But we analyze this in more general terms.
We introduce and investigate a.e.c. and also versions of limit
models, and prove some basic properties like representation by PC
class, for any a.e.c. For PC$_{\aleph_0}$-representable a.e.c. we
investigate the conclusion of having not too many non-isomorphic models
in $\aleph_1$ and $\aleph_2$, but have to assume $2^{\aleph_0}
< 2^{\aleph_1}$ and even $2^{\aleph_1} < 2^{\aleph_2}$.},
},
@article{Sh:89,
author = {Shelah, Saharon},
ams-subject = {(06E05)},
journal = {Proceedings of the American Mathematical Society},
review = {MR 82i:06017},
pages = {135--142},
title = {{Boolean algebras with few endomorphisms}},
volume = {74},
year = {1979},
},
@article{Sh:90,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
journal = {Topology Proceedings},
review = {MR 81f:03060},
nt = {Proceedings of the 1977 Topology Conference (Louisiana State
Univ., Baton Rouge, La., 1977), II.},
pages = {583--592},
title = {{Remarks on $\lambda $-collectionwise Hausdorff spaces}},
volume = {2},
year = {1977},
},
@article{HHSh:91,
author = {Hiller, Howard L. and Huber, Martin and Shelah, Saharon},
ams-subject = {(20K20)},
journal = {Mathematische Zeitschrift},
review = {MR 58:11171},
pages = {39--50},
title = {{The structure of ${\rm Ext}(A, {\bf Z})$ and $V=L$}},
volume = {162},
year = {1978},
},
@article{Sh:92,
author = {Shelah, Saharon},
ams-subject = {(06E05)},
journal = {Algebra Universalis},
review = {MR 82k:06016},
pages = {77--89},
title = {{Remarks on Boolean algebras}},
volume = {11},
year = {1980},
},
@article{Sh:93,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
journal = {Annals of Mathematical Logic},
review = {MR 82g:03055},
pages = {177--203},
title = {{Simple unstable theories}},
volume = {19},
year = {1980},
},
@article{Sh:94,
author = {Shelah, Saharon},
ams-subject = {(04A20)},
journal = {The Journal of Symbolic Logic},
review = {MR 81i:04009},
pages = {559--562},
title = {{Weakly compact cardinals: a combinatorial proof}},
volume = {44},
year = {1979},
},
@article{Sh:95,
author = {Shelah, Saharon},
ams-subject = {(04A20)},
journal = {The Journal of Symbolic Logic},
review = {MR 83j:04007},
pages = {345--353},
title = {{Canonization theorems and applications}},
volume = {46},
year = {1981},
},
@article{ShZi:96,
author = {Shelah, Saharon and Ziegler, Martin},
ams-subject = {(03C60)},
journal = {The Journal of Symbolic Logic},
review = {MR 80j:03048},
note = { arxiv:arXiv },
pages = {522--532},
title = {{Algebraically closed groups of large cardinality}},
volume = {44},
year = {1979},
},
@article{ShRd:97,
author = {Shelah, Saharon and Rudin, M. E. },
ams-subject = {(04A20)},
journal = {Topology Proceedings},
review = {MR 80k:04002},
nt = {Proceedings of the 1978 Topology Conference (Univ. Oklahoma,
Norman, Okla., 1978), I.},
pages = {199--204},
title = {{Unordered types of ultrafilters}},
volume = {3},
year = {1979},
},
@article{Sh:98,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
journal = {Israel Journal of Mathematics},
review = {MR 82h:03055},
pages = {257--285},
title = {{Whitehead groups may not be free, even assuming CH. II}},
volume = {35},
year = {1980},
},
@article{HrSh:99,
author = {Harrington, Leo and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {1,IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 86g:03079},
note = {Proceedings of the 1980/1 Jerusalem Model Theory year},
pages = {178--188},
title = {{Some exact equiconsistency results in set theory}},
volume = {26},
year = {1985},
},
@article{Sh:100,
author = {Shelah, Saharon},
ams-subject = {(03E40)},
journal = {The Journal of Symbolic Logic},
review = {MR 82b:03099},
pages = {563--573},
title = {{Independence results}},
volume = {45},
year = {1980},
},
@article{MwSh:101,
author = {Makowsky, Johann A. and Shelah, Saharon},
ams-subject = {(03C80)},
journal = {Archiv fur Mathematische Logik und Grundlagenforschung},
review = {MR 83g:03034},
note = {Proceedings of a Workshop, Berlin, July 1977},
pages = {13--35},
title = {{The theorems of Beth and Craig in abstract model theory. II.
Compact logics}},
volume = {21},
year = {1981},
},
@article{AbSh:102,
author = {Avraham (Abraham), Uri and Shelah, Saharon},
ams-subject = {(03E35)},
journal = {The Journal of Symbolic Logic},
review = {MR 83h:03071},
pages = {37--42},
title = {{Forcing with stable posets}},
volume = {47},
year = {1982},
},
@article{FrSh:103,
author = {Fremlin, David H. and Shelah, Saharon},
ams-subject = {(03E35)},
journal = {Israel Journal of Mathematics},
review = {MR 82b:03096},
pages = {299--304},
title = {{On partitions of the real line}},
volume = {32},
year = {1979},
},
@article{LvSh:104,
author = {Laver, Richard and Shelah, Saharon},
ams-subject = {(03E35)},
journal = {Transactions of the American Mathematical Society},
review = {MR 82e:03049},
pages = {411--417},
title = {{The $\aleph _{2}$-Souslin Hypothesis}},
volume = {264},
year = {1981},
},
@article{Sh:105,
author = {Shelah, Saharon},
ams-subject = {(03E60)},
journal = {Israel Journal of Mathematics},
review = {MR 82h:03054},
pages = {311--330},
title = {{On uncountable abelian groups}},
volume = {32},
year = {1979},
},
@article{AbSh:106,
author = {Avraham (Abraham), Uri and Shelah, Saharon},
ams-subject = {(03E35)},
journal = {Israel Journal of Mathematics},
review = {MR 82a:03048},
pages = {161--176},
title = {{Martin's axiom does not imply that every two $\aleph
_{1}$-dense sets of reals are isomorphic}},
volume = {38},
year = {1981},
},
@article{Sh:107,
author = {Shelah, Saharon},
ams-subject = {(03C80)},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 85j:03056},
pages = {183--212},
title = {{Models with second order properties. IV. A general method and
eliminating diamonds}},
volume = {25},
year = {1983},
},
@incollection{Sh:108,
author = {Shelah, Saharon},
booktitle = {Logic Colloquium '78 (Mons, 1978)},
ams-subject = {(03E10)},
review = {MR 82d:03079},
pages = {357--380},
publisher = {North-Holland, Amsterdam-New York},
series = {Stud. Logic Foundations Math},
title = {{On successors of singular cardinals}},
volume = {97},
year = {1979},
},
@article{HoSh:109,
author = {Hodges, Wilfrid and Shelah, Saharon},
ams-subject = {(03C20)},
journal = {Annals of Mathematical Logic},
review = {MR 82f:03025},
pages = {77--108},
title = {{Infinite games and reduced products}},
volume = {20},
year = {1981},
},
@article{Sh:110,
author = {Shelah, Saharon},
ams-subject = {(03E05)},
journal = {Israel Journal of Mathematics},
review = {MR 85b:03085},
pages = {177--226},
title = {{Better quasi-orders for uncountable cardinals}},
volume = {42},
year = {1982},
},
@article{Sh:111,
author = {Shelah, Saharon},
ams-subject = {(04A30)},
fromwhere = {IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 87j:04006},
pages = {263--299},
title = {{On power of singular cardinals}},
volume = {27},
year = {1986},
},
@article{ShSt:112,
author = {Shelah, Saharon and Stanley, Lee},
ams-subject = {(03E35)},
journal = {Israel Journal of Mathematics},
review = {MR 84h:03119},
pages = {185--224},
title = {{$S$-forcing. I. A ``black-box'' theorem for morasses, with
applications to super-Souslin trees}},
volume = {43},
year = {1982},
},
@article{Sh:113,
author = {Shelah, Saharon},
ams-subject = {(03C95)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 91i:03078},
pages = {193--213},
title = {{The theorems of Beth and Craig in abstract model theory. III.
$\Delta$-logics and infinitary logics}},
volume = {69},
year = {1990},
},
@article{AbSh:114,
author = {Abraham, Uri and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
review = {MR 86i:03063},
pages = {75--113},
title = {{Isomorphism types of Aronszajn trees}},
volume = {50},
year = {1985},
},
@article{ChSh:115,
author = {Cherlin, Gregory and Shelah, Saharon},
ams-subject = {(03C60)},
journal = {Annals of Mathematical Logic},
review = {MR 82c:03045},
pages = {227--270},
title = {{Superstable fields and groups}},
volume = {18},
year = {1980},
},
@article{MwSh:116,
author = {Makowsky, Johann A. and Shelah, Saharon},
ams-subject = {(03C95)},
fromwhere = {IL,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 85i:03125},
pages = {263--299},
title = {{Positive results in abstract model theory: a theory of compact
logics}},
volume = {25},
year = {1983},
},
@article{RuSh:117,
author = {Rubin, Matatyahu and Shelah, Saharon},
ams-subject = {(04A20)},
fromwhere = {IL,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 88h:04005},
pages = {43--81},
title = {{Combinatorial problems on trees: partitions, $\Delta$-systems
and large free subtrees}},
volume = {33},
year = {1987},
},
@article{RuSh:118,
author = {Rubin, Matatyahu and Shelah, Saharon},
ams-subject = {(03C80)},
fromwhere = {IL,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 85e:03090},
pages = {542--557},
title = {{On the expressibility hierarchy of Magidor-Malitz
quantifiers}},
volume = {48},
year = {1983},
},
@article{Sh:119,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
journal = {Israel Journal of Mathematics},
review = {MR 83g:03051},
pages = {1--32},
title = {{Iterated forcing and changing cofinalities}},
volume = {40},
year = {1981},
},
@article{Sh:120,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
journal = {Israel Journal of Mathematics},
review = {MR 83a:03047},
pages = {315--334},
title = {{Free limits of forcing and more on Aronszajn trees}},
volume = {38},
year = {1981},
},
@article{MShS:121,
author = {Magidor, Menachem and Shelah, Saharon and Stavi, Jonathan},
ams-subject = {(03C62)},
journal = {The Journal of Symbolic Logic},
review = {MR 84m:03058},
pages = {33--38},
title = {{On the standard part of nonstandard models of set theory}},
volume = {48},
year = {1983},
},
@article{Sh:122,
author = {Shelah, Saharon},
ams-subject = {(03E05)},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 83i:03075},
pages = {29--35},
title = {{On Fleissner's diamond}},
volume = {22},
year = {1981},
},
@article{GuSh:123,
author = {Gurevich, Yuri and Shelah, Saharon},
ams-subject = {(03C85)},
fromwhere = {1,IL},
journal = {Annals of Mathematical Logic},
review = {MR 85d:03080},
pages = {179--198},
title = {{Monadic theory of order and topology in ${\rm ZFC}$}},
volume = {23},
year = {1982},
},
@article{Sh:124,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
journal = {Israel Journal of Mathematics},
review = {MR 83a:03048},
pages = {283--288},
title = {{$\aleph _{\omega }$ may have a strong partition relation}},
volume = {38},
year = {1981},
},
@article{Sh:125,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
journal = {Israel Journal of Mathematics},
review = {MR 82k:03084},
pages = {74--82},
title = {{The consistency of ${\rm Ext}(G,\,{\bf Z})={\bf Q}$}},
volume = {39},
year = {1981},
},
@article{Sh:126,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 83b:03036},
pages = {239--248},
title = {{On saturation for a predicate}},
volume = {22},
year = {1981},
},
@article{Sh:127,
author = {Shelah, Saharon},
ams-subject = {(03E50)},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 83d:03060},
pages = {301--308},
title = {{On uncountable Boolean algebras with no uncountable pairwise
comparable or incomparable sets of elements}},
volume = {22},
year = {1981},
},
@article{Sh:128,
author = {Shelah, Saharon},
ams-subject = {(03C50)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 87d:03096},
pages = {273--297},
title = {{Uncountable constructions for B.A., e.c. groups and Banach
spaces}},
volume = {51},
year = {1985},
},
@article{Sh:129,
author = {Shelah, Saharon},
ams-subject = {(03C75)},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 82k:03054},
pages = {5--10},
title = {{On the number of nonisomorphic models of cardinality
$\lambda$, $L_{\infty \lambda }$-equivalent to a fixed model}},
volume = {22},
year = {1981},
},
@article{PiSh:130,
author = {Pillay, Anand and Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {1,IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 87d:03095},
pages = {361--376},
title = {{Classification theory over a predicate. I}},
volume = {26},
year = {1985},
},
@article{Sh:131,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
journal = {Israel Journal of Mathematics},
review = {MR 84j:03070a},
pages = {324--356},
title = {{The spectrum problem. I. $\aleph _{\varepsilon }$-saturated
models, the main gap}},
volume = {43},
year = {1982},
},
@article{Sh:132,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
journal = {Israel Journal of Mathematics},
review = {MR 84j:03070b},
pages = {357--364},
title = {{The spectrum problem. II. Totally transcendental and infinite
depth}},
volume = {43},
year = {1982},
},
@article{Sh:133,
author = {Shelah, Saharon},
ams-subject = {(03C80)},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 84h:03093},
pages = {21--26},
title = {{On the number of nonisomorphic models in $L_{\infty ,\kappa }$
when $\kappa $ is weakly compact}},
volume = {23},
year = {1982},
},
@incollection{GPShS:134,
author = {Gabbai, D. and Pnueli, A. and Shelah, Saharon and Stavi,
Jonathan},
booktitle = {Proc.~of the seventh Annual SIG ACT --- SIG PLAN Symposium
on Principles of Programming Languages, January 23- 30, 1980},
fromwhere = {IL},
pages = {163--173},
publisher = {Association Comp. Machinery, NY},
title = {{On the temporal analysis of fairness}},
year = {1980},
},
@article{GGHSh:135,
author = {Glass, A. M. W. and Gurevich, Yuri and Holland, W. Charles and
Shelah, Saharon},
ams-subject = {(06A05)},
journal = {Mathematical Proceedings of the Cambridge Philosophical
Society},
review = {MR 82c:06001},
pages = {7--17},
title = {{Rigid homogeneous chains}},
volume = {89},
year = {1981},
},
@article{Sh:136,
author = {Shelah, Saharon},
ams-subject = {(06E05)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 86k:06010},
pages = {100--146},
title = {{Constructions of many complicated uncountable structures and
Boolean algebras}},
volume = {45},
year = {1983},
},
@incollection{Sh:137,
author = {Shelah, Saharon},
booktitle = {Surveys in set theory},
ams-subject = {(03E35)},
fromwhere = {IL},
review = {MR 87b:03114},
note = {Proceedings of Symp. in Set Theory, Cambridge, August 1978; ed.
Mathias, A.R.D.},
pages = {116--134},
publisher = {Cambridge Univ. Press, Cambridge-New York},
series = {London Math. Soc. Lecture Note Ser},
title = {{The singular cardinals problem: independence results}},
volume = {87},
year = {1983},
},
@article{SgSh:138,
author = {Sageev, Gershon and Shelah, Saharon},
ams-subject = {(20K35)},
fromwhere = {1,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 87c:20097},
pages = {302--315},
title = {{On the structure of ${\rm Ext}(A,{\bf Z})$ in ${\rm ZFC}^
+$}},
volume = {50},
year = {1985},
},
@article{Sh:139,
author = {Shelah, Saharon},
ams-subject = {(20A15)},
journal = {Algebra Universalis},
review = {MR 84i:20005},
pages = {131--146},
title = {{On the number of nonconjugate subgroups}},
volume = {16},
year = {1983},
},
@article{Sh:140,
author = {Shelah, Saharon},
ams-subject = {(20K26)},
journal = {Israel Journal of Mathematics},
review = {MR 83f:20042},
pages = {291--295},
title = {{On endo-rigid, strongly $\aleph _{1}$-free abelian groups in
$\aleph _{1}$}},
volume = {40},
year = {1981},
},
@article{GMSh:141,
author = {Gurevich, Yuri and Magidor, Menachem and Shelah, Saharon},
ams-subject = {(03C85)},
fromwhere = {IL,IL,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 84i:03076},
pages = {387--398},
title = {{The monadic theory of $\omega _{2}$}},
volume = {48},
year = {1983},
},
@article{BlSh:142,
author = {Baldwin, John T. and Shelah, Saharon},
ams-subject = {(03C50)},
fromwhere = {1,IL},
journal = {Algebra Universalis},
review = {MR 85h:03032},
note = {A volume in honour of Tarski},
pages = {191--199},
title = {{The structure of saturated free algebras}},
volume = {17},
year = {1983},
},
@article{GuSh:143,
author = {Gurevich, Yuri and Shelah, Saharon},
ams-subject = {(03C85)},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
review = {MR 86m:03064},
note = {Proceedings of the 1980/1 Jerusalem Model Theory year},
pages = {55--68},
title = {{The monadic theory and the ``next world''}},
volume = {49},
year = {1984},
},
@article{MShS:144,
author = {Magidor, Menachem and Shelah, Saharon and Stavi, Jonathan},
ams-subject = {(03C70)},
fromwhere = {IL,IL,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 86i:03048},
note = {Proceedings of the 1980/1 Jerusalem Model Theory year},
pages = {287--361},
title = {{Countably decomposable admissible sets}},
volume = {26},
year = {1984},
},
@article{EMSh:145,
author = {Eklof, Paul C. and Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(20K20)},
fromwhere = {IL,IL,IL},
journal = {Israel Journal of Mathematics},
review = {MR 86m:20062},
note = {Proceedings of the 1980/1 Jerusalem Model Theory year},
pages = {34--54},
title = {{Almost disjoint abelian groups}},
volume = {49},
year = {1984},
},
@article{AbSh:146,
author = {Abraham, Uri and Shelah, Saharon},
ams-subject = {(03C62)},
fromwhere = {IL,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 85i:03112},
pages = {643--657},
title = {{Forcing closed unbounded sets}},
volume = {48},
year = {1983},
},
@article{HrSh:147,
author = {Harrington, Leo and Shelah, Saharon},
ams-subject = {(03D25)},
journal = {American Mathematical Society. Bulletin. New Series},
review = {MR 83i:03067},
pages = {79--80},
title = {{The undecidability of the recursively enumerable degrees}},
volume = {6},
year = {1982},
},
@incollection{SgSh:148,
author = {Sageev, Gershon and Shelah, Saharon},
booktitle = {Abelian group theory (Oberwolfach, 1981)},
ams-subject = {(20K40)},
review = {MR 83e:20062},
note = {ed. Goebel, R. and Walker, A.E.},
pages = {87--92},
publisher = {Springer, Berlin-New York},
series = {Lecture Notes in Mathematics},
title = {{Weak compactness and the structure of {\rm Ext}$(A,\,{\bf
Z})$}},
volume = {874},
year = {1981},
},
@article{FdSh:149,
author = {Friedman, Sy D. and Shelah, Saharon},
ams-subject = {(03C70)},
fromwhere = {1,IL},
journal = {Proceedings of the American Mathematical Society},
review = {MR 85g:03057},
pages = {672--678},
title = {{Tall $\alpha $-recursive structures}},
volume = {88},
year = {1983},
},
@article{ShKf:150,
author = {Shelah, Saharon and Kaufmann, Matt},
ams-subject = {(03C80)},
fromwhere = {IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 87d:03105},
pages = {111--123},
title = {{The Hanf number of stationary logic}},
volume = {27},
year = {1986},
},
@article{GuSh:151,
author = {Gurevich, Yuri and Shelah, Saharon},
ams-subject = {(03B15)},
fromwhere = {1,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 85f:03007},
pages = {816--828},
title = {{Interpreting second-order logic in the monadic theory of
order}},
volume = {48},
year = {1983},
},
@incollection{HrSh:152,
author = {Harrington, Leo and Shelah, Saharon},
booktitle = {Logic Colloquium '80 (Prague, 1980)},
ams-subject = {(03E15)},
review = {MR 84c:03088},
note = {eds. van Dalen, ~D., Lascar, D. and Smiley, T.J.},
pages = {147--152},
publisher = {North-Holland, Amsterdam-New York},
series = {Stud. Logic Foundations Math},
title = {{Counting equivalence classes for co-$\kappa $-Souslin
equivalence relations}},
volume = {108},
year = {1982},
},
@article{ARSh:153,
author = {Abraham, Uri and Rubin, Matatyahu and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL,IL,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 87d:03132},
pages = {123--206},
title = {{On the consistency of some partition theorems for continuous
colorings, and the structure of $\aleph_ 1$-dense real order types}},
volume = {29},
year = {1985},
},
@article{ShSt:154,
author = {Shelah, Saharon and Stanley, Lee},
ams-subject = {(03E35)},
journal = {Israel Journal of Mathematics},
review = {MR 84h:03120},
note = {Corrections in [Sh:154a]},
pages = {225--236},
title = {{Generalized Martin's axiom and Souslin's hypothesis for higher
cardinals}},
volume = {43},
year = {1982},
},
@article{ShSt:154a,
author = {Shelah, Saharon and Stanley, Lee},
ams-subject = {(03E35)},
fromwhere = {IL,1},
journal = {Israel Journal of Mathematics},
review = {MR 87m:03069},
pages = {304--314},
title = {{Corrigendum to: ``Generalized Martin's axiom and Souslin's
hypothesis for higher cardinals'' [Israel Journal of Mathematics 43
(1982), no. 3, 225--236; MR 84h:03120]}},
volume = {53},
year = {1986},
},
@article{Sh:155,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 90b:03048},
pages = {229--256},
title = {{The spectrum problem. III. Universal theories}},
volume = {55},
year = {1986},
},
@article{BlSh:156,
author = {Baldwin, John T. and Shelah, Saharon},
ams-subject = {(03C85)},
fromwhere = {1,IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 87h:03053},
note = {Proceedings of the 1980/1 Jerusalem Model Theory year},
pages = {229--303},
title = {{Second-order quantifiers and the complexity of theories}},
volume = {26},
year = {1985},
},
@article{LaSh:157,
author = {Lachlan, Alistair H. and Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
review = {MR 87h:03047b},
note = {Proceedings of the 1980/1 Jerusalem Model Theory year},
pages = {155--180},
title = {{Stable structures homogeneous for a finite binary language}},
volume = {49},
year = {1984},
},
@article{ShHM:158,
author = {Shelah, Saharon and Harrington, Leo and Makkai, Michael},
ams-subject = {(03C15)},
fromwhere = {IL,IL,IL},
journal = {Israel Journal of Mathematics},
review = {MR 86j:03029b},
note = {Proceedings of the 1980/1 Jerusalem Model Theory year},
pages = {259--280},
title = {{A proof of Vaught's conjecture for $\omega$-stable theories}},
volume = {49},
year = {1984},
},
@article{ShWd:159,
author = {Shelah, Saharon and Woodin, Hugh},
ams-subject = {(03E35)},
fromwhere = {IL,1},
journal = {The Journal of Symbolic Logic},
review = {MR 86h:03087},
pages = {1185--1189},
title = {{Forcing the failure of CH by adding a real}},
volume = {49},
year = {1984},
},
@article{HoSh:160,
author = {Hodges, Wilfrid and Shelah, Saharon},
ams-subject = {(03E75)},
fromwhere = {4,1},
journal = {Journal of the London Mathematical Society. Second Series},
review = {MR 87i:03115},
pages = {1--12},
title = {{Naturality and definability. I}},
volume = {33},
year = {1986},
},
@article{Sh:161,
author = {Shelah, Saharon},
ams-subject = {(03C60)},
fromwhere = {IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 87f:03095},
pages = {195--228},
title = {{Incompactness in regular cardinals}},
volume = {26},
year = {1985},
},
@article{HLSh:162,
author = {Hart, Bradd and Laflamme, Claude and Shelah, Saharon},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9311211 },
pages = {169--194},
title = {{Models with second order properties, V: A General principle}},
volume = {64},
year = {1993},
abstract = {We present a general framework for carrying out
some constructions. The unifying factor is a combinatorial
principle which we present in terms of a game in which the first
player challenges the second player to carry out constructions which
would be much easier in a generic extension of the universe, and
the second player cheats with the aid of $\Diamond$. Section 1
contains an axiomatic framework suitable for the description of a
number of related constructions, and the statement of the main theorem
in terms of this framework. In \S2 we illustrate the use of
our combinatorial principle. The proof of the main result is
then carried out in \S\S3-5.},
},
@incollection{GuSh:163,
author = {Gurevich, Yuri and Shelah, Saharon},
booktitle = {Foundations of logic and linguistics (Salzburg, 1983)},
ams-subject = {(03B45)},
fromwhere = {1,IL},
review = {MR 87b:03034},
note = {Proceedings of the Seventh International Congress for Logic,
Methology and Philosophy of Science, Salzburg, July 1983; eds. Dorn, G.
and Weingartner, P.},
pages = {181--198},
publisher = {Plenum, New York-London},
title = {{To the decision problem for branching time logic}},
year = {1985},
},
@article{JaSh:164,
author = {Jarden, Moshe and Shelah, Saharon},
ams-subject = {(12F20)},
journal = {Proceedings of the American Mathematical Society},
review = {MR 84c:12015},
pages = {223--228},
title = {{Pseudo-algebraically closed fields over rational function
fields}},
volume = {87},
year = {1983},
},
@article{ShWe:165,
author = {Shelah, Saharon and Weiss, Benjamin},
ams-subject = {(28D05)},
journal = {Israel Journal of Mathematics},
review = {MR 84d:28025},
pages = {154--160},
title = {{Measurable recurrence and quasi-invariant measures}},
volume = {43},
year = {1982},
},
@article{MkSh:166,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(03C80)},
fromwhere = {3,IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 87c:03081a},
note = {Proceedings of the 1980/1 Jerusalem Model Theory year},
pages = {129--138},
title = {{Stationary logic and its friends. I}},
volume = {26},
year = {1985},
},
@article{ShSt:167,
author = {Shelah, Saharon and Stanley, Lee},
ams-subject = {(03E35)},
fromwhere = {IL,1},
journal = {Israel Journal of Mathematics},
review = {MR 88e:03077},
nt = {With an appendix by John P. Burgess.},
pages = {1--65},
title = {{$S$-forcing. IIa. Adding diamonds and more applications:
coding sets, Arhangelskii's problem and ${\scr L}[Q^ {<\omega}_ 1,Q^ 1_
2]$}},
volume = {56},
year = {1986},
},
@article{GuSh:168,
author = {Gurevich, Yuri and Shelah, Saharon},
ams-subject = {(03F25)},
fromwhere = {1,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 90c:03053},
pages = {305--323},
title = {{On the strength of the interpretation method}},
volume = {54},
year = {1989},
},
@article{EMSh:169,
author = {Eklof, Paul C. and Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(20K30)},
fromwhere = {1,3,IL},
journal = {Israel Journal of Mathematics},
review = {MR 89c:20080},
pages = {283--298},
title = {{On strongly nonreflexive groups}},
volume = {59},
year = {1987},
},
@incollection{Sh:170,
author = {Shelah, Saharon},
booktitle = {Logic colloquium '82 (Florence, 1982)},
ams-subject = {(03F30)},
fromwhere = {IL},
review = {MR 86g:03097},
pages = {145--160},
publisher = {North-Holland, Amsterdam-New York},
series = {Stud. Logic Found. Math},
title = {{On logical sentences in {\rm PA}}},
volume = {112},
year = {1984},
},
@incollection{Sh:171,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
pages = {1-46},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{Classifying generalized quantifiers}},
volume = {1182},
year = {1986},
abstract = {We address the question of classifying
generalized quantifier ranging on a family of relations of fixed arity
on a fixed set U. We consider two notions of order (and equivalence) by
interpretability. In essence, every such family is equivalent to one
consisting of equivalence relations but we have to add a well ordering
of length $\lambda$. In the weaker equivalence, there is a gap in
cardinality between the size of equivalence relations which can be
interpreted and the one needed to interpret, which consistently may
occurs. Of course, the problem of classifying families of equivalence
relations is clear. The proof proceed by cases, summed up in section 7,
where we concentrate on the less fine equivalence relation. [Note
pages 24,25 where interchanged]},
abstract1 = {\endgraf NOTE: if U is countable, the gap in cardinals
disappear even for the weaker (i.e. finer) equivalence every such
family is equivalent to a family of equivalence relations (if you want
just to follow a proof of this then read):\\ 0.6: definition\\ 1.5 page
5: how to represent equivalence relations, hence later we can deal with
a family with one isomorphism type\\ 2.2 page 7: finish the analysis
for interpreting cases of monadic quantifier\\ 3.5 page 15: finish the
analysis of interpreting one to one functions, concerning uniformity
see the remark at the bottom of page 15\\ 4.5 page 16: definition of
$\lambda_2 (R)$\\ 4.17 page 23: reduction on a relation of cardinality
$\le \lambda_2 (R)$\\ Def 5.1 page 26: defining $\lambda_3$\\
Definition 5.11A(?) defining $K^{word}$ page 3(?)\\ Fact 5.2 page 26(?)
$\lambda_2 \in \{\lambda_3,\;lambda_2^+\}$\\ 5.11,5.12, 5.14 pages
31,32,33: analyses the case $\lambda_2\neq\lambda_3$,\\ rest on the
case of equality page 35 line -16; defining $\chi$ page 35 line -2\\ },
abstract2 = { 6.11 page 40: finishing this case and from now on
$\chi\ge cardinality(R)$ when U has cardinality $\aleph_0$\\
Explanation of the from of the paper:\\ The original aim of this
article was to prove that for every $K$ (a family of relations on U on
a fixed arity) its quantifier is equivalent to one for $KU$, a family
of equivalence relations (all such classes are assumed to be closed
under isomorphism). Sections 1-6 were written for this and realize it
to large extent. Clearly it suffice to deal with $I$ with one
isomorphism type as long as the interpretations are uniform. But two
essential difficulties arise (1) the
quantifier $\exist^{word}_{\lambda}$ (a well ordering of length
$\lambda$), provably is not biinterpretable with $\exist_K$ for any
family $K$ of equivalence classes. This is not so serious: just
add another case. (2) It is consistent that there are
cardinals $\chi,\lambda$ satisfying $\chi\le\lambda \le 2^\chi $ such
that $R$ is e.g. a family of $\chi$ sunsets of $\lambda$, again
we cannot reduce this to equivalence relations in general (see section
8)\\ Only under the assumptions $V=L$ and considering more liberal
notion of bi-interpretability ($\equivalence_{exp}$ rather than
$\equivalence_{int}$ the desired result is gotten. However when the gap
degenerates for any reason we get the original hope (note
$\exist^{word}_\w$ is second order quantification).},
},
@article{Sh:172,
author = {Shelah, Saharon},
ams-subject = {(16A99)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 86i:16044},
note = {Proceedings of the 1980/1 Jerusalem Model Theory year},
pages = {239--257},
title = {{A combinatorial principle and endomorphism rings. I. On
$p$-groups}},
volume = {49},
year = {1984},
},
@incollection{ANSh:173,
author = {Aharoni, R. and Nash Williams, C. St. J. A. and Shelah,
Saharon},
booktitle = {Progress in graph theory (Waterloo, Ont., 1982)},
ams-subject = {(04A20)},
fromwhere = {IL,4,IL},
review = {MR 86h:04002},
note = {Proceedings of the Conference in Waterloo, July 1982},
pages = {71--79},
publisher = {Academic Press, Toronto, Ont.},
title = {{Marriage in infinite societies}},
year = {1984},
},
@article{GrSh:174,
author = {Grossberg, Rami and Shelah, Saharon},
ams-subject = {(20A15)},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
review = {MR 85c:20001},
pages = {289--302},
title = {{On universal locally finite groups}},
volume = {44},
year = {1983},
},
@article{Sh:175,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 85h:03054},
note = {See also [Sh:175a]},
pages = {75--87},
title = {{On universal graphs without instances of CH}},
volume = {26},
year = {1984},
},
@article{Sh:175a,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {1},
journal = {Israel Journal of Mathematics},
review = {MR 94e:03048},
pages = {69--81},
title = {{Universal graphs without instances of {\rm CH}: revisited}},
volume = {70},
year = {1990},
},
@article{Sh:176,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 86g:03082a},
pages = {1--47},
title = {{Can you take Solovay's inaccessible away?}},
volume = {48},
year = {1984},
},
@article{Sh:177,
author = {Shelah, Saharon},
ams-subject = {(03E40)},
fromwhere = {IL},
journal = {The Journal of Symbolic Logic},
review = {MR 86m:03082},
pages = {1034--1038},
title = {{More on proper forcing}},
volume = {49},
year = {1984},
},
@article{GuSh:178,
author = {Gurevich, Yuri and Shelah, Saharon},
ams-subject = {(03B25)},
fromwhere = {1,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 85d:03019},
pages = {1120--1124},
title = {{Random models and the Godel case of the decision problem}},
volume = {48},
year = {1983},
},
@article{ShSn:179,
author = {Shelah, Saharon and Steinhorn, Charles},
ams-subject = {(03C80)},
fromwhere = {IL,1},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 87d:03104},
pages = {1--11},
title = {{On the nonaxiomatizability of some logics by finitely many
schemas}},
volume = {27},
year = {1986},
},
@article{ShSn:180,
author = {Shelah, Saharon and Steinhorn, Charles},
ams-subject = {(03C80)},
fromwhere = {IL,1},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 91a:03084},
pages = {1--13},
title = {{The nonaxiomatizability of $L(Q^ 2_ {\aleph_ 1})$ by finitely
many schemata}},
volume = {31},
year = {1990},
},
@article{KfSh:181,
author = {Kaufmann, Matt and Shelah, Saharon},
ams-subject = {(03C80)},
fromwhere = {1,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 86a:03037},
pages = {209--214},
title = {{A nonconservativity result on global choice}},
volume = {27},
year = {1984},
},
@article{AbSh:182,
author = {Abraham, Uri and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 87e:03117},
pages = {180--189},
title = {{On the intersection of closed unbounded sets}},
volume = {51},
year = {1986},
},
@article{GuSh:183,
author = {Gurevich, Yuri and Shelah, Saharon},
ams-subject = {(03C65)},
fromwhere = {1,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 85g:03055},
pages = {1105--1119},
title = {{Rabin's uniformization problem}},
volume = {48},
year = {1983},
},
@article{GGSh:184,
author = {Goldfarb, Warren D. and Gurevich, Yuri and Shelah, Saharon},
ams-subject = {(03B10)},
fromwhere = {1,1,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 86g:03015b},
pages = {1253--1261},
title = {{A decidable subclass of the minimal Godel class with
identity}},
volume = {49},
year = {1984},
},
@article{Sh:185,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 85b:03092},
pages = {90--96},
title = {{Lifting problem of the measure algebra}},
volume = {45},
year = {1983},
},
@article{Sh:186,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL},
journal = {The Journal of Symbolic Logic},
review = {MR 86g:03083},
pages = {1022--1033},
title = {{Diamonds, uniformization}},
volume = {49},
year = {1984},
},
@article{MkSh:187,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(03C80)},
fromwhere = {3,IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 87c:03081b},
pages = {39--50},
title = {{Stationary logic and its friends. II}},
volume = {27},
year = {1986},
},
@article{Sh:188,
author = {Shelah, Saharon},
ams-subject = {(03C75)},
fromwhere = {IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 85j:03055},
pages = {97--104},
title = {{A pair of nonisomorphic $\equiv _{\infty \lambda }$ models of
power $\lambda $ for $\lambda $ singular with $\lambda ^{\omega
}=\lambda $}},
volume = {25},
year = {1984},
},
@article{Sh:189,
author = {Shelah, Saharon},
ams-subject = {(03C75)},
fromwhere = {IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 86h:03072},
pages = {36--50},
title = {{On the possible number ${\rm no}(M)=$ the number of
nonisomorphic models $L_ {\infty,\lambda}$-equivalent to $M$ of power
$\lambda$, for $\lambda$ singular}},
volume = {26},
year = {1985},
},
@article{GbSh:190,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
ams-subject = {(20K20)},
fromwhere = {D,IL},
journal = {Journal of Algebra},
review = {MR 86d:20061},
pages = {136--150},
title = {{Semirigid classes of cotorsion-free abelian groups}},
volume = {93},
year = {1985},
},
@article{GiSh:191,
author = {Gitik, Moti and Shelah, Saharon},
ams-subject = {(03E40)},
fromwhere = {1,IL},
journal = {Israel Journal of Mathematics},
review = {MR 87c:03104},
pages = {148--158},
title = {{On the $\bbfI$-condition}},
volume = {48},
year = {1984},
},
@article{Sh:192,
author = {Shelah, Saharon},
ams-subject = {(20A15)},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 89d:20001},
pages = {153--206},
title = {{Uncountable groups have many nonconjugate subgroups}},
volume = {36},
year = {1987},
},
@article{LhSh:193,
author = {Lehmann, Daniel and Shelah, Saharon},
ams-subject = {(68Q55)},
fromwhere = {IL,IL},
journal = {Information and Control},
review = {MR 85c:68046},
pages = {165--198},
title = {{Reasoning with time and chance}},
volume = {53},
year = {1982},
},
@article{ANSh:194,
author = {Aharoni, R. and Nash Williams, C. St. J. A. and Shelah,
Saharon},
ams-subject = {(04A20)},
journal = {Proceedings of the London Mathematical Society. Third
Series},
review = {MR 85g:04001},
pages = {43--68},
title = {{A general criterion for the existence of transversals}},
volume = {47},
year = {1983},
},
@article{DrSh:195,
author = {Droste, Manfred and Shelah, Saharon},
ams-subject = {(20F29)},
fromwhere = {D,IL},
journal = {Israel Journal of Mathematics},
review = {MR 87d:20055},
pages = {223--261},
title = {{A construction of all normal subgroup lattices of
$2$-transitive automorphism groups of linearly ordered sets}},
volume = {51},
year = {1985},
},
@article{ANSh:196,
author = {Aharoni, R. and Nash Williams, C. St. J. A. and Shelah,
Saharon},
ams-subject = {(04A20)},
fromwhere = {IL,4,IL},
journal = {Journal of the London Mathematical Society. Second Series},
review = {MR 85i:04004},
pages = {193--203},
title = {{Another form of a criterion for the existence of
transversals}},
volume = {29},
year = {1984},
},
@incollection{Sh:197,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-26},
pages = {203--223},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{Monadic logic: Hanf numbers}},
volume = {1182},
year = {1986},
},
@article{LMSh:198,
author = {Levinski, Jean Pierre and Magidor, Menachem and Shelah,
Saharon},
ams-subject = {(03C52)},
fromwhere = {1,IL,IL},
journal = {Israel Journal of Mathematics},
review = {MR 91g:03071},
pages = {161--172},
title = {{Chang's conjecture for $\aleph_ \omega$}},
volume = {69},
year = {1990},
},
@article{Sh:199,
author = {Shelah, Saharon},
ams-subject = {(03C95)},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 87g:03040},
pages = {255--288},
title = {{Remarks in abstract model theory}},
volume = {29},
year = {1985},
},
@article{Sh:200,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL},
journal = {American Mathematical Society. Bulletin. New Series},
review = {MR 86h:03058},
pages = {227--232},
title = {{Classification of first order theories which have a structure
theorem}},
volume = {12},
year = {1985},
},
@article{KfSh:201,
author = {Kaufmann, Matt and Shelah, Saharon},
ams-subject = {(03C13)},
fromwhere = {1,IL},
journal = {Discrete Mathematics},
review = {MR 86m:03049},
pages = {285--293},
title = {{On random models of finite power and monadic logic}},
volume = {54},
year = {1985},
},
@article{Sh:202,
author = {Shelah, Saharon},
ams-subject = {(03E15)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 86a:03054},
pages = {139--153},
title = {{On co-$\kappa $-Souslin relations}},
volume = {47},
year = {1984},
},
@article{BdSh:203,
author = {Ben David, Shai and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 87h:03078},
pages = {207--217},
title = {{Souslin trees and successors of singular cardinals}},
volume = {30},
year = {1986},
},
@article{MgSh:204,
author = {Magidor, Menachem and Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of the American Mathematical Society},
pages = {769--830},
title = {{When does almost free imply free? (For groups, transversal
etc.)}},
volume = {7},
year = {1994},
},
@article{Sh:205,
author = {Shelah, Saharon},
ams-subject = {(03C85)},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 87i:03075},
pages = {203--216},
title = {{Monadic logic and Lowenheim numbers}},
volume = {28},
year = {1985},
},
@article{Sh:206,
author = {Shelah, Saharon},
ams-subject = {(54G99)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 90e:54088},
pages = {183--211},
title = {{Decomposing topological spaces into two rigid homeomorphic
subspaces}},
volume = {63},
year = {1988},
},
@incollection{Sh:207,
author = {Shelah, Saharon},
booktitle = {Axiomatic set theory (Boulder, Colo., 1983)},
ams-subject = {(03E35)},
fromwhere = {IL},
review = {MR 86b:03064},
note = {Proceedings of the Conference in Set Theory, Boulder, June 1983;
ed. Baumgartner J., Martin, D. and Shelah, S.},
pages = {183--207},
publisher = {Amer. Math. Soc., Providence, RI},
series = {Contemp. Mathematics},
title = {{On cardinal invariants of the continuum}},
volume = {31},
year = {1984},
},
@article{Sh:208,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 87d:03136},
pages = {315--318},
title = {{More on the weak diamond}},
volume = {28},
year = {1985},
},
@article{ShTo:209,
author = {Shelah, Saharon and Todorcevic, Stevo},
ams-subject = {(54A35)},
fromwhere = {IL,1},
journal = {Canadian Journal of Mathematics. Journal Canadien de
Mathematiques},
review = {MR 88c:54005},
pages = {659--665},
title = {{A note on small Baire spaces}},
volume = {38},
year = {1986},
},
@article{BoSh:210,
author = {Bonnet, Robert and Shelah, Saharon},
ams-subject = {(06E99)},
fromwhere = {F,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 86g:06024},
pages = {1--12},
title = {{Narrow Boolean algebras}},
volume = {28},
year = {1985},
},
@article{Sh:211,
author = {Shelah, Saharon},
ams-subject = {(03C75)},
fromwhere = {IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 93h:03054},
note = { arxiv:math.LO/9201243 },
pages = {1--12},
title = {{The Hanf numbers of stationary logic. II. Comparison with
other logics}},
volume = {33},
year = {1992},
abstract = {We show that the ordering of the Hanf number
of $L_{\omega,\omega}(wo)$ (well ordering),
$L^c_{\omega,\omega}$ (quantification on countable sets), $L_{\omega,
\omega}(aa)$ (stationary logic) and second order logic, have no more
restraints provable in $ZFC$ than previously known (those independence
proofs assume $CON(ZFC)$ only). We also get results on corresponding
logics for $L_{\lambda,\mu}$.},
},
@incollection{Sh:212,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-27},
pages = {188--202},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{The existence of coding sets}},
volume = {1182},
year = {1986},
},
@article{DGSh:213,
author = {Denenberg, Larry and Gurevich, Yuri and Shelah, Saharon},
ams-subject = {(03G05)},
fromwhere = {1,1,IL},
journal = {Information and Control},
review = {MR 88b:03094},
pages = {216--240},
title = {{Definability by constant-depth polynomial-size circuits}},
volume = {70},
year = {1986},
abstract = {We investigate the expressive power of constant-depth
polynomial-size circuit models. In particular, we construct a circuit
model whose expressive power is precisely that of first-order logic.},
},
@article{MkSh:214,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(20K10)},
fromwhere = {3,IL},
journal = {Pacific Journal of Mathematics},
review = {MR 87f:20073a},
note = {See also [MkSh:214a]},
pages = {121--132},
title = {{$\omega$-elongations and Crawley's problem}},
volume = {121},
year = {1986},
},
@article{MkSh:214a,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(20K10)},
fromwhere = {3,IL},
journal = {Pacific Journal of Mathematics},
review = {MR 87f:20073b},
pages = {133--134},
title = {{The solution to Crawley's problem}},
volume = {121},
year = {1986},
},
@article{HMSh:215,
author = {Harrington, Leo and Marker, David and Shelah, Saharon},
ams-subject = {(03E15)},
fromwhere = {1,1,IL},
journal = {Transactions of the American Mathematical Society},
review = {MR 90c:03041},
pages = {293--302},
title = {{Borel orderings}},
volume = {310},
year = {1988},
},
@article{HMSh:216,
author = {Holland, W. Charles and Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(06F15)},
fromwhere = {3,3,IL},
journal = {Order},
review = {MR 86m:06032},
note = {See also [HMSh:216a]},
pages = {383--397},
title = {{Lawless order}},
volume = {1},
year = {1985},
},
@incollection{HMSh:216a,
author = {Holland, W. Charles and Mekler, Alan H. and Shelah, Saharon},
booktitle = {Algebra and order (Luminy-Marseille, 1984)},
ams-subject = {(06F15)},
fromwhere = {1,3,IL},
review = {MR 88h:06023},
note = {Proceedings of the First International Symposium on Ordered
Algebraic Structures, Luminy-Marseilles, July 1984; ed. Wolfenstein,
S.},
pages = {29--33},
publisher = {Heldermann, Berlin},
series = {R \& E Res. Exp. Math},
title = {{Total orders whose carried groups satisfy no laws}},
volume = {14},
year = {1986},
},
@article{SgSh:217,
author = {Sageev, Gershon and Shelah, Saharon},
fromwhere = {IL},
journal = {Abstracts of the American Mathematical Society},
note = { arxiv:0705.4132 },
pages = {369},
title = {{Noetherian ring with free additive groups}},
volume = {7},
year = {1986},
},
@article{Sh:218,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 87h:03080},
pages = {110--114},
title = {{On measure and category}},
volume = {52},
year = {1985},
},
@article{GbSh:219,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
ams-subject = {(13C13)},
fromwhere = {D,IL},
journal = {Mathematische Zeitschrift},
review = {MR 86d:13011},
pages = {325--337},
title = {{Modules over arbitrary domains}},
volume = {188},
year = {1985},
},
@article{Sh:220,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 89c:03058},
note = {Proceedings of the Model Theory Conference, Trento, June 1986},
nt = {Stability in model theory (Trento, 1984).},
pages = {291--310},
title = {{Existence of many $L_ {\infty,\lambda}$-equivalent,
nonisomorphic models of $T$ of power $\lambda$}},
volume = {34},
year = {1987},
},
@article{AShS:221,
author = {Abraham, Uri and Shelah, Saharon and Solovay, R. M. },
ams-subject = {(03E05)},
fromwhere = {IL,IL,1},
journal = {Fundamenta Mathematicae},
review = {MR 88d:03092},
pages = {133--162},
title = {{Squares with diamonds and Souslin trees with special
squares}},
volume = {127},
year = {1987},
},
@article{GrSh:222,
author = {Grossberg, Rami and Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {1,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 87j:03037},
pages = {302--322},
title = {{On the number of nonisomorphic models of an infinitary theory
which has the infinitary order property. I}},
volume = {51},
year = {1986},
},
@article{DrSh:223,
author = {Droste, Manfred and Shelah, Saharon},
ams-subject = {(20F10)},
fromwhere = {D,IL},
journal = {Pacific Journal of Mathematics},
review = {MR 88b:20055},
pages = {321--328},
title = {{On the universality of systems of words in permutation
groups}},
volume = {127},
year = {1987},
},
@article{GbSh:224,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
ams-subject = {(13C13)},
fromwhere = {D,IL},
journal = {Fundamenta Mathematicae},
review = {MR 88d:13021},
pages = {217--243},
title = {{Modules over arbitrary domains. II}},
volume = {126},
year = {1986},
},
@article{Sh:225,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 89b:03057},
note = {See also [Sh:225a]},
pages = {279--287},
title = {{On the number of strongly $\aleph_ \epsilon$-saturated models
of power $\lambda$}},
volume = {36},
year = {1987},
},
@article{Sh:225a,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 89i:03060},
pages = {89--91},
title = {{Number of strongly $\aleph_ \epsilon$ saturated models---an
addition}},
volume = {40},
year = {1988},
},
@article{FMSh:226,
author = {Foreman, Matthew and Magidor, Menachem and Shelah, Saharon},
ams-subject = {(03E40)},
fromwhere = {1,IL,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 87i:03102},
pages = {39--46},
title = {{$0^ \sharp$ and some forcing principles}},
volume = {51},
year = {1986},
},
@incollection{Sh:227,
author = {Shelah, Saharon},
booktitle = {Abelian groups and modules (Udine, 1984)},
ams-subject = {(20K30)},
fromwhere = {IL},
review = {MR 86i:20075},
note = {Proceedings of the Conference on Abelian Groups, Undine, April
9-14, 1984); ed. Goebel, R., Metelli, C., Orsatti, A. and Solce, L.},
pages = {37--86},
publisher = {Springer, Vienna},
series = {CISM Courses and Lectures},
title = {{A combinatorial theorem and endomorphism rings of abelian
groups. II}},
volume = {287},
year = {1984},
},
@incollection{Sh:228,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-30},
pages = {120--134},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{On the ${\rm no}(M)$ for $M$ of singular power}},
volume = {1182},
year = {1986},
},
@incollection{Sh:229,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-31},
note = { arxiv:math.LO/9201238 },
pages = {91--119},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{Existence of endo-rigid Boolean Algebras}},
volume = {1182},
year = {1986},
abstract = {In [Sh:89] we, answering a question of Monk, have explicated
the notion of ``a Boolean algebra with no endomorphisms except the
ones induced by ultrafilters on it'' (see \S 2 here) and proved
the existence of one with character density $\aleph_0$, assuming
first $\diamondsuit_{\aleph_1}$ and then only $CH$. The idea was that
if $h$ is an endomorphism of $B$, not among the ``trivial'' ones,
then there are pairwise disjoint $D_n\in B$ with
$h(d_n)\not\subset d_n$. Then we can, for some $S\subset\omega$, add an
element $x$ such that $d\leq x$ for $n\in S$, $x\cap d_n=0$ for
$n\not\in S$ while forbidding a solution for $\{y\cap h(d_n):n\in
S\}\cup\{y\cap h(d_n)=0:n\not\in S\}$. Further analysis showed that the
point is that we are omitting positive quantifier free types.
Continuing this Monk succeeded to prove in ZFC, the existence of such
Boolean algebras of cardinality $2^{\aleph_0}$. \endgraf We prove (in
ZFC) the existence of such $B$ of density character $\lambda$ and
cardinality $\lambda^{\aleph_0}$ whenever $\lambda>\aleph_0$. We can
conclude answers to some questions from Monk's list. We use a
combinatorial method from [Sh:45],[Sh:172], that is represented in
Section 1.},
},
@article{GuSh:230,
author = {Gurevich, Yuri and Shelah, Saharon},
ams-subject = {(03B45)},
fromwhere = {1,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 87c:03033},
pages = {668--681},
title = {{The decision problem for branching time logic}},
volume = {50},
year = {1985},
},
@article{JuSh:231,
author = {Juhasz, Istvan and Shelah, Saharon},
trueauthor = {Juh\'asz, Istv\'an and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {H,IL},
journal = {Israel Journal of Mathematics},
review = {MR 87f:03143},
pages = {355--364},
title = {{How large can a hereditarily separable or hereditarily
Lindelof space be?}},
volume = {53},
year = {1986},
},
@incollection{Sh:232,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-29},
pages = {135--150},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{Nonstandard uniserial module over a uniserial domain exists}},
volume = {1182},
year = {1986},
},
@incollection{Sh:233,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-28},
pages = {151--187},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{Remarks on the numbers of ideals of Boolean algebra and open
sets of a topology}},
volume = {1182},
year = {1986},
},
@incollection{Sh:234,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-32},
pages = {47--90},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{Classification over a predicate. II}},
volume = {1182},
year = {1986},
},
@article{ShSf:235,
author = {Shelah, Saharon and Soifer, Alexander},
ams-subject = {(20K35)},
fromwhere = {IL,1},
journal = {Journal of Algebra},
review = {MR 87f:20078},
pages = {359--369},
title = {{Two problems on $\aleph_ 0$-indecomposable abelian groups}},
volume = {99},
year = {1986},
},
@article{BdSh:236,
author = {Ben David, Shai and Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
review = {MR 87k:03051},
pages = {93--96},
title = {{Nonspecial Aronszajn trees on $\aleph_ {\omega+1}$}},
volume = {53},
year = {1986},
},
@article{Sh:237,
author = {Shelah, Saharon},
fromwhere = {IL},
title = {{See 237a, 237b, 237c, 237d, 237e}},
},
@incollection{Sh:237a,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-24},
pages = {247--259},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{On normal ideals and Boolean algebras}},
volume = {1182},
year = {1986},
},
@incollection{Sh:237b,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-23},
pages = {260--268},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{A note on $\kappa$-freeness of abelian groups}},
volume = {1182},
year = {1986},
},
@incollection{Sh:237c,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-22},
pages = {269--271},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{On countable theories with models---homogeneous models only}},
volume = {1182},
year = {1986},
},
@incollection{Sh:237d,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-21},
pages = {272--275},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{On decomposable sentences for finite models}},
volume = {1182},
year = {1986},
},
@incollection{Sh:237e,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-20},
pages = {276--279},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{Remarks on squares}},
volume = {1182},
year = {1986},
},
@article{GrSh:238,
author = {Grossberg, Rami and Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL,IL},
journal = {Illinois Journal of Mathematics},
review = {MR 87j:03036},
note = {Volume dedicated to the memory of W.W.~Boone; ed. Appel, K.,
Higman, G., Robinson, D. and Jockush, C.},
pages = {364--390},
title = {{A nonstructure theorem for an infinitary theory which has the
unsuperstability property}},
volume = {30},
year = {1986},
},
@article{ShSf:239,
author = {Shelah, Saharon and Soifer, Alexander},
ams-subject = {(20K21)},
fromwhere = {IL,1},
journal = {Journal of Algebra},
review = {MR 87k:20092},
pages = {421--429},
title = {{Countable $\aleph_ 0$-indecomposable mixed abelian groups of
finite torsion-free rank}},
volume = {100},
year = {1986},
},
@article{FMSh:240,
author = {Foreman, Matthew and Magidor, Menachem and Shelah, Saharon},
ams-subject = {(03E50)},
fromwhere = {1,IL,IL},
journal = {Annals of Mathematics. Second Series},
review = {MR 89f:03043},
note = {See also ANN. of Math. (2) 129 (1989)},
pages = {1--47},
title = {{Martin's maximum, saturated ideals, and nonregular
ultrafilters. I}},
volume = {127},
year = {1988},
},
@article{ShWd:241,
author = {Shelah, Saharon and Woodin, Hugh},
ams-subject = {(03E55)},
fromwhere = {IL,1},
journal = {Israel Journal of Mathematics},
review = {MR 92m:03087},
pages = {381--394},
title = {{Large cardinals imply that every reasonably definable set of
reals is Lebesgue measurable}},
volume = {70},
year = {1990},
},
@article{BsSh:242,
author = {Blass, Andreas and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {1,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 88e:03073},
pages = {213--243},
title = {{There may be simple $P_ {\aleph_ 1}$- and $P_ {\aleph_
2}$-points and the Rudin-Keisler ordering may be downward directed}},
volume = {33},
year = {1987},
},
@article{GuSh:243,
author = {Gurevich, Yuri and Shelah, Saharon},
ams-subject = {(05C80)},
fromwhere = {1,IL},
journal = {SIAM Journal on Computing},
review = {MR 88i:05162},
pages = {486--502},
title = {{Expected computation time for Hamiltonian path problem}},
volume = {16},
year = {1987},
},
@incollection{GuSh:244,
author = {Gurevich, Yuri and Shelah, Saharon},
booktitle = {Proceedings of 26th Annual Symp. on Foundation of Computer
Science},
fromwhere = {1,IL},
note = {See also [GuSh:244a] below},
pages = {346--353},
publisher = {IEEE Computer Science Society Press},
title = {{The fix point extensions of first order logic}},
year = {1985},
},
@article{GuSh:244a,
author = {Gurevich, Yuri and Shelah, Saharon},
ams-subject = {(03C80)},
fromwhere = {1,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 88b:03056},
pages = {265--280},
title = {{Fixed-point extensions of first-order logic}},
volume = {32},
year = {1986},
},
@article{CHSh:245,
author = {Compton, Kevin J. and Henson, C. Ward and Shelah, Saharon},
ams-subject = {(03C13)},
fromwhere = {1,1,1},
journal = {Annals of Pure and Applied Logic},
review = {MR 89f:03021},
pages = {207--224},
title = {{Nonconvergence, undecidability, and intractability in
asymptotic problems}},
volume = {36},
year = {1987},
},
@incollection{Sh:246,
author = {Shelah, Saharon},
booktitle = {Algebraic logic (Budapest, 1988)},
ams-subject = {(03G15)},
fromwhere = {IL},
review = {MR 93a:03073},
note = {Proceedings of Conference of Cylindrical Algebras, Budapest,
8.1988},
pages = {645--664},
publisher = {North-Holland, Amsterdam},
series = {Colloq. Math. Soc. Janos Bolyai},
title = {{On a problem in cylindric algebra}},
volume = {54},
year = {1991},
},
@incollection{Sh:247,
author = {Shelah, Saharon},
booktitle = {Around classification theory of models},
fromwhere = {IL},
review = {MR 18-15-25},
pages = {224--246},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{More on stationary coding}},
volume = {1182},
year = {1986},
},
@article{Sh:248,
author = {Shelah, Saharon},
fromwhere = {IL},
note = {moved to F45},
title = {{TBA}},
},
@article{HJSh:249,
author = {Hajnal, Andras and Juhasz, Istvan and Shelah, Saharon},
trueauthor = {Hajnal, Andras and Juh\'asz, Istv\'an and Shelah,
Saharon},
ams-subject = {(03E05)},
fromwhere = {H,H,IL},
journal = {Transactions of the American Mathematical Society},
review = {MR 87i:03098},
pages = {369--387},
title = {{Splitting strongly almost disjoint families}},
volume = {295},
year = {1986},
},
@article{Sh:250,
author = {Shelah, Saharon},
ams-subject = {(03E40)},
fromwhere = {IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 89a:03093},
pages = {1--17},
title = {{Some notes on iterated forcing with $2^ {\aleph_ 0}>\aleph_
2$}},
volume = {29},
year = {1988},
},
@incollection{MkSh:251,
author = {Mekler, Alan H. and Shelah, Saharon},
booktitle = {Abelian group theory (Oberwolfach, 1985)},
ams-subject = {(20K20)},
fromwhere = {3,IL},
review = {MR 90f:20082},
note = {Proceedings of the third conference on Abelian Groups Theory,
Oberwolfach},
pages = {137--148},
publisher = {Gordon and Breach, New York},
title = {{When $\kappa$-free implies strongly $\kappa$-free}},
year = {1987},
},
@article{FMSh:252,
author = {Foreman, Matthew and Magidor, Menachem and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {1,IL,IL},
journal = {Annals of Mathematics. Second Series},
review = {MR 90a:03077},
pages = {521--545},
title = {{Martin's maximum, saturated ideals and nonregular
ultrafilters. II}},
volume = {127},
year = {1988},
},
@article{Sh:253,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 90g:03050},
note = {reappeared (in a revised form) in chapter XIII of [Sh:f]},
pages = {345--380},
title = {{Iterated forcing and normal ideals on $\omega_ 1$}},
volume = {60},
year = {1987},
},
@article{BaSh:254,
author = {Baumgartner, James E. and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {1,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 88d:03100},
pages = {109--129},
title = {{Remarks on superatomic Boolean algebras}},
volume = {33},
year = {1987},
},
@incollection{EkSh:255,
author = {Eklof, Paul C. and Shelah, Saharon},
booktitle = {Abelian group theory (Oberwolfach, 1985)},
ams-subject = {(20K25)},
fromwhere = {1,IL},
review = {MR 91d:20063},
note = {Proceedings of the third conference on Abelian Groups Theory,
Oberwolfach, Aug. 1983},
pages = {149--163},
publisher = {Gordon and Breach, New York},
title = {{On groups $A$ such that $A\oplus {\bf Z}^ n\cong A$}},
year = {1987},
},
@article{Sh:256,
author = {Shelah, Saharon},
ams-subject = {(03E10)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 89h:03082},
pages = {299--326},
title = {{More on powers of singular cardinals}},
volume = {59},
year = {1987},
},
@article{BsSh:257,
author = {Blass, Andreas and Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {1,1},
journal = {Israel Journal of Mathematics},
review = {MR 90e:03057},
pages = {259--271},
title = {{Ultrafilters with small generating sets}},
volume = {65},
year = {1989},
},
@article{ShSt:258,
author = {Shelah, Saharon and Stanley, Lee},
ams-subject = {(03E05)},
fromwhere = {IL,1},
journal = {Annals of Pure and Applied Logic},
review = {MR 89d:03045},
pages = {119--152},
title = {{A theorem and some consistency results in partition
calculus}},
volume = {36},
year = {1987},
},
@article{GrSh:259,
author = {Grossberg, Rami and Shelah, Saharon},
fromwhere = {1,L},
journal = {Mathematica Japonica},
note = { arxiv:math.LO/9809196 },
title = {{On Hanf numbers of the infinitary order property}},
volume = {submitted},
abstract = {We study several cardinal, and ordinal--valued
functions that are relatives of Hanf numbers. Let $\kappa$ be
an infinite cardinal, and let $T\subseteq L_{\kappa^+,\omega}$ be
a theory of cardinality $\leq\kappa$, and let $\gamma$ be an ordinal
$\geq\kappa^+$. For example we look at (1)
$\mu_{T}^*(\gamma,\kappa):=\min\{\mu^*\forall\phi\in L_{\infty,\omega}$
, with $rk(\phi)<\gamma$, if $T$ has the $(\phi,\mu^*)$-order property
then there exists a formula $\phi'(x;y)\in L_{\kappa^+,\omega}$, such
that for every $\chi\geq\kappa$, $T$ has the $(\phi',\chi)$-order
property$\}$; and (2)
$\mu^*(\gamma,\kappa):=\sup\{\mu_{T}^*(\gamma,\kappa)\;|\;T\in L_{\kapp
a^+,\omega}\}$.},
},
@article{ShSr:260,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
ams-subject = {(20F24)},
fromwhere = {IL,3},
journal = {Annals of Pure and Applied Logic},
review = {MR 88d:20058},
pages = {87--97},
title = {{Extraspecial $p$-groups}},
volume = {34},
year = {1987},
},
@article{Sh:261,
author = {Shelah, Saharon},
ams-subject = {(05C99)},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 89k:05104},
pages = {171--183},
title = {{A graph which embeds all small graphs on any large set of
vertices}},
volume = {38},
year = {1988},
},
@article{Sh:262,
author = {Shelah, Saharon},
ams-subject = {(03C52)},
fromwhere = {IL},
journal = {The Journal of Symbolic Logic},
review = {MR 91h:03042},
pages = {1431--1455},
title = {{The number of pairwise non-elementarily-embeddable models}},
volume = {54},
year = {1989},
},
@article{Sh:263,
author = {Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL},
journal = {The Journal of Symbolic Logic},
review = {MR 89g:03072},
note = {Represented in [Sh:f], Chapter 17},
pages = {360--367},
title = {{Semiproper forcing axiom implies Martin maximum but not ${\rm
PFA}^ +$}},
volume = {52},
year = {1987},
},
@article{ShSr:264,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
ams-subject = {(46B99)},
fromwhere = {IL,3},
journal = {Proceedings of the American Mathematical Society},
review = {MR 90a:46047},
pages = {101--105},
title = {{A Banach space on which there are few operators}},
volume = {104},
year = {1988},
},
@article{DFSh:265,
author = {Dugas, M. and Fay, T. H. and Shelah, Saharon},
ams-subject = {(20K99)},
fromwhere = {1,1,IL},
journal = {Journal of Algebra},
review = {MR 88g:20120},
pages = {127--137},
title = {{Singly cogenerated annihilator classes}},
volume = {109},
year = {1987},
},
@article{Sh:266,
author = {Shelah, Saharon},
fromwhere = {IL},
note = { arxiv:math.LO/1401.3175 },
title = {{Compactness in singular cardinals revisited}},
},
@article{FlSh:267,
author = {Fleissner, William G. and Shelah, Saharon},
ams-subject = {(54D15)},
fromwhere = {1,IL},
journal = {Topology and its Applications},
review = {MR 90d:54043},
pages = {101--107},
title = {{Collectionwise Hausdorff: incompactness at singulars}},
volume = {31},
year = {1989},
},
@article{HKSh:268,
author = {Hajnal, Andras and Kanamori, Akihiro and Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {H,1,IL},
journal = {Transactions of the American Mathematical Society},
review = {MR 88f:03041},
pages = {145--154},
title = {{Regressive partition relations for infinite cardinals}},
volume = {299},
year = {1987},
},
@article{Sh:269,
author = {Shelah, Saharon},
ams-subject = {(03C80)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 91a:03083},
pages = {133--152},
title = {{``Gap $1$'' two-cardinal principles and the omitting types
theorem for ${\scr L} (Q)$}},
volume = {65},
year = {1989},
},
@article{Sh:270,
author = {Shelah, Saharon},
ams-subject = {(54A35)},
fromwhere = {IL},
journal = {Topology and its Applications},
review = {MR 91c:54009},
pages = {217--221},
title = {{Baire irresolvable spaces and lifting for a layered ideal}},
volume = {33},
year = {1989},
},
@article{HoSh:271,
author = {Hodges, Wilfrid and Shelah, Saharon},
ams-subject = {(03C80)},
fromwhere = {4,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 93h:03055},
pages = {300--322},
title = {{There are reasonably nice logics}},
volume = {56},
year = {1991},
},
@incollection{Sh:272,
author = {Shelah, Saharon},
booktitle = {Classification theory (Chicago, IL, 1985)},
ams-subject = {(03C45)},
fromwhere = {IL},
review = {MR 90m:03071},
note = {Proceedings of the USA--Israel Conference on Classification
Theory, Chicago, December 1985; ed. Baldwin, J.T.},
pages = {498--500},
publisher = {Springer, Berlin-New York},
series = {Lecture Notes in Mathematics},
title = {{On almost categorical theories}},
volume = {1292},
year = {1987},
},
@article{Sh:273,
author = {Shelah, Saharon},
ams-subject = {(55Q05)},
fromwhere = {IL},
journal = {Proceedings of the American Mathematical Society},
review = {MR 89g:55021},
pages = {627--632},
title = {{Can the fundamental (homotopy) group of a space be the
rationals?}},
volume = {103},
year = {1988},
},
@article{MkSh:274,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {3,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 90j:03089},
pages = {441--459},
title = {{Uniformization principles}},
volume = {54},
year = {1989},
},
@article{MkSh:275,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(08B20)},
fromwhere = {3,IL},
journal = {Algebra Universalis},
review = {MR 91j:08011},
pages = {351--366},
title = {{$L_ {\infty\omega}$-free algebras}},
volume = {26},
year = {1989},
},
@article{Sh:276,
author = {Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 89m:03037},
pages = {355--380},
title = {{Was Sierpi\'nski right? I}},
volume = {62},
year = {1988},
},
@incollection{GuSh:277,
author = {Gurevich, Yuri and Shelah, Saharon},
booktitle = {Logic at Botik '89 (Pereslavl-Zalesskiy, 1989)},
ams-subject = {(68Q15)},
fromwhere = {1},
review = {MR 91a:68091},
pages = {108--118},
publisher = {Springer, Berlin-New York},
series = {Lecture Notes in Comput. Sci},
title = {{Nearly linear time}},
volume = {363},
year = {1989},
},
@incollection{CCShSW:278,
author = {Chatzidakis, Z. and Cherlin, G. and Shelah, Saharon and Srour,
G. and Wood, C. },
booktitle = {Classification theory (Chicago, IL, 1985)},
ams-subject = {(03C60)},
fromwhere = {1,1,IL,3,1},
review = {MR 91f:03074},
note = {Proceedings of the USA--Israel Conference on Classification
Theory, Chicago, December 1985; ed. Baldwin, J.T.},
pages = {72--88},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{Orthogonality of types in separably closed fields}},
volume = {1292},
year = {1987},
},
@article{ShSt:279,
author = {Shelah, Saharon and Stanley, Lee},
ams-subject = {(03E05)},
fromwhere = {IL,1},
journal = {Proceedings of the American Mathematical Society},
review = {MR 90e:03060},
pages = {887--897},
title = {{Weakly compact cardinals and nonspecial Aronszajn trees}},
volume = {104},
year = {1988},
},
@article{Sh:280,
author = {Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {IL},
journal = {The Journal of Symbolic Logic},
review = {MR 91b:03083},
pages = {21--31},
title = {{Strong negative partition above the continuum}},
volume = {55},
year = {1990},
},
@article{DzSh:281,
author = {Drezner, Zvi and Shelah, Saharon},
ams-subject = {(90B10)},
fromwhere = {1,IL},
journal = {Mathematics of Operations Research},
review = {MR 88e:90030},
pages = {255--261},
title = {{On the complexity of the Elzinga-Hearn algorithm for the
$1$-center problem}},
volume = {12},
year = {1987},
},
@article{Sh:282,
author = {Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 90c:03040},
pages = {213--256},
title = {{Successors of singulars, cofinalities of reduced products of
cardinals and productivity of chain conditions}},
volume = {62},
year = {1988},
},
@incollection{Sh:282a,
author = {Shelah, Saharon},
booktitle = {Cardinal Arithmetic},
fromwhere = {IL},
nt = {General Editors: Dov M. Gabbai, Angus Macintyre, Dana Scott},
publisher = {Oxford University Press},
series = {Oxford Logic Guides},
title = {{Colorings}},
volume = {29},
year = {1994},
},
@article{Sh:283,
author = {Shelah, Saharon},
ams-subject = {(20K10)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 90g:20079},
pages = {146--166},
title = {{On reconstructing separable reduced $p$-groups with a given
socle}},
volume = {60},
year = {1987},
},
@article{Sh:284,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
note = {See [Sh:284a] [Sh:284b] [Sh:284c] [Sh:284d] below},
title = {{}},
},
@article{Sh:284a,
author = {Shelah, Saharon},
ams-subject = {(03F25)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 90b:03088},
pages = {335--352},
title = {{Notes on monadic logic. Part A. Monadic theory of the real
line}},
volume = {63},
year = {1988},
},
@article{Sh:284b,
author = {Shelah, Saharon},
ams-subject = {(03C85)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 91m:03041},
pages = {94--116},
title = {{Notes on monadic logic. B. Complexity of linear orders in
ZFC}},
volume = {69},
year = {1990},
},
@article{Sh:284c,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 92e:03046},
pages = {353--364},
title = {{More on monadic logic. Part C. Monadically interpreting in
stable unsuperstable ${\scr T}$ and the monadic theory of ${}^
\omega\lambda$}},
volume = {70},
year = {1990},
},
@article{Sh:284d,
author = {Shelah, Saharon},
ams-subject = {(03C85)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 91c:03035},
pages = {302--306},
title = {{More on monadic logic. D. A note on addition of theories}},
volume = {68},
year = {1989},
},
@article{MaSh:285,
author = {Makkai, Michael and Shelah, Saharon},
ams-subject = {(03C75)},
fromwhere = {IL,3},
journal = {Annals of Pure and Applied Logic},
review = {MR 92a:03054},
pages = {41--97},
title = {{Categoricity of theories in $L_ {\kappa\omega},$ with $\kappa$
a compact cardinal}},
volume = {47},
year = {1990},
},
@article{JdSh:286,
author = {Ihoda (Haim Judah), Jaime and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {1,IL},
journal = {Proceedings of the American Mathematical Society},
review = {MR 89a:03091},
pages = {681--683},
title = {{$Q$-sets do not necessarily have strong measure zero}},
volume = {102},
year = {1988},
},
@article{BsSh:287,
author = {Blass, Andreas and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {1,IL},
journal = {Notre Dame Journal of Formal Logic},
review = {MR 90m:03087},
pages = {530--538},
title = {{Near coherence of filters. III. A simplified consistency
proof}},
volume = {30},
year = {1989},
},
@incollection{Sh:288,
author = {Shelah, Saharon},
booktitle = {Proceedings of the Conference on Set Theory and its
Applications in honor of A.Hajnal and V.T.Sos, Budapest, 1/91},
fromwhere = {IL},
note = { arxiv:math.LO/9201244 },
pages = {637--668},
series = {Colloquia Mathematica Societatis Janos Bolyai. Sets, Graphs,
and Numbers},
title = {{Strong Partition Relations Below the Power Set: Consistency,
Was Sierpi\'nski Right, II?}},
volume = {60},
year = {1991},
abstract = {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 Erd\H os 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 $\lambda$.},
},
@incollection{Sh:289,
author = {Shelah, Saharon},
booktitle = {Set theory and its applications (Toronto, ON, 1987)},
ams-subject = {(03E35)},
fromwhere = {IL},
review = {MR 91a:03102},
note = {ed. Steprans, J. and Watson, S.},
pages = {167--193},
publisher = {Springer, Berlin-New York},
series = {Lecture Notes in Mathematics},
title = {{Consistency of positive partition theorems for graphs and
models}},
volume = {1401},
year = {1989},
},
@article{BiSh:290,
author = {Biro, B. and Shelah, Saharon},
ams-subject = {(03G15)},
fromwhere = {H,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 89k:03084},
pages = {846--853},
title = {{Isomorphic but not lower base-isomorphic cylindric set
algebras}},
volume = {53},
year = {1988},
},
@article{MNSh:291,
author = {Mekler, Alan H. and Nelson, E. and Shelah, Saharon},
ams-subject = {(03B25)},
fromwhere = {1,?,IL},
journal = {Proceedings of the London Mathematical Society},
review = {MR 93m:03018},
note = { arxiv:math.LO/9301203 },
pages = {225-256},
title = {{A variety with solvable, but not uniformly solvable, word
problem}},
volume = {66},
year = {1993},
abstract = {In the literature two notions of the word problem for
a variety occur. A variety has a {\em decidable word problem} if every
finitely presented algebra in the variety has a decidable word problem.
It has a {\em 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.},
},
@article{JdSh:292,
author = {Ihoda (Haim Judah), Jaime and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {RCH,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 90h:03035},
pages = {1188--1207},
title = {{Souslin forcing}},
volume = {53},
year = {1988},
},
@article{ShSt:293,
author = {Shelah, Saharon and Stanley, Lee},
fromwhere = {IL,1},
journal = {Israel Journal of Mathematics},
pages = {97--110},
title = {{More consistency results in partition calculus}},
volume = {81},
year = {1993},
},
@incollection{ShSt:294,
author = {Shelah, Saharon and Stanley, Lee},
booktitle = {Set Theory of the Continuum},
fromwhere = {IL,1},
note = {ed. Judah, H., Just, W. and Woodin, H.. arxiv:math.LO/9201249
},
pages = {407--416},
publisher = {Springer Verlag},
series = {Mathematical Sciences Research Institute Publications},
title = {{Coding and reshaping when there are no sharps}},
volume = {26},
year = {1992},
abstract = {Assuming $0^\sharp$ does not exist, $\kappa$ is
an uncountable cardinal and for all cardinals $\lambda$ with $\kappa
\leq \lambda \lt\kappa^{+\omega},\ 2^\lambda = \lambda^+$, we present a
\lq\lq 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, \lq\lq reshapes the
interval $[\kappa,\ \kappa^+)$, i.e., for all $\kappa \lt \delta
\lt \kappa^+, \ \kappa = (card\ \delta)^{L[W \cap \delta]}$. 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.},
},
@article{Sh:295,
author = {Shelah, Saharon},
fromwhere = {A, IL},
journal = {in preparation},
title = {{Projective measurability does not imply projective Baire
property}},
abstract = {From an inaccessible cardinal we construct a model where all
projective sets are measurable, but there is a projective set without
the Baire property. We use the coding from ShSt:340, and a new
iteration alng a partial order.},
},
@article{ShSr:296,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
ams-subject = {(54C05)},
fromwhere = {IL,3},
journal = {Fundamenta Mathematicae},
review = {MR 90h:54015},
pages = {135--141},
title = {{Nontrivial homeomorphisms of $\beta {\bf N}\setminus {\bf N}$
without the continuum hypothesis}},
volume = {132},
year = {1989},
},
@article{HdSh:297,
author = {Hodkinson, Ion and Shelah, Saharon},
fromwhere = {?,IL},
journal = {Proceedings of the London Mathematical Society},
pages = {449--492},
title = {{A construction of many uncountable rings using SFP domains and
Aronszajn trees}},
volume = {67},
year = {1993},
},
@article{EkSh:298,
author = {Eklof, Paul C. and Shelah, Saharon},
ams-subject = {(13D05)},
fromwhere = {1,IL},
journal = {Rendiconti del Seminario Matematico dell'Universita di
Padova},
review = {MR 89c:13017},
pages = {279--284},
title = {{A calculation of injective dimension over valuation domains}},
volume = {78},
year = {1987},
},
@incollection{Sh:299,
author = {Shelah, Saharon},
booktitle = {Proceedings of the International Congress of Mathematicians
(Berkeley, Calif., 1986)},
ams-subject = {(03-02)},
fromwhere = {IL},
review = {MR 89e:03006},
note = {ed. Gleason, A.M.},
pages = {154--162},
publisher = {Amer. Math. Soc., Providence, RI},
title = {{Taxonomy of universal and other classes}},
volume = {1},
year = {1987},
},
@incollection{Sh:300,
author = {Shelah, Saharon},
booktitle = {Classification theory (Chicago, IL, 1985)},
ams-subject = {(03C52)},
fromwhere = {IL},
review = {MR 91k:03088},
note = {Proceedings of the USA--Israel Conference on Classification
Theory, Chicago, December 1985; ed. Baldwin, J.T.},
pages = {264--418},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{Universal classes}},
volume = {1292},
year = {1987},
},
@inbook{Sh:300a,
author = {Shelah, Saharon},
booktitle = {Classification Theory for Abstract Elementary Classes II},
fromwhere = {IL},
note = {Chapter V (A), in series Studies in Logic, vol. 20,
College Publications},
title = {{Stability theory for a model}},
abstract = { We deal with a universal class of models $K$, (i.e. a
structure $\in K$ iff any finitely generated substructure $\in K$). We
prove that either in $K$ there are long orders (hence many complicated
models) or $K$, under suitable order $\le_{\frak s}$ is an a.e.c. with
some stability theory built in. For this we deal with the existence of
indiscernible sets and (introduce and prove existence) of convergence
sets. Moreover, improve the results on the existence of indiscernible
sets such that for some first order theories, we get strong existence
results for set of elements, whereas possibly for some sets of
$n$-tuples this fails. In later sub-chapters we continue going up in
a spiral - getting either non-structure or showing closed affinity to
stable, but the dividing lines are in general missing for first order
classes.},
},
@inbook{Sh:300b,
author = {Shelah, Saharon},
booktitle = {Classification Theory for Abstract Elementary Classes II},
fromwhere = {IL},
note = {Chapter V (B)},
title = {{Universal Classes: Axiomatic Framework [Sh:h]}},
},
@inbook{Sh:300c,
author = {Shelah, Saharon},
booktitle = {Classification Theory for Abstract Elementary Classes II},
fromwhere = {IL},
note = {Chapter V (C)},
title = {{Universal Classes: A frame is not smooth or not
$\chi$-based}},
},
@inbook{Sh:300d,
author = {Shelah, Saharon},
booktitle = {Classification Theory for Abstract Elementary Classes II},
fromwhere = {IL},
note = {Chapter V (D)},
title = {{Universal Classes: Non-Forking and Prime Modes}},
},
@inbook{Sh:300e,
author = {Shelah, Saharon},
booktitle = {Classification Theory for Abstract Elementary Classes II},
fromwhere = {IL},
note = {Chapter V (E)},
title = {{Universal Classes: Types of finite sequences}},
},
@inbook{Sh:300f,
author = {Shelah, Saharon},
booktitle = {Classification Theory for Abstract Elementary Classes II},
fromwhere = {IL},
note = {Chapter V (F)},
title = {{Universal Classes: the heart of the matter}},
},
@inbook{Sh:300g,
author = {Shelah, Saharon},
booktitle = {Classification Theory for Abstract Elementary Classes II},
fromwhere = {IL},
note = {Chapter V (G)},
title = {{Universal Classes: Changing the framework}},
},
@inbook{Sh:300x,
author = {Shelah, Saharon},
booktitle = {Universal Classes [Sh:h]},
fromwhere = {IL},
title = {{Bibliography}},
},
@inbook{Sh:300y,
author = {Shelah, Saharon},
booktitle = {Universal Classes [Sh:h]},
fromwhere = {IL},
title = {{Glossary}},
},
@inbook{Sh:300z,
author = {Shelah, Saharon},
booktitle = {Universal Classes [Sh:h]},
fromwhere = {IL},
title = {{Annotated Contents}},
},
@article{HoSh:301,
author = {Hodges, Wilfrid and Shelah, Saharon},
fromwhere = {4,IL},
journal = {Preprint},
note = { arxiv:math.LO/0102060 },
title = {{Naturality and Definability II}},
abstract = {In two papers we noted that in common practice many
algebraic constructions are defined only `up to isomorphism' rather
than explicitly. We mentioned some questions raised by this fact, and
we gave some partial answers. The present paper provides much
fuller answers, though some questions remain open. Our main result
says that there is a transitive model of Zermelo-Fraenkel set theory
with choice (ZFC) in which every fully definable construction is
`weakly natural' (a weakening of the notion of a natural
transformation). A corollary is that there are models of ZFC in which
some well-known constructions, such as algebraic closure of fields, are
not explicitly definable. We also show that there is no model of ZFC
in which the explicitly definable constructions are precisely
the natural ones.},
},
@article{GrSh:302,
author = {Grossberg, Rami and Shelah, Saharon},
ams-subject = {(20K35)},
fromwhere = {1,1},
journal = {Journal of Algebra},
review = {MR 90d:20101},
note = {See also [GrSh:302a] below},
pages = {117--128},
title = {{On the structure of ${\rm Ext}_ p(G,{\Bbb Z})$}},
volume = {121},
year = {1989},
},
@article{GrSh:302a,
author = {Grossberg, Rami and Shelah, Saharon},
trueauthor = {Grossberg, Rami and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Mathematica Japonica},
note = { arxiv:math.LO/9911225 },
pages = {189--197},
title = {{On cardinalities in quotients of inverse limits of groups}},
volume = {47},
year = {1998},
abstract = {Let $\lambda$ be $\aleph_0$ or a strong limit of cofinality
$\aleph_0$. Suppose that $\langle G_m,\pi_{m,n}\;:\;m\leq
n\lt\omega\rangle$ and $\langle H_m,\pi^t_{m,n}\;:\;m\leq
n\lt\omega\rangle$ are projective systems of groups of cardinality less
than $\lambda$ and suppose that for every $n\lt \omega$ there is a
homorphism $\sigma:H_n\rightarrow G_n$ such that all the
diagrams commute. If for every $\mu\lt \lambda$ there exists
$\langle f_i\in G_{\omega} \;:\;i\lt \mu\rangle$ such that
$i\neq j\Longrightarrow f_if_j^{-1}\not\in\sigma_{\omega}(H_{\omega})$
then there exists $\langle f_i\in G_{\omega} \;:\;i
\lt 2^{\lambda}\rangle$ such that $i\neq
j\Longrightarrow f_if_j^{-1}\not\in\sigma_{\omega}(H_{\omega})$.},
},
@article{KoSh:303,
author = {Komjath, Peter and Shelah, Saharon},
trueauthor = {Komj\'{a}th, P\'eter and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {H,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 89m:03044},
pages = {696--707},
title = {{Forcing constructions for uncountably chromatic graphs}},
volume = {53},
year = {1988},
},
@article{ShSp:304,
author = {Shelah, Saharon and Spencer, Joel},
ams-subject = {(05C80)},
fromwhere = {IL,1},
journal = {Journal of the American Mathematical Society},
review = {MR 89i:05249},
pages = {97--115},
title = {{Zero-one laws for sparse random graphs}},
volume = {1},
year = {1988},
},
@article{ShTh:305,
author = {Shelah, Saharon and Thomas, Simon},
ams-subject = {(03E05)},
fromwhere = {IL,1},
journal = {The Journal of Symbolic Logic},
review = {MR 90b:03065},
pages = {95--99},
title = {{Subgroups of small index in infinite symmetric groups. II}},
volume = {54},
year = {1989},
},
@article{MkSh:306,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(20K10)},
fromwhere = {3,IL},
journal = {Communications in Algebra},
review = {MR 91b:20074},
pages = {287--307},
title = {{Determining abelian $p$-groups from their $n$-socles}},
volume = {18},
year = {1990},
},
@article{BeSh:307,
author = {Buechler, Steven and Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL,1},
journal = {Annals of Pure and Applied Logic},
review = {MR 91f:03068},
pages = {277--308},
title = {{On the existence of regular types}},
volume = {45},
year = {1989},
},
@article{JdSh:308,
author = {Judah, Haim and Shelah, Saharon},
ams-subject = {(03E15)},
fromwhere = {IL,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 91g:03097},
pages = {909--927},
title = {{The Kunen-Miller chart (Lebesgue measure, the Baire property,
Laver reals and preservation theorems for forcing)}},
volume = {55},
year = {1990},
},
@article{Sh:309,
author = {Shelah, Saharon},
fromwhere = {IL},
note = {0812.0656. 0812.0656. arxiv:0812.0656 },
title = {{Black Boxes}},
abstract = {We shall deal comprehensively with Black Boxes, the
intention being that provably in ZFC we have a sequence of guesses of
extra structure on small subsets, the guesses are pairwise with
quite little interaction, are far but together are ``dense''. We first
deal with the simplest case, were the existence comes from winning a
game by just writing down the opponent's moves. We show how it help
when instead orders we have trees with boundedly many levels,
having freedom in the last. After this we quite systematically look
at existence of black boxes, and make connection to non-saturation
of natural ideals and diamonds on them. },
},
@article{GiSh:310,
author = {Gitik, Moti and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9605234 },
pages = {1527--1551},
title = {{Cardinal preserving ideals}},
volume = {64},
year = {1999},
abstract = {We give some general criteria, when
$\kappa$-complete forcing preserves largeness properties --
like $\kappa$-presaturation of normal ideals on $\lambda$ (even
when they concentrate on small cofinalities). Then we quite accurately
obtain the consistency strength ``$NS_\lambda$
is $\aleph_1$-preserving'', for $\lambda >\aleph_2$.},
},
@article{Sh:311,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
note = { arxiv:math.LO/0404221 },
title = {{A more general iterable condition ensuring $\aleph_1$ is not
collapsed}},
abstract = {In a self-contained way, we deal with revised countable
support iterated forcing for the reals. We improve theorems on
preservation of the property UP, weaker than semi proper, and we
hopefully improve the presentation. We continue \cite[Ch.X,XI]{Sh:b}
(or see \cite[Ch.X,XI]{Sh:f}), and Gitik and Shelah \cite{GiSh:191}
and \cite[Ch.XIII,XIV]{Sh:f} and particularly Ch.XV. Concerning ``no
new reals'' see lately Larson and Shelah \cite{LrSh:746}.
In particular, we fulfill some promises from \cite{Sh:f} and give
a more streamlined version. },
},
@article{Sh:312,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
note = { arxiv:math.LO/1102.5578v3 },
title = {{Existentially closed locally finite groups}},
volume = {Beyond First Order Model Theory},
abstract = {We investigate this class of groups originally called
ulf (universal locally finite groups) of cardinality $\lambda$. We
prove that for every locally finite group $G$ there is a
canonical existentially closed extention of the same cardinality,
unique up to isomorphism and increasing with $G$. Also we get, e.g.
existence of complete members (i.e. with no non-inner automorphisms) in
many cardinals (provably in ZFC). We also get a parallel to
stability theory in the sense of investigating definable types.},
},
@article{MkSh:313,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {3,IL},
journal = {Fundamenta Mathematicae},
review = {MR 90i:03055},
pages = {45--51},
title = {{Diamond and $\lambda$-systems}},
volume = {131},
year = {1988},
},
@article{MRSh:314,
author = {Mekler, Alan H. and Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Mekler, Alan H. and Ros{\l}anowski, Andrzej and Shelah,
Saharon},
fromwhere = {3,PL,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9806165 },
pages = {327--356},
title = {{On the $p$-rank of Ext}},
volume = {112},
year = {1999},
abstract = {Assume $V=L$ and $\lambda$ is regular smaller than the first
weakly compact cardinal. Under those circumstances and with arbitrary
requirements on the structure of $Ext(G,{\Bbb Z})$ (under well known
limitations), we construct an abelian group $G$ of cardinality
$\lambda$ such that for no $G'\subseteq G$, $|G'|\lt \lambda$ is $G/G'$
free and $Ext(G,{\Bbb Z})$ realizes our requirements.},
},
@article{ShSr:315,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
ams-subject = {(03E05)},
fromwhere = {1,3},
journal = {Proceedings of the American Mathematical Society},
review = {MR 89e:03080},
pages = {1220--1225},
title = {{PFA implies all automorphisms are trivial}},
volume = {104},
year = {1988},
},
@article{FuSh:316,
author = {Fuchs, Laszlo and Shelah, Saharon},
ams-subject = {(13L05)},
fromwhere = {1,IL},
journal = {Proceedings of the American Mathematical Society},
review = {MR 89e:13030},
pages = {25--30},
title = {{Kaplansky's problem on valuation rings}},
volume = {105},
year = {1989},
},
@article{BFSh:317,
author = {Becker, Thomas and Fuchs, Laszlo and Shelah, Saharon},
ams-subject = {(13C05)},
fromwhere = {1,1,IL},
journal = {Forum Mathematicum},
review = {MR 90a:13017},
pages = {53--68},
title = {{Whitehead modules over domains}},
volume = {1},
year = {1989},
},
@article{MMSh:318,
author = {Macpherson, Dugald and Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(03C55)},
fromwhere = {4,3,IL},
journal = {Mathematical Proceedings of the Cambridge Philosophical
Society},
review = {MR 92a:03049},
pages = {193--209},
title = {{The number of infinite substructures}},
volume = {109},
year = {1991},
},
@article{JdSh:319,
author = {Ihoda (Haim Judah), Jaime and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {IL,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 90f:03083},
pages = {78--94},
title = {{Martin's axioms, measurability and equiconsistency results}},
volume = {54},
year = {1989},
},
@article{JShS:320,
author = {Juhasz, Istvan and Shelah, Saharon and Soukup, Lajos},
trueauthor = {Juh\'asz, Istv\'an and Shelah, Saharon and Soukup, Lajos},
ams-subject = {(03E35)},
fromwhere = {H,IL,H},
journal = {Israel Journal of Mathematics},
review = {MR 89i:03097},
pages = {302--310},
title = {{More on countably compact, locally countable spaces}},
volume = {62},
year = {1988},
},
@article{JdSh:321,
author = {Ihoda (Haim Judah), Jaime and Shelah, Saharon},
ams-subject = {(03E15)},
fromwhere = {1,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 90f:03081},
pages = {207--223},
title = {{$\Delta^ 1_ 2$-sets of reals}},
volume = {42},
year = {1989},
},
@article{Sh:322,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
title = {{Classification over a predicate}},
abstract = {Alex adopt this. The first task is about
$P^-_(n)$-amalgamtion all are model with the same P or half of
them have P-part only, as in teh no two cardinal model case. This part
is like the schizophrenia of stablity: should be like [705, \S12], we
should analyse the definable/undefinable types},
},
@article{HaSh:323,
author = {Hart, Bradd and Shelah, Saharon},
ams-subject = {(03C35)},
fromwhere = {IL,1},
journal = {Israel Journal of Mathematics},
review = {MR 91m:03033},
note = { arxiv:math.LO/9201240 },
pages = {219--235},
title = {{Categoricity over $P$ for first order $T$ or categoricity for
$\phi\in{\rm L}_ {\omega_ 1\omega}$ can stop at $\aleph_ k$ while
holding for $\aleph_ 0,\cdots,\aleph_ {k-1}$}},
volume = {70},
year = {1990},
abstract = {Suppose $L$ is a relational language and $P\in L$ is a
unary predicate. If $M$ is an $L$-structure then $P(M)$ is
the $L$-structure formed as the substructure of $M$ with domain
$\{a: M\models P(a)\}$. Now suppose $T$ is a complete first order
theory in $L$ with infinite models. Following Hodges, we say that T
is relatively $\lambda$-categorical if whenever $M$, $N\models
T$, $P(M)=P(N)$, $|P(M)|=\lambda$ then there is an
isomorphism $i:M\rightarrow N$ which is the identity on $P(M)$. $T$
is relatively categorical if it is relatively $\lambda$-categorical
for every $\lambda$. The question arises whether the
relative $\lambda$-categoricity of $T$ for some $\lambda>|T|$ implies
that $T$ is relatively categorical. \endgraf 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 $n0$. 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.},
},
@article{MkSh:367,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(03E55)},
fromwhere = {3,IL},
journal = {Israel Journal of Mathematics},
review = {MR 91b:03090},
pages = {353--366},
title = {{The consistency strength of ``every stationary set
reflects''}},
volume = {67},
year = {1989},
},
@article{BJSh:368,
author = {Bartoszynski, Tomek and Judah, Haim and Shelah, Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Judah, Haim and Shelah, Saharon
},
fromwhere = {1,IL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9905122 },
pages = {401--423},
title = {{The Cicho\'n diagram}},
volume = {58},
year = {1993},
},
@article{GJSh:369,
author = {Goldstern, Martin and Judah, Haim and Shelah, Saharon},
ams-subject = {(54A25)},
fromwhere = {1,IL,IL},
journal = {Proceedings of the American Mathematical Society},
review = {MR 91g:54008},
pages = {1151--1159},
title = {{A regular topological space having no closed subsets of
cardinality $\aleph_ 2$}},
volume = {111},
year = {1991},
abstract = {We show in ZFC that there is a regular
(even zerodimensional) topological space of size $> \aleph_2$ in
which there are no closed sets of size $\aleph_2$. The proof starts by
noticing that if $\beta\omega$ does not work, then we can use a
$\diamondsuit$.},
},
@article{ShSo:370,
author = {Shelah, Saharon and Soukup, Lajos},
fromwhere = {Il,H},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9401210 },
pages = {349--371},
title = {{On the number of non-isomorphic subgraphs}},
volume = {86},
year = {1994},
abstract = {Let $\cal K$ be the family of graphs on $\omega_1$ without
cliques or independent subsets of size $\omega_1$. We prove that:
\endgraf 1) it is consistent with CH that every $G\in{\cal K}$
has $2^{\omega_1}$ many pairwise non-isomorphic subgraphs, \endgraf 2)
the following proposition holds in L: $(*)$ there is a $G\in{\cal K}$
such that for each partition $(A,B)$ of $\omega_1$ either $G\cong G[A]$
or $G\cong G[B]$, \endgraf 3) the failure of $(*)$ is consistent with
ZFC.},
},
@incollection{Sh:371,
author = {Shelah, Saharon},
booktitle = {Cardinal Arithmetic},
fromwhere = {IL},
nt = {General Editors: Dov M. Gabbai, Angus Macintyre, Dana Scott},
publisher = {Oxford University Press},
series = {Oxford Logic Guides},
title = {{Advanced: cofinalities of small reduced products}},
volume = {29},
year = {1994},
},
@article{JMSh:372,
author = {Judah, Haim and Miller, Arnold W. and Shelah, Saharon},
ams-subject = {(03E50)},
fromwhere = {IL,1,IL},
journal = {Archive for Mathematical Logic},
review = {MR 93e:03074},
pages = {145--161},
title = {{Sacks forcing, Laver forcing, and Martin's axiom}},
volume = {31},
year = {1992},
},
@article{JRSh:373,
author = {Judah, Haim and Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Judah, Haim and Ros{\l}anowski, Andrzej and Shelah,
Saharon},
fromwhere = {IL,PL,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9310224 },
pages = {23--42},
title = {{Examples for Souslin Forcing}},
volume = {144},
year = {1994},
abstract = {We give a model where there is a ccc Souslin forcing which
does not satisfy the Knaster condition. Next, we present a model
where there is a $\sigma$-linked not $\sigma$-centered Souslin
forcing such that all its small subsets are $\sigma$-centered but
Martin Axiom fails for this order. Furthermore, we construct a
totally nonhomogeneous Souslin forcing and we build a Souslin forcing
which is proper but not ccc that does not contain a perfect set
of mutually incompatible conditions. Finally we show that
ccc $\Sigma^1_2$-notions of forcing may not be indestructible ccc.},
},
@article{JdSh:374,
author = {Judah, Haim and Shelah, Saharon},
ams-subject = {(03E40)},
fromwhere = {1,IL},
journal = {Proceedings of the American Mathematical Society},
review = {MR 93k:03049},
pages = {267--273},
title = {{Adding dominating reals with the random algebra}},
volume = {119},
year = {1993},
},
@article{MkSh:375,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(03C80)},
fromwhere = {3,IL},
journal = {Annals of Mathematics},
review = {MR 94b:03068},
note = {Dedicated to the memory of Alan. arxiv:math.LO/9301204 },
pages = {221-248},
title = {{Some compact logics --- results in ZFC}},
volume = {137},
year = {1993},
abstract = {We show that if we enrich first order logic by
allowing quantification over isomorphisms between definable ordered
fields the resulting logic, $L(Q_{\rm Of})$, is fully compact. In
this logic, we can give standard compactness proofs of various results.
Next, we attempt to get compactness results for some other logics
without recourse to $\diamondsuit$, i.e., all our results are in ZFC.
We get the full result for the language where we quantify over
automorphisms (isomorphisms) of ordered fields in Theorem
6.4. Unfortunately we are not able to show that the language
with quantification over automorphisms of Boolean algebras is
compact, but will have to settle for a close relative of that logic.
This is theorem 5.1. In section 4 we prove we can construct models in
which all relevant automorphism are somewhat definable: 4.1, 4.8 for
BA, 4.13 for ordered fields. We also give a new proof of
the compactness of another logic -- the one which is obtained when
a quantifier $Q_{{\rm Brch}}$ is added to first order logic which
says that a level tree (definitions will be given later) has an
infinite branch. This logic was previously shown to be compact, but our
proof yields a somewhat stronger result and provides a nice
illustration of one of our methods.},
},
@article{ShSo:376,
author = {Shelah, Saharon and Soukup, Lajos},
fromwhere = {IL,H},
journal = {Periodica Hung. Mathematica},
pages = {155--163},
title = {{Some remarks on a question of Monk}},
volume = {30},
year = {1995},
},
@article{ShTV:377,
author = {Shelah, Saharon and Tuuri, Heikki and Vaananen, Jouko},
trueauthor = {Shelah, Saharon and Tuuri, Heikki and
V{\"{a}}{\"{a}}n{\"{a}}nen, Jouko},
fromwhere = {IL,SF,SF},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9301205 },
pages = {1402--1418},
title = {{On the number of automorphisms of uncountable models}},
volume = {58},
year = {1993},
abstract = {Let $s({\cal A})$ denote the number of automorphisms of a
model ${\cal A}$ of power $\omega_1$. We derive a necessary and
sufficient condition in terms of trees for the existence of an ${\cal
A}$ with $\omega_1 \lt s({\cal A}) \lt 2^{\omega_1}$. We study
the sufficiency of some conditions for $s({\cal A})=2^{\omega_1}$.
These conditions are analogous to conditions studied by D.Kueker
in connection with countable models.},
},
@article{JeSh:378,
author = {Jech, Thomas and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {1,IL},
journal = {Israel Journal of Mathematics},
review = {MR 90m:03088},
note = { arxiv:math.LO/9201239 },
pages = {376--380},
title = {{A note on canonical functions}},
volume = {68},
year = {1989},
abstract = {We construct a generic extension in which the $\aleph_2$
nd canonical function on $\aleph_1$ exists.},
},
@article{EkSh:379,
author = {Eklof, Paul C. and Shelah, Saharon},
ams-subject = {(03E35)},
fromwhere = {1,IL},
journal = {Journal of Algebra},
review = {MR 92h:03077},
pages = {492--510},
title = {{On Whitehead modules}},
volume = {142},
year = {1991},
},
@incollection{Sh:380,
author = {Shelah, Saharon},
booktitle = {Cardinal Arithmetic},
fromwhere = {IL},
nt = {General Editors: Dov M. Gabbai, Angus Macintyre, Dana Scott},
publisher = {Oxford University Press},
series = {Oxford Logic Guides},
title = {{Jonsson Algebras in an inaccessible $\lambda $ not $\lambda
$-Mahlo}},
volume = {29},
year = {1994},
},
@article{Sh:381,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 93e:03048},
pages = {91--127},
title = {{Kaplansky test problem for $R$-modules}},
volume = {74},
year = {1991},
},
@article{ShSp:382,
author = {Shelah, Saharon and Spencer, Joel},
fromwhere = {IL,1},
journal = {Random Structures and Algorithms},
note = {Proceedings of the Random Graph Conference, Pozna\'n.
arxiv:math.LO/9401211 },
pages = {191--204},
title = {{Can You Feel the Double Jump?}},
volume = {5},
year = {1994},
abstract = {Paul Erdos and Alfred Renyi considered the evolution of
the random graph $G(n,p)$ as $p$ ``evolved'' from $0$ to $1$. At
$p=1/n$ a sudden and dramatic change takes place in $G$. When $p=c/n$
with $c<1$ the random $G$ consists of small components, the largest
of size $\Theta(\log n)$. But by $p=c/n$ with $c>1$ many of
the components have ``congealed'' into a ``giant component'' of
size $\Theta (n)$. Erdos and Renyi called this the {\em 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. },
},
@article{JeSh:383,
author = {Jech, Thomas and Shelah, Saharon},
fromwhere = {1,IL},
journal = {American Journal of Mathematics},
note = { arxiv:math.LO/9204218 },
pages = {435-455},
title = {{Full reflection of stationary sets at regular cardinals}},
volume = {115},
year = {1993},
abstract = {A stationary subset $S$ of a regular uncountable
cardinal $\kappa$ reflects fully at regular cardinals if for every
stationary set $T\subseteq\kappa$ of higher order consisting of
regular cardinals there exists an $\alpha\in T$ such that $S\cap\alpha
$ is a stationary subset of $\alpha$. We prove that the Axiom of
Full Reflection which states that every stationary set reflects fully
at regular cardinals, together with the existence of
$n$-Mahlo cardinals is equiconsistent with the existence
of $\Pi^1_n$-indescribable cardinals. We also state the
appropriate generalization for greatly Mahlo cardinals.},
},
@incollection{Sh:384,
author = {Shelah, Saharon},
booktitle = {Non structure theory, Ch X},
fromwhere = {IL},
title = {{Compact logics in ZFC: Constructing complete embeddings of
atomless Boolean rings}},
abstract = {A debt.high , together with 482. Now [F1649] for casanovas
call for proving: there for enough cardinals $yk$, preferably
$uk=yk^yk$, we have more then compactness and LST, but together $n$
cardianlity $yk$},
},
@article{JeSh:385,
author = {Jech, Thomas and Shelah, Saharon},
ams-subject = {(03E10)},
fromwhere = {1,IL},
journal = {Proceedings of the American Mathematical Society},
review = {MR 91j:03066},
note = { arxiv:math.LO/9201247 },
pages = {1117--1124},
title = {{On a conjecture of Tarski on products of cardinals}},
volume = {112},
year = {1991},
abstract = {We look at an old conjecture of A. Tarski on
cardinal arithmetic and show that if a counterexample exists, then
there exists one of length $\omega_1 + \omega$.},
},
@incollection{Sh:386,
author = {Shelah, Saharon},
booktitle = {Cardinal Arithmetic},
fromwhere = {IL},
nt = {General Editors: Dov M. Gabbai, Angus Macintyre, Dana Scott},
publisher = {Oxford University Press},
series = {Oxford Logic Guides},
title = {{Bounding $pp(\mu )$ when $cf(\mu ) > \mu > \aleph _0$ using
ranks and normal ideals}},
volume = {29},
year = {1994},
},
@article{JeSh:387,
author = {Jech, Thomas and Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {1,IL},
journal = {The Journal of Symbolic Logic},
review = {MR 91i:03096},
note = { arxiv:math.LO/9201242 },
pages = {822--830},
title = {{Full reflection of stationary sets below $\aleph_ \omega$}},
volume = {55},
year = {1990},
abstract = {It is consistent that for every $n \ge 2$, every stationary
subset of $\omega_n$ consisting of ordinals of cofinality $\omega_k$
where $k = 0$ or $k \le n -3$ reflects fully in the set of ordinals of
cofinality $\omega_{n-1}$. We also show that this result is best
possible.},
},
@article{GoSh:388,
author = {Goldstern, Martin and Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {1,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 91m:03050},
pages = {121--142},
title = {{Ramsey ultrafilters and the reaping number---${\rm Con}({\germ
r}<{\germ u})$}},
volume = {49},
year = {1990},
abstract = {We show that the reaping number $r$ is consistenly
smaller than the smallest base of an ultrafilter. We use a
forcing notion $P_U$ that destroys a selected ultrafilter $U$ and
all ultrafilters below it, but preserves all Ramsey ultrafilters that
are not below $U$ in the Rudin-Keisler order.},
},
@article{ShSo:389,
author = {Shelah, Saharon and Soukup, Lajos},
fromwhere = {IL,H},
journal = {Journal of the London Mathematical Society},
pages = {193--203},
title = {{The Existence of large $\omega_1$-homogeneous but not
$\omega$-homogeneous permutation groups is consistent with ZFC + GCH}},
volume = {48},
year = {1993},
},
@article{KaSh:390,
author = {Kanamori, Akihiro and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.LO/9401212 },
pages = {1963--1979},
title = {{Complete quotient Boolean Algebras}},
volume = {347},
year = {1995},
abstract = {For $I$ a proper, countably complete ideal on ${\cal P}(X)$
for some set $X$, can the quotient Boolean algebra ${\cal P}(X)/I$
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 ${\cal P}(\kappa )$ containing
all singletons. In this paper we provide consequences from
and consistency results about completeness.},
},
@article{HHLSh:391,
author = {Hodges, Wilfrid and Hodkinson, Ion and Lascar, Daniel and
Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {4,?,F,IL},
journal = {Journal of the London Mathematical Society},
review = {MR 94d:03063},
pages = {204--218},
title = {{The small index property for $\omega $-stable,
$\omega$-categorical structures and the random graph}},
volume = {48},
year = {1993},
},
@article{JeSh:392,
author = {Jech, Thomas and Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {1,IL},
journal = {Journal of the American Mathematical Society},
review = {MR 93a:03049},
note = { arxiv:math.LO/9201248 },
pages = {647--656},
title = {{A partition theorem for pairs of finite sets}},
volume = {4},
year = {1991},
abstract = {Every partition of $[[\omega_1]^{\lt\omega}]^2$
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.},
},
@article{BlSh:393,
author = {Baldwin, John T. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9502231 },
pages = {246--265},
title = {{Abstract classes with few models have `homogeneous-universal'
models}},
volume = {60},
year = {1995},
abstract = {This paper is concerned with a class {\bf 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 {\bf K}. In the
original constructions of Jonsson and Fraisse, {\bf 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 {\bf K} obeying certain
minimal restrictions satisfies a fundamental dichotomy: For arbitrarily
large $\lambda$, either {\bf K} has the maximal number of models in
power $\lambda$ or {\bf 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 {\bf K}-homogeneous-universal
model in the more normal sense.},
},
@article{Sh:394,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9809197 },
pages = {261--294},
title = {{Categoricity for abstract classes with amalgamation}},
volume = {98},
year = {1999},
abstract = {Let ${\frak K}$ be an abstract elementary class
with amalgamation, and Lowenheim Skolem number LS$({\frak K})$. We
prove that for a suitable Hanf number $\chi_0$ if $\chi_0 <
\lambda_0\le \lambda_1$, and ${\frak K}$ is categorical in
$\lambda^+_1$ then it is categorical in $\lambda_0$.},
},
@incollection{Sh:395,
author = {Shelah, Saharon},
booktitle = {Open problems in topology},
fromwhere = {IL},
note = {ed. van Mill, J. and Reed, G.M.},
pages = {217--218},
publisher = {Elsvier Science Publishers, B.V. North Holland},
title = {{$\germ{d} \leq \germ{i} ( = Min \{|P| : P \subseteq [ \omega
]^\omega$ maximal independent $\}$); Appendix to J.E.~Vaughan, ``Small
Uncountable Cardinals in Topology''}},
year = {1990},
},
@article{FShZ:396,
author = {Frankiewicz, Ryszard and Shelah, Saharon and Zbierski, Pawel},
trueauthor = {Frankiewicz, Ryszard and Shelah, Saharon and Zbierski,
Pawe{\l}},
fromwhere = {PL,IL,PL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9303207 },
pages = {1171--1176},
title = {{On closed $P$-sets with ccc in the space $\omega ^*$}},
volume = {58},
year = {1993},
abstract = {It is proved that -- consistently -- there can be no ccc
closed $P$-sets in the remainder space $\omega^*$.},
},
@article{Sh:397,
author = {Shelah, Saharon},
ams-subject = {(06E05)},
fromwhere = {IL},
journal = {Mathematica Japonica},
review = {MR 93e:06011},
note = { arxiv:math.LO/9201250 },
pages = {385--400},
title = {{Factor = quotient, uncountable Boolean algebras, number of
endomorphism and width}},
volume = {37},
year = {1992},
abstract = {We prove that assuming suitable cardinal arithmetic, if $B$
is a Boolean algebra every homomorphic image of which is isomorphic to
a factor, then $B$ has locally small density. We also prove that for an
(infinite) Boolean algebra $B$, the number of subalgebras is not
smaller than the number of endomorphisms, and other related
inequalities. Lastly we deal with the obtainment of the supremum of the
cardinalities of sets of pairwise incomparable elements of a Boolean
algebra.},
},
@article{MkSh:398,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {3,IL},
journal = {Canadian Journal of Mathematics. Journal Canadien de
Mathematiques},
review = {MR 94d:03102},
note = { arxiv:math.LO/9308210 },
pages = {209-215},
title = {{The canary tree}},
volume = {36},
year = {1993},
abstract = {A {\em 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.},
},
@article{GJSh:399,
author = {Goldstern, Martin and Judah, Haim and Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {1,IL,IL},
journal = {Proceedings of the American Mathematical Society},
review = {MR 91g:03093},
pages = {1095--1104},
title = {{Saturated families}},
volume = {111},
year = {1991},
},
@incollection{Sh:400,
author = {Shelah, Saharon},
booktitle = {Cardinal Arithmetic},
fromwhere = {IL},
note = {Note: See also [Sh400a] below},
nt = {General Editors: Dov M. Gabbai, Angus Macintyre, Dana Scott},
publisher = {Oxford University Press},
series = {Oxford Logic Guides},
title = {{Cardinal Arithmetic}},
volume = {29},
year = {1994},
},
@article{Sh:400a,
author = {Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {IL},
journal = {American Mathematical Society. Bulletin. New Series},
review = {MR 92h:03071},
note = { arxiv:math.LO/9201251 },
pages = {197--210},
title = {{Cardinal arithmetic for skeptics}},
volume = {26},
year = {1992},
abstract = {We pressent a survey of some results of the pcf-theory
and their applications to cardinal arithmetic. We review basics
notions (in \S1), briefly look at history in \S2 (and some personal
history in \S3). We present main results on pcf in \S5 and
describe applications to cardinal arithmetic in \S6. The limitations
on independence proofs are discussed in \S7, and in \S8 we discuss
the status of two axioms that arise in the new setting. Applications
to other areas are found in \S9.},
},
@article{Sh:401,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9609215 },
pages = {61--111},
title = {{Characterizing an $\aleph_\epsilon $-saturated model of
superstable NDOP theories by its $\Bbb
L_{\infty,\aleph_\epsilon}$-theory}},
volume = {140},
year = {2004},
abstract = {After the main gap theorem was proved (see \cite{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
${\frak C}$ 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
$M\equiv_{{\Cal L}_{\infty,\aleph_\epsilon}(d.q.)}N \Leftrightarrow M
\cong N$ for $\aleph_\epsilon$-saturated models of a superstable
countable (first order) theory $T$ without dop. },
},
@article{Sh:402,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Mathematica Japonica},
note = { arxiv:math.LO/9809198 },
pages = {121--130},
title = {{Borel Whitehead groups}},
volume = {50},
year = {1999},
abstract = {We investigate the Whiteheadness of Borel abelian
groups ($\aleph_1$-free, wlog, as otherwise this is trivial). We show
that CH (and even WCH) implies any such abelian group is free, and
always $\aleph_2$-free.},
},
@article{AbSh:403,
author = {Abraham, Uri and Shelah, Saharon},
ams-subject = {(03E15)},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 94b:03085},
note = { arxiv:math.LO/9812115 },
pages = {1-32},
title = {{A $\Delta ^2_2$ well-order of the reals and incompactness of
$L(Q^{MM})$}},
volume = {59},
year = {1993},
abstract = {A forcing poset of size $2^{2^{\aleph_1}}$ which adds no
new reals is described and shown to provide a $\Delta^2_2$
definable well-order of the reals (in fact, any given relation of the
reals may be so encoded in some generic extension). The encoding of
this well-order is obtained by playing with products of Aronszajn
trees: Some products are special while other are Suslin trees. The
paper also deals with the Magidor-Malitz logic: it is consistent that
this logic is highly non compact.},
},
@article{GvSh:404,
author = {Givant, Steven and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9401213 },
pages = {27--51},
title = {{Universal theories categorical in power and $\kappa$-generated
models}},
volume = {69},
year = {1994},
abstract = {We investigate a notion called {\it 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 $T$.},
},
@article{Sh:405,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9304207 },
pages = {351--390},
title = {{Vive la diff\'erence II. The Ax-Kochen isomorphism theorem}},
volume = {85},
year = {1994},
abstract = {We show in \S1 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 \S2 we give an
unrelated result on cuts in models of Peano arithmetic which answers
a question on the ideal structure of countable ultraproducts of ${\Bbb
Z}$. In \S1 we also answer a question of Keisler and Schmerl regarding
Scott complete ultrapowers of ${\Bbb R}$.},
},
@article{FrSh:406,
author = {Fremlin, David H. and Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9209218 },
pages = {435--455},
title = {{Pointwise compact and stable sets of measurable functions}},
volume = {58},
year = {1993},
abstract = {In a series of papers, M.Talagrand, the second author
and others investigated at length the properties and structure
of pointwise compact sets of measurable functions. A number
of problems, interesting in themselves and important for the theory
of Pettis integration, were solved subject to various special axioms.
It was left unclear just how far the special axioms were necessary. In
particular, several results depended on the fact that it is consistent
to suppose that every countable relatively pointwise compact set of
Lebesgue measurable functions is `stable' in Talagrand's sense; the
point being that stable sets are known to have a variety of properties
not shared by all pointwise compact sets. In the present paper we
present a model of set theory in which there is a countable relatively
pointwise compact set of Lebesgue measurable functions which is not
stable, and discuss the significance of this model in relation to the
original questions. A feature of our model which may be of independent
interest is the following: in it, there is a closed negligible set
$Q\subseteq [0,1]^2$ such that whenever $D\subseteq [0,1]$ has outer
measure 1 then the set $Q^{-1}[D]=\{x:(\exists y\in D)((x,y)\in Q)\}$
has inner measure 1.},
},
@unpublished{FrSh:406a,
author = {Fremlin, David H. and Shelah, Saharon},
fromwhere = {IL},
title = {{Postscript to Shelah \& Fremlin [FrSh:406]}},
},
@article{Sh:407,
author = {Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
review = {MR 93i:03066},
pages = {433--443},
title = {{${\rm CON}({\germ u}>{\germ i})$}},
volume = {31},
year = {1992},
},
@article{KPSh:408,
author = {Kojman, Menachem and Perles, M. A. and Shelah, Saharon},
ams-subject = {(52A27)},
fromwhere = {IL,IL,IL},
journal = {Israel Journal of Mathematics},
review = {MR 92e:52006},
pages = {313--342},
title = {{Sets in a Euclidean space which are not a countable union
of convex subsets}},
volume = {70},
year = {1990},
},
@article{KjSh:409,
author = {Kojman, Menachem and Shelah, Saharon},
ams-subject = {(03C55)},
fromwhere = {IL,IL},
journal = {Journal of Symbolic Logic},
review = {MR 94b:03064},
note = { arxiv:math.LO/9209201 },
pages = {875--891},
title = {{Non-existence of Universal Orders in Many Cardinals}},
volume = {57},
year = {1992},
abstract = {We give an example of a first order theory $T$
with countable $D(T)$ which cannot have a universal model at $\aleph_1$
without CH; we prove in $ZFC$ a covering theorem from the hypothesis of
the existence of a universal model for some theory; and we prove --
again in ZFC -- that for a large class of cardinals there is no
universal linear order (e.g. in every $\aleph_1\lt \lambda\lt
2^{\aleph_0}$). 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).},
},
@article{Sh:410,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0406550 },
pages = {399--428},
title = {{More on Cardinal Arithmetic}},
volume = {32},
year = {1993},
},
@article{LeSh:411,
author = {Lifsches, Shmuel and Shelah, Saharon},
ams-subject = {(03D15)},
fromwhere = {IL,IL},
journal = {Archive for Mathematical Logic},
review = {MR 93b:03064},
pages = {207--213},
title = {{The monadic theory of $(\omega_ 2,<)$ may be complicated}},
volume = {31},
year = {1992},
},
@article{GiSh:412,
author = {Gitik, Moti and Shelah, Saharon},
ams-subject = {(03E40)},
fromwhere = {IL,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 94c:03070},
pages = {219--238},
title = {{More on simple forcing notions and forcings with ideals}},
volume = {59},
year = {1993},
},
@article{Sh:413,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9809199 },
pages = {1--44},
title = {{More Jonsson Algebras}},
volume = {42},
year = {2003},
abstract = {We prove that on many inaccessible there is a
Jonsson algebra, so e.g. the first regular Jonsson cardinal $\lambda$
is $\lambda\times\omega$-Mahlo. We give further restrictions
on successor of singulars which are Jonsson cardinals. E.g. there is a
Jonsson algebra of cardinality $\beth^+_\omega$. Lastly, we
give further information on guessing of clubs.},
},
@article{KoSh:414,
author = {Komjath, Peter and Shelah, Saharon},
trueauthor = {Komj\'{a}th, P\'eter and Shelah, Saharon},
fromwhere = {IL},
journal = {Acta Mathematica Hungarica},
pages = {115--120},
title = {{A consistent partition theorem for infinite graphs}},
volume = {61},
year = {1993},
},
@article{KpSh:415,
author = {Koppelberg, Sabine and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Canadian Journal Of Mathematics. Journal Canadien de
Mathematiques},
note = { arxiv:math.LO/9404226 },
pages = {132--145},
title = {{Densities of ultraproducts of Boolean algebras}},
volume = {47},
year = {1995},
abstract = {We answer three problems by J. D. Monk on cardinal
invariants of Boolean algebras. Two of these are whether taking the
algebraic density $\pi(A)$ resp. the topological density $d(A)$ of a
Boolean algebra $A$ commutes with formation of ultraproducts; the third
one compares the number of endomorphisms and of ideals of a
Boolean algebra.},
},
@article{MShV:416,
author = {Mekler, Alan H. and Shelah, Saharon and Vaananen, Jouko},
trueauthor = {Mekler, Alan H. and Shelah, Saharon and
V{\"{a}}{\"{a}}n{\"{a}}nen, Jouko},
ams-subject = {(03C55)},
fromwhere = {3,IL,SF},
journal = {Transactions of the American Mathematical Society},
review = {MR 94a:03058},
note = { arxiv:math.LO/9305204 },
pages = {567--580},
title = {{The Ehrenfeucht-Fra{\"\i}ss{\'e}-game of length $\omega_1$}},
volume = {339},
year = {1993},
abstract = {Let $({\cal A})$ and $({\cal B})$ be two first order
structures of the same vocabulary. We shall consider the
Ehrenfeucht-Fra{\"\i}ss\'e-game of length $\omega_1$ of ${\cal A}$ and
${\cal B}$ which we denote by $G_{\omega_1}({\cal A},{\cal B})$. This
game is like the ordinary Ehrenfeucht-Fra{\"\i}ss\'e-game of
$L_{\omega\omega}$ except that there are $\omega_1$ moves. It is clear
that $G_{\omega_1}({\cal A},{\cal B})$ is determined if $\cal A$ and
${\cal B}$ are of cardinality $\leq\aleph_1$. We prove the following
results: \endgraf Theorem A: If V=L, then there are models $\cal A$
and $\cal B$ of cardinality $\aleph_2$ such that the game
$G_{\omega_1}({\cal A},{\cal B})$ is non-determined.
\endgraf Theorem B: If it is consistent that there is a measurable
cardinal, then it is consistent that $G_{\omega_1}({\cal A},{\cal B})$
is determined for all $\cal A$ and $\cal B$ of cardinality
$\le\aleph_2$. \endgraf Theorem C: For any $\kappa\geq\aleph_3$
there are $\cal A$ and $\cal B$ of cardinality $\kappa$ such that the
game $G_{\omega_1}({\cal A},{\cal B})$ is non-determined.},
},
@article{MShS:417,
author = {Mekler, Alan H. and Shelah, Saharon and Spinas, Otmar},
fromwhere = {3,IL,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9411234 },
pages = {1--8},
title = {{The essentially free spectrum of a variety}},
volume = {93},
year = {1996},
abstract = {We partially prove a conjecture from [MkSh:366] which
says that the spectrum of almost free, essentially free,
non-free algebras in a variety is either empty or consists of the
class of all successor cardinals.},
},
@article{MkSh:418,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(20K40)},
fromwhere = {3,IL},
journal = {Israel Journal of Mathematics},
review = {MR 94e:20074},
note = { arxiv:math.LO/9305205 },
pages = {161--178},
title = {{Every coseparable group may be free}},
volume = {81},
year = {1993},
abstract = {We show that if $2^{\aleph_0}$ Cohen reals are added to the
universe, then for every reduced non-free torsion-free abelian group
$A$ of cardinality less than the continuum, there is a prime $p$ so
that ${\rm Ext}_p(A, {\Bbb Z}) \neq 0$. 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.},
},
@article{ShSt:419,
author = {Shelah, Saharon and Stanley, Lee},
fromwhere = {IL,1},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9709228 },
pages = {259--271},
title = {{Filters, Cohen Sets and Consistent Extensions of
the Erd\H{o}s-Dushnik-Miller Theorem}},
volume = {65},
year = {2000},
abstract = {We present two different types of models where, for
certain singular cardinals $\lambda$ of uncountable
cofinality, $\lambda\rightarrow(\lambda,\omega+1)^2$, although
$\lambda$ is not a strong limit cardinal. We announce, here, and will
present in a subsequent paper, that, for example,
consistently, $\aleph_{\omega_1}\not\rightarrow
(\aleph_{\omega_1},\omega+1)^2$ and
consistently, $2^{\aleph_0}\not\rightarrow
(2^{\aleph_0},\omega+1)^2$.},
},
@incollection{Sh:420,
author = {Shelah, Saharon},
booktitle = {Finite and Infinite Combinatorics in Sets and Logic},
fromwhere = {IL},
note = {N.W. Sauer et al (eds.). arxiv:0708.1979 },
pages = {355-383},
publisher = {Kluwer Academic Publishers},
title = {{Advances in Cardinal Arithmetic}},
year = {1993},
},
@incollection{Sh:421,
author = {Shelah, Saharon},
fromwhere = {IL},
title = {{Kaplansky test problem for $R$-modules in ZFC}},
},
@article{EkSh:422,
author = {Eklof, Paul C. and Shelah, Saharon},
ams-subject = {(13C13)},
fromwhere = {1,IL},
journal = {Transactions of the American Mathematical Society},
review = {MR 94a:13007},
note = { arxiv:math.LO/9308211 },
pages = {337--351},
title = {{On a conjecture regarding nonstandard uniserial modules}},
volume = {340},
year = {1993},
abstract = {We consider the question of which valuation domains
(of cardinality $\aleph _1)$ have non-standard uniserial modules.
We show that a criterion conjectured by Osofsky is independent of ZFC +
GCH.},
},
@article{BnSh:423,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
note = {2013.12.03 ealier was a paper with Brelndle incorporated
into [BnSh:642] We intend to replace it with [F13]},
title = {{Compactness spectrum}},
},
@incollection{Sh:424,
author = {Shelah, Saharon},
booktitle = {Logic Colloquium'90. ASL Summer Meeting in Helsinki},
fromwhere = {IL},
note = { arxiv:math.LO/9308212 },
nt = {J.~Oikkonen, J.~V{\"{a}}{\"{a}}n{\"{a}}nen, eds},
pages = {281--289},
publisher = {Springer Verlag},
series = {Lecture Notes in Logic},
title = {{On $CH + 2^{\aleph_1}\rightarrow(\alpha)^2_2$ for
$\alpha<\omega_2$}},
volume = {2},
year = {1993},
abstract = {We prove the consistency of ``CH + $2^{\aleph_1}$
is arbitrarily large
+ $2^{\aleph_1}\not\rightarrow(\omega_1\times\omega)^2_2$''. If
fact, we can
get $2^{\aleph_1}\not\rightarrow[\omega_1\times\omega]^2_{\aleph_0}$.
In addition to this theorem, we give generalizations to
other cardinals.},
},
@article{ShSt:425,
author = {Shelah, Saharon and Stanley, Lee},
fromwhere = {IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9402214 },
pages = {36--57},
title = {{The combinatorics of combinatorial coding by a real}},
volume = {60},
year = {1995},
abstract = {We lay the combinatorial foundations for [ShSt:340]
by setting up and proving the essential properties of the
coding apparatus for singular cardinals. We also prove another
result concerning the coding apparatus for inaccessible cardinals.},
},
@article{EMSh:426,
author = {Eklof, Paul C. and Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(20K20)},
fromwhere = {1,1,IL},
journal = {Communications in Algebra},
review = {MR 93k:20078},
note = { arxiv:math.LO/9308213 },
pages = {343--353},
title = {{On coherent systems of projections for $\aleph_1$ separable
groups}},
volume = {21},
year = {1993},
abstract = {It is proved consistent with either CH or the negation of
CH that there is an $\aleph_1$-separable group of
cardinality $\aleph_1$ which does not have a coherent system of
projections. It had previously been shown that it is consistent with
$\neg$CH that every $\aleph_1$-separable group of cardinality
$\aleph_1$ does have a coherent system of projections.},
},
@article{ShSr:427,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
fromwhere = {IL,3},
journal = {Journal of the London Mathematical Society},
note = { arxiv:math.LO/9308214 },
pages = {569--580},
title = {{Somewhere trivial automorphisms}},
volume = {49},
year = {1994},
abstract = {It is shown to be consistent that there is a non-trivial
autohomeomorphism of $\beta{\bf N}$ while all such autohomeomorphisms
are trivial on some open set. The model used is one due to Velickovic
in which, coincidentally, Martin's Axiom also holds.},
},
@article{HShT:428,
author = {Hyttinen, Tapani and Shelah, Saharon and Tuuri, Heikki},
fromwhere = {SF,IL,SF},
journal = {Notre Dame Journal of Formal Logic},
pages = {157--168},
title = {{Remarks on Strong Nonstructure Theorems}},
volume = {34},
year = {1993},
},
@article{Sh:429,
author = {Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 92j:03032},
pages = {281--288},
title = {{Multi-dimensionality}},
volume = {74},
year = {1991},
},
@article{Sh:430,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9610226 },
pages = {61--114},
title = {{Further cardinal arithmetic}},
volume = {95},
year = {1996},
abstract = {We continue the investigations in the author's book on
cardinal arithmetic, assuming some knowledge of it. We deal with
the cofinality of $({\cal S}_{\le\aleph_0}(\kappa),\subseteq)$
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).},
},
@article{KoSh:431,
author = {Komjath, Peter and Shelah, Saharon},
trueauthor = {Komj\'{a}th, P\'eter and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Periodica Mathematica Hungarica},
pages = {39--42},
title = {{A note on a set-mapping problem of Hajnal and Mate}},
volume = {28},
year = {1994},
},
@article{ShSp:432,
author = {Shelah, Saharon and Spencer, Joel},
fromwhere = {IL,1},
journal = {Random Structures \& Algorithms},
note = { arxiv:math.LO/9401214 },
pages = {375--394},
title = {{Random Sparse Unary Predicates}},
volume = {5},
year = {1994},
abstract = {The main result is the following \endgraf Theorem: Let
$p=p(n)$ be such that $p(n)\in[0,1]$ for all $n$ and either $p(n)\ll
n^{-1}$ or for some positive integer $k$, $n^{-1/k}\ll p(n)\ll
n^{-1/(k+1)}$ or for all $\epsilon>0$, $n^{-\epsilon}\ll p(n)$ and
$n^{-\epsilon}\ll 1-p(n)$ or for some positive integer $k$,
$n^{-1/k}\ll 1-p(n)\ll n^{-1/(k+1)}$ or $1-p(n)\ll n^{-1}$. Then $p(n)$
satisfies the Zero-One Law for circular unary predicates. Inversely, if
$p(n)$ falls into none of the above categories then it does not satisfy
the Zero-One Law for circular unary predicates. },
},
@article{MgSh:433,
author = {Magidor, Menachem and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Mathematica Japonica},
note = { arxiv:math.LO/9805145 },
pages = {301--307},
title = {{Length of Boolean algebras and ultraproducts}},
volume = {48},
year = {1998},
abstract = {We prove the consistency with ZFC of ``the length of
an ultraproduct of Boolean algebras is smaller than the ultraproduct
of the lengths''. Similarly for some other cardinal invariants of
Boolean algebras. },
},
@article{BGJSh:434,
author = {Bartoszynski, Tomek and Goldstern, Martin and Judah, Haim and
Shelah, Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Goldstern, Martin and Judah,
Haim and Shelah, Saharon },
ams-subject = {(03E35)},
fromwhere = {1,IL,IL,IL},
journal = {Proceedings of the American Mathematical Society},
review = {MR 93d:03055},
note = { arxiv:math.LO/9301206 },
pages = {515--521},
title = {{All meager filters may be null}},
volume = {117},
year = {1993},
abstract = {We show that it is consistent with ZFC that all
filters which have the Baire property are Lebesgue measurable. We
also show that the existence of a Sierpinski set implies that
there exists a nonmeasurable filter which has the Baire property.},
},
@article{LuSh:435,
author = {Luczak, Tomasz and Shelah, Saharon},
trueauthor = {{\L}uczak, Tomasz and Shelah, Saharon},
fromwhere = {PL,IL},
journal = {Random Structures \& Algorithms},
note = { arxiv:math.LO/9501221 },
pages = {371--391},
title = {{Convergence in homogeneous random graphs}},
volume = {6},
year = {1995},
abstract = {For a sequence $\bar{p}=(p(1),p(2),\dots)$
let $G(n,\bar{p})$ denote the random graph with vertex
set $\{1,2,\dots,n\}$ in which two vertices $i$, $j$ are adjacent with
probability $p(|i-j|)$, independently for each pair. We study how the
convergence of probabilities of first order properties of
$G(n,\bar{p})$, can be affected by the behaviour of $\bar{p}$ and the
strength of the language we use.},
},
@article{BrSh:436,
author = {Bartoszynski, Tomek and Shelah, Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Shelah, Saharon },
ams-subject = {(03E35)},
fromwhere = {1,IL},
journal = {Archive for Mathematical Logic},
review = {MR 93e:03073},
note = { arxiv:math.LO/9904068 },
pages = {221--226},
title = {{Intersection of $<2^{\aleph_0}$ ultrafilters may have measure
zero}},
volume = {31},
year = {1992},
},
@article{BuSh:437,
author = {Burke, Max R. and Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9201252 },
pages = {289--296},
title = {{Linear liftings for non complete probability space}},
volume = {79},
year = {1992},
abstract = {We show that it is consistent with ZFC
that $L^\infty(Y,{\cal B},\nu)$ has no linear lifting for
many non-complete probability spaces $(Y,{\cal B},\nu)$, in particular
for $Y=[0,1]^A$, ${\cal B}=$ Borel subsets of $Y$, $\nu=$ usual Radon
measure on ${\cal B}$.},
},
@article{GJSh:438,
author = {Goldstern, Martin and Judah, Haim and Shelah, Saharon},
fromwhere = {IL,IL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9306214 },
pages = {1323--1341},
title = {{Strong measure zero sets without Cohen reals}},
volume = {58},
year = {1993},
abstract = {If ZFC is consistent, then each of the following
are consistent with ZFC + $2^{{\aleph_0}}=\aleph_2$: \endgraf 1.) $X
subseteq R$ is of strong measure zero iff $|X| \leq \aleph_1$ + there
is a generalized Sierpinski set. \endgraf 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 $\aleph_2$.},
},
@article{BrSh:439,
author = {Bartoszynski, Tomek and Shelah, Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {1,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 94b:03084},
note = { arxiv:math.LO/9905123 },
pages = {93--110},
title = {{Closed measure zero sets}},
volume = {58},
year = {1992},
},
@incollection{CKSh:440,
author = {Comfort, W. Wistar and Kato, Akio and Shelah, Saharon},
booktitle = {Proceedings of the Seventh Summer Conference at the
University of Wisconsin, Papers on General Topology and Applications},
fromwhere = {1, J, IL},
note = { arxiv:math.LO/9305206 },
pages = {70--80},
publisher = {New York Acad. Sci.},
series = {Annals New York Acad. Sciences},
title = {{Topological Partition Relations to the Form
$\omega^*\rightarrow(Y)^1_2$}},
volume = {704},
year = {1993},
abstract = {Theorem: The topological partition relation
$\omega^{*}\rightarrow(Y)^{1}_{2}$ \endgraf (a) fails for every space
$Y$ with $|Y|\geq 2^{\rm \bf c}$; \endgraf (b) holds for $Y$ discrete
if and only if $|Y|\leq$ {\bf c}; \endgraf (c) holds for certain
non-discrete $P$-spaces $Y$; \endgraf (d) fails for
$Y=\omega\cup\{p\}$ with $p\in\omega^{*}$; \endgraf (e) fails for $Y$
infinite and countably compact.},
},
@article{EMSh:441,
author = {Eklof, Paul C. and Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(20K20)},
fromwhere = {1,1,IL},
journal = {Israel Journal of Mathematics},
review = {MR 94f:20105},
note = { arxiv:math.LO/9204219 },
pages = {301--321},
title = {{Uniformization and the diversity of Whitehead groups}},
volume = {80},
year = {1992},
abstract = {The connections between Whitehead groups and
uniformization properties were investigated by the third author in
[Sh:98]. In particular it was essentially shown there that there is a
non-free Whitehead (respectively, $\aleph_1$-coseparable) group
of cardinality $\aleph_1$ if and only if there is a ladder system on
a stationary subset of $\omega_1$ which satisfies
$2$-uniformization (respectively, $omega$-uniformization). These
techniques allowed also the proof of various independence and
consistency results about Whitehead groups, for example that it is
consistent that there is a non-free Whitehead group of cardinality
$\aleph_1$ but no non-free $\aleph_1$-coseparable group. However, some
natural questions remained open, among them the following two: (i) Is
it consistent that the class of W-groups of cardinality $\aleph_1$ is
exactly the class of strongly $\aleph_1$-free groups of cardinality
$\aleph_1$? (ii) If every strongly $\aleph_1$-free group of
cardinality $\aleph_1$ is a W-group, are they also all
$\aleph_1$-coseparable? In this paper we use the techniques of
uniformization to answer the first question in the negative and give a
partial affirmative answer to the second question.},
},
@article{EMSh:442,
author = {Eklof, Paul C. and Mekler, Alan H. and Shelah, Saharon},
fromwhere = {1,3,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/0406552 },
pages = {213--235},
title = {{Hereditarily separable groups and monochromatic
uniformization}},
volume = {88},
year = {1994},
abstract = {We give a combinatorial equivalent to the existence of
a non-free hereditarily separable group of cardinality $\aleph_1$.
This can be used, together with a known combinatorial equivalent of
the existence of a non-free Whitehead group, to prove that it
is consistent that every Whitehead group is free but not
every hereditarily separable group is free. We also show that the
fact that ${\mathbb Z}$ is a p.i.d. with infinitely many primes
is essential for this result. },
},
@article{DShS:443,
author = {Diestel, Reinhard and Shelah, Saharon and Steprans, Juris},
trueauthor = {Diestel, Reinhard and Shelah, Saharon and Stepr\={a}ns,
Juris},
fromwhere = {D,IL,3},
journal = {Journal of the London Mathematical Society},
note = { arxiv:math.LO/9308215 },
pages = {16--24},
title = {{Dominating Functions and Graphs}},
volume = {49},
year = {1994},
abstract = {A graph is called dominating if its vertices can be
labelled with integers in such a way that for every
function $f:\omega\to\omega$ the graph contains a ray whose sequence
of labels eventually exceeds $f$. We obtain a characterization of
these graphs by producing a small family of dominating graphs with
the property that every dominating graph must contain some member of
the family.},
},
@article{HNSh:444,
author = {Huck, A. and Niedermeyer, F. and Shelah, Saharon},
fromwhere = {D,D,IL},
journal = {Journal of Graph Theory},
pages = {413--426},
title = {{Large $\kappa$-preserving sets in infinite graphs}},
volume = {18},
year = {1994},
},
@article{Sh:445,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9406228 },
pages = {357--376},
title = {{Every null additive set of reals is meager additive}},
volume = {89},
year = {1995},
abstract = {We show that every null-additive set is meager-additive,
where: \endgraf (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;
\endgraf (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.},
},
@article{JdSh:446,
author = {Judah, Haim and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9211213 },
pages = {435--450},
title = {{Withdrawn, was: Baire Property and Axiom of Choice}},
volume = {84},
year = {1993},
abstract = {Was withdrawn. Old abstract: We show that \endgraf (1) If
ZF is consistent then the following theory is consistent ``ZF +
DC($\omega_{1}$) + Every set of reals has Baire property'' and
\endgraf (2) If ZF is consistent then the following theory is
consistent ``ZFC + `every projective set of reals has Baire property' +
`any union of $\omega_{1}$ meager sets is meager' ''.},
},
@article{KjSh:447,
author = {Kojman, Menachem and Shelah, Saharon},
ams-subject = {(03C45)},
fromwhere = {IL,IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 93h:03043},
note = { arxiv:math.LO/9201253 },
pages = {57--72},
title = {{The universality spectrum of stable unsuperstable theories}},
volume = {58},
year = {1992},
abstract = {It is shown that if $T$ is stable unsuperstable,
and $\aleph_1\lt \lambda=cf(\lambda)\lt 2^{\aleph_0}$, or
$2^{\aleph_0} \lt \mu^+\lt \lambda=cf(\lambda)\lt \mu^{\aleph_0}$ then
$T$ has no universal model in cardinality $\lambda$, and if e.g.
$\aleph_\omega \lt 2^{\aleph_0}$ then $T$ has no universal model
in $\aleph_\omega$. These results are generalized to
$\kappa=cf(\kappa) \lt \kappa(T)$ in the place of $\aleph_0$. Also: if
there is a universal model in $\lambda>|T|$, $T$ stable and
$\kappa\lt \kappa(T)$ then there is a universal tree of height
$\kappa+1$ in cardinality $\lambda$.},
},
@article{GoSh:448,
author = {Goldstern, Martin and Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {D,IL},
journal = {Archive for Mathematical Logic},
review = {MR 94c:03064},
note = { arxiv:math.LO/9205208 },
pages = {203--221},
title = {{Many simple cardinal invariants}},
volume = {32},
year = {1993},
abstract = {For $g \lt f$ 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.},
},
@article{KjSh:449,
author = {Kojman, Menachem and Shelah, Saharon},
ams-subject = {(03E05)},
fromwhere = {IL,IL},
journal = {Archive for Mathematical Logic},
review = {MR 94e:03045},
note = { arxiv:math.LO/9306215 },
pages = {195--201},
title = {{$\mu $-complete Suslin trees on $\mu ^+$}},
volume = {32},
year = {1993},
abstract = {We prove that $\mu=\mu^{\lt\mu}$, $2^\mu=\mu^+$ and
``there is a non reflecting stationary subset of $\mu^+$ composed
of ordinals of cofinality $\lt \mu$'' imply that there is
a $\mu$-complete Souslin tree on $\mu^+$.},
},
@article{MeSh:450,
author = {Melles, Garvin and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Proceedings of the London Mathematical Society},
note = { arxiv:math.LO/9308216 },
pages = {449--463},
title = {{A saturated model of an unsuperstable theory of cardinality
greater than its theory has the small index property}},
volume = {69},
year = {1994},
abstract = {A model $M$ of cardinality $\lambda$ is said to have the
small index property if for every $G\subseteq Aut(M)$ such
that $[Aut(M):G]\leq\lambda$ there is an $A\subseteq M$ with
$|A|\lt \lambda$ 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.},
},
@article{LsSh:451,
author = {Lascar, Daniel and Shelah, Saharon},
ams-subject = {(03C50)},
fromwhere = {?,IL},
journal = {Bulletin of the London Mathematical Society},
review = {MR 94d:03068},
pages = {125--131},
title = {{Uncountable saturated structures have the small index
property}},
volume = {25},
year = {1993},
},
@article{MeSh:452,
author = {Melles, Garvin and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Bulletin of the London Mathematical Society},
note = { arxiv:math.LO/9304201 },
pages = {339--344},
title = {{$Aut(M)$ has a large dense free subgroup for saturated $M$}},
volume = {26},
year = {1994},
abstract = {We prove that for a stable theory $T,$ if $M$ is a
saturated model of $T$ of cardinality $\lambda$ where $\lambda>|T|,$
then $Aut(M)$ has a dense free subgroup on $2^{\lambda}$ generators.
This affirms a conjecture of Hodges.},
},
@article{MSShT:453,
author = {Mekler, Alan H. and Schipperus, R. and Shelah, Saharon and
Truss, J.K.},
ams-subject = {(20B27)},
fromwhere = {2, ?, IL, GB},
journal = {Bulletin of the London Mathematical Society},
review = {MR 94a:20010},
pages = {343--346},
title = {{The Random Graph and Automorphisms of the Rational World}},
volume = {25},
year = {1993},
},
@article{Sh:454,
author = {Shelah, Saharon},
ams-subject = {(54A25)},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
review = {MR 94f:54007},
note = {Note: See also [Sh454a] below. arxiv:math.LO/9308217 },
pages = {369--374},
title = {{Number of open sets for a topology with a countable basis}},
volume = {83},
year = {1993},
abstract = {Let $T$ be the family of open subsets of a topological
space (not necessarily Hausdorff or even $T_0$). We prove that if $T$
has a countable base and is not countable, then $T$ has cardinality
at least continuum.},
},
@article{Sh:454a,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9403219 },
pages = {95--113},
title = {{Cardinalities of topologies with small base}},
volume = {68},
year = {1994},
abstract = {Let $T$ be the family of open subsets of a topological space
(not necessarily Hausdorff or even $T_0$). We prove that if $T$ has a
base of cardinality $\leq \mu$, $\lambda\leq \mu\lt 2^\lambda$,
$\lambda$ 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].},
},
@article{KjSh:455,
author = {Kojman, Menachem and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9409207 },
pages = {113--124},
title = {{Universal Abelian Groups}},
volume = {92},
year = {1995},
abstract = {We examine the existence of universal elements in classes
of infinite abelian groups. The main method is using group
invariants which are defined relative to club guessing sequences. We
prove, for example: \endgraf Theorem: For $n\ge 2$, there is a purely
universal separable $p$-group in $\aleph_n$ if, and only if,
${2^{\aleph_0}}\leq \aleph_n$.},
},
@article{Sh:456,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Mathematica Japonica},
note = { arxiv:math.LO/9509225 },
pages = {1--11},
title = {{Universal in $(<\lambda)$-stable abelian group}},
volume = {43},
year = {1996},
abstract = {A characteristic result is that if $2^{\aleph_0}\lt \mu\lt
\mu^+\lt \lambda= cf(\lambda)\lt \mu^{\aleph_0}$, then among the
separable reduced $p$-groups of cardinality $\lambda$ which are $(\lt
\lambda)$-stable there is no universal one.},
},
@incollection{Sh:457,
author = {Shelah, Saharon},
booktitle = {Combinatorics, Paul Erd\H{o}s is Eighty},
fromwhere = {IL},
note = {Proceedings of the Meeting in honour of P.Erd\H{o}s, Keszthely,
Hungary 7.1993; A corrected version available as ftp:
//ftp.math.ufl.edu/pub/settheory/shelah/457.tex. arxiv:math.LO/9412229
},
pages = {403--420},
publisher = {Bolyai Society Mathematical Studies},
title = {{The Universality Spectrum: Consistency for more classes}},
volume = {1},
year = {1993},
abstract = {We deal with consistency results for the existence of
universal models in natural classes of models (more exactly--a somewhat
weaker version). We apply a result on quite general family to $T_{\rm
feq}$ and to the class of triangle-free graphs.},
},
@article{AbSh:458,
author = {Abraham, Uri and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Archive for Mathematical Logic},
note = {A special issue in honour of Prof. Azriel Levy.
arxiv:math.LO/9408214 },
title = {{Martin's Axiom and $\Delta ^2_1$ well-ordering of the reals}},
volume = {36},
year = {1997},
abstract = {Assuming an inaccessible cardinal $\kappa$, there is
a generic extension in which $MA + 2^{\aleph_0} = \kappa$ holds and the
reals have a $\Delta^2_1$ well-ordering.},
},
@article{BShT:459,
author = {Baumgartner, James E. and Shelah, Saharon and Thomas, Simon},
fromwhere = {1,IL,1},
journal = {Notre Dame Journal of Formal Logic},
pages = {1--11},
title = {{Maximal subgroups of infinite symmetric groups}},
volume = {34},
year = {1993},
},
@article{Sh:460,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9809200 },
pages = {285--321},
title = {{The Generalized Continuum Hypothesis revisited}},
volume = {116},
year = {2000},
abstract = {We argue that we solved Hilbert's first problem
positively (after reformulating it just to avoid the known consistency
results) and give some applications. Let $\lambda$ to the revised
power of $\kappa$ be the minimal cardinality of a family of subsets
of $\lambda$ each of cardinality $\kappa$ such that any other subset
of $\lambda$ of cardinality $\kappa$ is included in the union
of $<\kappa$ members of the family. The main theorem says that
almost always this revised power is equal to $\lambda$. Our main result
is \endgraf The Revised GCH Theorem: \endgraf 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:
\endgraf (a)
$\kappa\leq\theta<\mu\Rightarrow\lambda^{[\theta]}=\lambda$
and \endgraf (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 ${\mathcal P}$.},
},
@article{EkSh:461,
author = {Eklof, Paul C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Annals of Pure and Applied Algebra},
note = { arxiv:math.LO/9301207 },
pages = {35--50},
title = {{Explicitly nonstandard uniserial modules}},
volume = {86},
year = {1993},
abstract = {A new construction is given of non-standard
uniserial modules over certain valuation domains; the
construction resembles that of a special Aronszajn tree in set theory.
A consequence is the proof of a sufficient condition for the existence
of non-standard uniserial modules; this is a theorem of ZFC which
complements an earlier independence result.},
},
@article{Sh:462,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9609216 },
pages = {199--275},
title = {{$\sigma $-entangled linear orders and narrowness of
products of Boolean algebras}},
volume = {153},
year = {1997},
abstract = {We investigate $\sigma$-entangled linear orders
and narrowness of Boolean algebras. We show existence
of $\sigma$-entangled linear orders in many cardinals, and we
build Boolean algebras with neither large chains nor large pies.
We study the behavior of these notions in ultraproducts.},
},
@article{Sh:463,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Logic and Computation},
note = { arxiv:math.LO/9507221 },
pages = {137--159},
title = {{On the very weak $0-1$ law for random graphs with orders}},
volume = {6},
year = {1996},
abstract = {Let us draw a graph R on {0,1,...,n-1} by having an
edge {i,j} with probability $p_(|i-j|)$, where $\sum_i p_i$ is
finite and let $M_n=(n,\lt,R)$. 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.},
},
@article{BLSh:464,
author = {Baldwin, John T. and Laskowski, Michael C. and Shelah,
Saharon},
fromwhere = {1,1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9301208 },
pages = {1291--1301},
title = {{Forcing Isomorphism}},
volume = {58},
year = {1993},
abstract = {A forcing extension may create new isomorphisms between
two models of a first order theory. Certain model theoretic
constraints on the theory and other constraints on the forcing can
prevent this pathology. A countable first order theory is classifiable
if it is superstable and does not have either the dimensional order
property or the omitting types order property. Shelah [Sh:c] showed
that if a theory $T$ is classifiable then each model of cardinality
$\lambda$ is described by a sentence of $L_{\infty,\lambda}$. In fact
this sentence can be chosen in the $L^*_{\lambda}$. ($L^*_{\lambda}$
is the result of enriching the language $L_{\infty,\beth^+}$ by
adding for each $\mu<\lambda$ a quantifier saying the dimension of
a dependence structure is greater than $\mu$.) The truth of
such sentences will be preserved by any forcing that does not
collapse cardinals $\leq\lambda$ and that adds no new countable subsets
of $\lambda$. Hence, if two models of a classifiable theory of
power $\lambda$ are non-isomorphic, they are non-isomorphic after
a $\lambda$-complete forcing. Here we show that the hypothesis of
the forcing adding no new countable subsets of $\lambda$ cannot
be eliminated. In particular, we show that non-isomorphism of models
of a classifiable theory need not be preserved by ccc forcings.},
},
@article{ShSr:465,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
fromwhere = {IL,3},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9204205 },
pages = {305--319},
title = {{Maximal Chains in ${}^\omega\omega$ and Ultrapowers of the
Integers}},
volume = {32},
year = {1993},
abstract = {Various questions posed by P. Nyikos concerning ultrafilters
on $\omega$ and chains in the partial order $(\omega,\lt ^*)$
are answered. The main tool is the oracle chain condition and
variations of it. (Note: Corrections in [Sh:465a])},
},
@article{ShSr:465a,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
fromwhere = {IL,3},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9308202 },
pages = {167--168},
title = {{Erratum: ``Maximal Chains in ${}^\omega\omega$ and Ultrapowers
of the Integers''}},
volume = {33},
year = {1994},
abstract = {This note is intended as a supplement and clarification to
the proof of Theorem 3.3 of [ShSr:465]; namely, it is consistent
that ${\frak b}=\aleph_1$ yet for every ultrafilter $U$ on $\omega$
there is a $\leq^*$ chain $\{f_\xi:\xi\in\omega_2\}$ such that
$\{f_\xi/U:\xi\in\omega_2\}$ is cofinal in $\omega/U$.},
},
@article{JiSh:466,
author = {Jin, Renling and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9308218 },
pages = {1--16},
title = {{A model in which there are Jech-Kunen trees but there are no
Kurepa trees}},
volume = {84},
year = {1993},
abstract = {By an $\omega_1$--tree we mean a tree of power $\omega_1$
and height $\omega_1$. We call an $\omega_1$--tree a Jech--Kunen tree
if it has $\kappa$--many branches for some $\kappa$ strictly
between $\omega_1$ and $2^{\omega_1}$. In this paper we construct the
models of CH plus $2^{\omega_1}>\omega_2$, in which there are
Jech--Kunen trees and there are no Kurepa trees.},
},
@article{Sh:467,
author = {Shelah, Saharon},
ams-subject = {(03C13); (60F20); (03C10)},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9606226 },
pages = {195--239},
title = {{Zero-one laws for graphs with edge probabilities decaying with
distance. Part I}},
volume = {175},
year = {2002},
abstract = {Let $G_n$ be the random graph on $[n]=\{1,\ldots,n\}$
with the possible edge $\{i,j\}$ having probability being $p_{|i-j|}=
1/|i-j|^\alpha$, $\alpha\in (0,1)$ irrational. We prove that the zero
one law (for first order logic) holds. The paper is continued in
[Sh:517]},
},
@article{ShSi:468,
author = {Shelah, Saharon and Spinas, Otmar},
fromwhere = {IL,CH},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.LO/9510215 },
pages = {4257--4277},
title = {{On Gross Spaces}},
volume = {348},
year = {1996},
abstract = {A Gross space is a vector space $E$ of infinite
dimension over some field $F$, which is endowed with a symmetric
bilinear form $\Phi:E^2 \rightarrow F$ and has the property that
every infinite dimensional subspace $U\subseteq E$
satisfies dim$U^\perp \lt $ dim$E$. 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 {\bf 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.},
},
@article{JiSh:469,
author = {Jin, Renling and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9211214 },
pages = {287--296},
title = {{Planting Kurepa trees and killing Jech-Kunen trees in a model
by using one inaccessible cardinal}},
volume = {141},
year = {1992},
abstract = {By an $\omega_1$--tree we mean a tree of power $\omega_1$
and height $\omega_1$. Under CH and $2^{\omega_1}>\omega_2$ we call
an $\omega_1$--tree a Jech--Kunen tree if it has $\kappa$ many
branches for some $\kappa$ strictly between $\omega_1$ and
$2^{\omega_1}$. In this paper we prove that, assuming the existence of
one inaccessible cardinal, \endgraf (1) it is consistent with CH plus
$2^{\omega_1}>\omega_2$ that there exist Kurepa trees and there are no
Jech--Kunen trees, \endgraf (2) it is consistent with CH plus
$2^{\omega_1}=\omega_4$ that only Kurepa trees with $\omega_3$ many
branches exist.},
},
@article{RoSh:470,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {PL,IL},
journal = {Memoirs of the American Mathematical Society},
note = { arxiv:math.LO/9807172 },
pages = {xii + 167},
title = {{Norms on possibilities I: forcing with trees and creatures}},
volume = {141},
year = {1999},
abstract = {In this paper we present a systematic study of the method
of {\em 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. },
},
@article{LeSh:471,
author = {Lifsches, Shmuel and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9308219 },
pages = {848--872},
title = {{Peano Arithmetic may not be interpretable in the
monadic theory of linear orders}},
volume = {62},
year = {1997},
abstract = {Gurevich and Shelah have shown that Peano Arithmetic cannot
be interpreted in the monadic second-order theory of short chains
(hence, in the monadic second-order theory of the real line). We show
here that it is consistent that there is no interpretation even in the
monadic second-order theory of all chains.},
},
@article{Sh:472,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9604241 },
pages = {165--196},
title = {{Categoricity of Theories in $L_{\kappa^* \omega}$,
when $\kappa^*$ is a measurable cardinal. Part II}},
volume = {170},
year = {2001},
abstract = {We continue the work of [KlSh:362] and prove that
for $\lambda$successor, 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.},
},
@article{Sh:473,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Publications de L'Institute Math\'ematique - Beograd,
Nouvelle S\'erie},
note = { arxiv:math.LO/9511220 },
pages = {47--60},
title = {{Possibly every real function is continuous on a non--meagre
set}},
volume = {57(71)},
year = {1995},
abstract = {We prove consistency of the following sentence: ``ZFC +
every real function is continuous on a non-meagre set'', answering
a question of Fremlin.},
},
@article{HySh:474,
author = {Hyttinen, Tapani and Shelah, Saharon},
fromwhere = {SF, IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0406587 },
pages = {984--996},
title = {{Constructing strongly equivalent nonisomorphic models for
unsuperstable theories, Part A}},
volume = {59},
year = {1994},
abstract = {We study how equivalent nonisomorphic models an
unsuperstable theory can have. We measure the equivalence by
Ehrenfeucht-Fraisse games.},
},
@article{RoSh:475,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {PL,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9408215 },
pages = {299--313},
title = {{More forcing notions imply diamond}},
volume = {35},
year = {1996},
abstract = {We prove that the Sacks forcing collapses the continuum onto
the dominating number ${\frak d}$, 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 $\diamondsuit_{\omega_1}$.},
},
@article{JeSh:476,
author = {Jech, Thomas and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9412208 },
pages = {313--317},
title = {{Possible pcf algebras}},
volume = {61},
year = {1996},
abstract = {There exists a family $\{B_{\alpha}\}_{\alpha\lt\omega_1}$
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) $\beta\lt \alpha$, 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.},
},
@article{BJSh:477,
author = {Brendle, Joerg and Judah, Haim and Shelah, Saharon},
trueauthor = {Brendle, J{\"{o}}rg and Judah, Haim and Shelah, Saharon},
ams-subject = {(03E40)},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
review = {MR 93k:03048},
note = { arxiv:math.LO/9211202 },
pages = {185--199},
title = {{Combinatorial properties of Hechler forcing}},
volume = {58},
year = {1992},
abstract = {Using a notion of rank for Hechler forcing we show:
\endgraf 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 $V$; \endgraf 2) adding one Hechler real
makes the invariants on the left-hand side of Cicho\'n'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 $\omega_1$; \endgraf 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 $V[r]$.},
},
@article{JdSh:478,
author = {Judah, Haim and Shelah, Saharon},
ams-subject = {(03E15)},
fromwhere = {IL,IL},
journal = {Proceedings of the American Mathematical Society},
review = {MR 94e:03046},
note = { arxiv:math.LO/9401215 },
pages = {917--920},
title = {{Killing Luzin and Sierpi\'nski sets}},
volume = {120},
year = {1994},
abstract = {We will kill the old Luzin and Sierpinski sets in order to
build a model where $U(Meager) = U(Null)=\aleph_1$ and there are
neither Luzin nor Sierpinski sets. Thus we answer a question of J.
Steprans, communicated by S. Todorcevic on route from Evans to MSRI.},
},
@article{Sh:479,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9601218 },
pages = {1-19},
title = {{On Monk's questions}},
volume = {151},
year = {1996},
abstract = {Monk asks (problems 13, 15 in his list; $\pi$ is the
algebraic density): ``For a Boolean algebra $B$,
$\aleph_0\le\theta\le\pi(B)$, does $B$ have a subalgebra $B'$ with
$\pi(B')=\theta$?'' If $\theta$ is regular the answer is easily
positive, we show that in general it may be negative, but for quite
many singular cardinals - it is positive; the theorems are quite
complementary. Next we deal with $\pi\chi$ and we show that the
$\pi\chi$ of an ultraproduct of Boolean algebras is not necessarily the
ultraproduct of the $\pi\chi$'s. We also prove that for infinite
Boolean algebras $A_i$ ($i<\kappa$) and a non-principal ultrafilter $D$
on $\kappa$: if $n_i<\aleph_0$ for $i<\kappa$ and $\mu=\prod_{i<\kappa}
n_i/D$ is regular, then $\pi\chi(A)\ge \mu$. Here
$A=\prod_{i<\kappa}A_i/D$. By a theorem of Peterson the regularity of
$\mu$ is needed.},
},
@article{Sh:480,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9303208 },
pages = {159--174},
title = {{How special are Cohen and random forcings i.e. Boolean
algebras of the family of subsets of reals modulo meagre or null}},
volume = {88},
year = {1994},
abstract = {The feeling that those two forcing notions -Cohen
and Random-(equivalently the corresponding Boolean
algebras Borel(R)/(meager sets), Borel(R)/(null sets)) are special,
was probably old and widespread. A reasonable interpretation is to
show them unique, or ``minimal'' or at least characteristic in a
family of ``nice forcing'' like Borel. We shall interpret ``nice''
as Souslin as suggested by Judah Shelah [JdSh 292]. We divide
the family of Souslin forcing to two, and expect that: among the
first part, i.e. those adding some non-dominated real, Cohen is
minimal (=is below every one), while among the rest random is
quite characteristic even unique. Concerning the second class we
have weak results, concerning the first class, our results
look satisfactory. We have two main results: one (1.14) says that
Cohen forcing is ``minimal'' in the first class, the other (1.10)
says that all c.c.c. Souslin forcing have a property shared by
Cohen forcing and Random real forcing, so it gives a weak answer to
the problem on how special is random forcing, but says much on
all c.c.c. Souslin forcing. },
},
@article{Sh:481,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9509226 },
pages = {259--269},
title = {{Was Sierpi\'nski right? III Can continuum--c.c. times c.c.c.
be continuum--c.c.?}},
volume = {78},
year = {1996},
abstract = {We prove the consistency of: if $B_1$, $B_2$ are
Boolean algebra satisfying the c.c.c. and the
$2^{\aleph_0}$-c.c. respectively then $B_1 \times B_2$ satisfies
the $2^{\aleph_0}$-c.c.},
},
@incollection{Sh:482,
author = {Shelah, Saharon},
fromwhere = {IL},
note = { arxiv:math.LO/1601.03596 },
title = {{Compactness of the Quantifier on ``Complete embedding of
BA's''}},
},
@article{LVSh:483,
author = {Louveau, Alain and Velickovic, Boban and Shelah, Saharon},
trueauthor = {Louveau, Alain and Shelah, Saharon and Veli\v{c}kovi\'c,
Boban},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9301209 },
pages = {271--281},
title = {{Borel partitions of infinite subtrees of a perfect tree}},
volume = {63},
year = {1993},
abstract = {A theorem of Galvin asserts that if the unordered pairs
of reals are partitioned into finitely many Borel classes then there
is a perfect set $P$ such that all pairs from $P$ lie in the
same class. The generalization to $n$-tuples for $n\geq 3$ is false.
Let us identify the reals with $2^\omega$ ordered by the
lexicographical ordering and define for distinct $x,y\in 2^\omega$,
$D(x,y)$ to be the least $n$ such that $x(n)\neq y(n)$. Let the type of
an increasing $n$-tuple $\{x_0,\ldots x_{n-1}\}_<$ be the
ordering $<^*$ on $\{0,\ldots,n-2\}$ defined by $i<^*j$ iff
$D(x_i,x_{i+1})< D(x_j,x_{j+1})$. Galvin proved that for any Borel
coloring of triples of reals there is a perfect set $P$ such that the
color of any triple from $P$ depends only on its type. Blass proved
an analogous result is true for any $n$. As a corollary it follows
that if the unordered $n$-tuples of reals are colored into finitely
many Borel classes there is a perfect set $P$ such that the
$n$-tuples from $P$ meet at most $(n-1)!$ classes. We consider
extensions of this result to partitions of infinite increasing
sequences of reals. We show, that for any Borel or even analytic
partition of all increasing sequences of reals there is a perfect set
$P$ such that all strongly increasing sequences from $P$ lie in the
same class.},
},
@article{LiSh:484,
author = {Liu, Kecheng and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9604242 },
pages = {189-205},
title = {{Cofinalities of elementary substructures of structures on
$\aleph_\omega$}},
volume = {99},
year = {1997},
abstract = {Let $02^{\aleph_0}$ the Erd\H{o}s-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
$\aleph_1\leq\kappa\leq\aleph_\omega$},
},
@article{KoSh:492,
author = {Komjath, Peter and Shelah, Saharon},
trueauthor = {Komj\'{a}th, P\'eter and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Journal of Combinatorial Theory. Ser. B},
note = { arxiv:math.LO/9308221 },
pages = {125--135},
title = {{Universal graphs without large cliques}},
volume = {63},
year = {1995},
abstract = {We give some existence/nonexistence statements on
universal graphs, which under GCH give a necessary and sufficient
condition for the existence of a universal graph of size $\lambda$ with
no $K(\kappa)$, namely, if either $\kappa$ is finite
or $cf(\kappa)>cf(\lambda)$. (Here $K(\kappa)$ denotes the
complete graph on $\kappa$ vertices.) The special case
when $\lambda^{<\kappa}=\lambda$ was first proved by F. Galvin. Next,
we investigate the question that if there is no
universal $K(\kappa)$-free graph of size $\lambda$ then how many of
these graphs embed all the other. It was known, that
if $\lambda^{<\lambda}= \lambda$ (e.g., if $\lambda$ is regular and
the GCH holds below $\lambda$), and $\kappa=\omega$, then this number
is $\lambda^+$. We show that this holds for every
$\kappa\leq\lambda$ of countable cofinality. On the other hand, even
for $\kappa=\omega_1$, and any regular $\lambda\geq\omega_1$ it
is consistent that the GCH holds below $\lambda$, $2^{\lambda}$ is
as large as we wish, and the above number is either $\lambda^+$
or $2^{\lambda}$, so both extremes can actually occur.},
},
@article{JiSh:493,
author = {Jin, Renling and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9401216 },
pages = {292--301},
title = {{The strength of the isomorphism property}},
volume = {59},
year = {1994},
abstract = {In \S 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 \S 2,
several applications are given. One of the applications answers
a question of D. Ross about infinite Loeb measure spaces},
},
@article{ShSi:494,
author = {Shelah, Saharon and Spinas, Otmar},
fromwhere = {IL,CH},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.LO/9606227 },
pages = {2023--2047},
title = {{The distributivity numbers of ${\cal P}(\omega)$/fin and its
square}},
volume = {352},
year = {2000},
abstract = {We show that in a model obtained by forcing with a countable
support iteration of length $\omega_2$ of Mathias forcing ${\bf h}(2)$,
the distributivity number of r.o.$({\cal P}(\omega)$/fin)$^2$, is
$\omega_1$ but ${\bf h}$, the one of ${\cal P}(\omega)$/fin, is
$\omega_2$.},
},
@article{ApSh:495,
author = {Apter, Arthur W. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.LO/9502232 },
pages = {103-128},
title = {{On the Strong Equality between Supercompactness and Strong
Compactness''}},
volume = {349},
year = {1997},
abstract = {We show that supercompactness and strong compactness can be
equivalent even as properties of pairs of regular
cardinals. Specifically, we show that if $V \models$ ZFC + GCH is a
given model (which in interesting cases contains instances
of supercompactness), then there is some cardinal and
cofinality preserving generic extension $V[G]\models$ ZFC + GCH
in which, (a) (preservation) for $\kappa \le \lambda$ regular, if $V
\models ``\kappa$ is $\lambda$ supercompact'', then $V[G] \models
``\kappa$ is $\lambda$ supercompact'' and so that, (b) (equivalence)
for $\kappa \le \lambda$ regular, $V[G] \models ``\kappa$ is $\lambda$
strongly compact'' iff $V[G] \models ``\kappa$ is $\lambda$
supercompact'', except possibly if $\kappa$ is a measurable limit of
cardinals which are $\lambda $ supercompact.},
},
@article{ApSh:496,
author = {Apter, Arthur and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.LO/9512226 },
pages = {2007-2034},
title = {{Menas' Result is Best Possible}},
volume = {349},
year = {1997},
abstract = {Generalizing some earlier techniques due to the
second author, we show that Menas' theorem which states that the
least cardinal $\kappa$ which is a measurable limit of supercompact
or strongly compact cardinals is strongly compact but not $2^\kappa$
supercompact is best possible. Using these same techniques, we also
extend and give a new proof of a theorem of Woodin and extend and give
a new proof of an unpublished theorem due to the first author.},
},
@article{Sh:497,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = {A special volume dedicated to Prof. Azriel Levy.
arxiv:math.LO/9512227 },
pages = {81-125},
title = {{Set Theory without choice: not everything on cofinality is
possible}},
volume = {36},
year = {1997},
abstract = {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]},
},
@article{JiSh:498,
author = {Jin, Renling and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9401217 },
pages = {107--131},
title = {{Essential Kurepa trees versus essential Jech---Kunen trees}},
volume = {69},
year = {1994},
abstract = {By an $\omega_1$--tree we mean a tree of size $\omega_1$
and height $\omega_1$. An $\omega_1$--tree is called a Kurepa tree
if all its levels are countable and it has more than
$\omega_1$ branches. An $\omega_1$--tree is called a Jech--Kunen tree
if it has $\kappa$ branches for some $\kappa$ strictly between
$\omega_1$ and $2^{\omega_1}$. A Kurepa tree is called an essential
Kurepa tree if it contains no Jech--Kunen subtrees. A Jech--Kunen tree
is called an essential Jech--Kunen tree if it contains no Kurepa
subtrees. In this paper we prove that (1) it is consistent with CH
and $2^{\omega_1}>\omega_2$ that there exist essential Kurepa trees
and there are no essential Jech--Kunen trees, (2) it is consistent
with CH and $2^{\omega_1}>\omega_2$ plus the existence of a Kurepa
tree with $2^{\omega_1}$ branches that there exist essential
Jech--Kunen trees and there are no essential Kurepa trees. In the
second result we require the existence of a Kurepa tree with
$2^{\omega_1}$ branches in order to avoid triviality.},
},
@article{KjSh:499,
author = {Kojman, Menachem and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Journal of the London Mathematical Society},
note = { arxiv:math.LO/9409205 },
pages = {303--317},
title = {{Homogeneous families and their automorphism groups}},
volume = {52},
year = {1995},
abstract = {A homogeneous family of subsets over a given set is one with
a very ``rich'' automorphism group. We prove the existence of a
bi-universal element in the class of homogeneous families over a given
infinite set and give an explicit construction of $2^{\bf c}$
isomorphism types of homogeneous families over a countable set.},
},
@article{Sh:500,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9508205 },
pages = {229--255},
title = {{Toward classifying unstable theories}},
volume = {80},
year = {1996},
abstract = {The paper deals with two issues: the existence of universal
models of a theory $T$ and related properties when cardinal arithmetic
does not give this existence offhand. In the first section we prove
that simple theories (e.g., theories without the tree property, a class
properly containing the stable theories) behaves ``better'' than
theories with the strict order property, by criterion from [Sh:457].
In the second section we introduce properties $SOP_n$ such that
the strict order property implies $SOP_{n+1}$, which implies $SOP_n$,
which in turn implies the tree property. Now $SOP_4$ already implies
non-existence of universal models in cases where earlier the strict
order property was needed, and $SOP_3$ implies maximality in the
Keisler order, again improving an earlier result which had used the
strict order property.},
},
@article{RoSh:501,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {PL,IL},
journal = {Archive for Mathematical Logic},
note = {A special volume dedicated to Prof. Azriel Levy.
arxiv:math.LO/9506222 },
pages = {315--339},
title = {{Localizations of infinite subsets of $\omega$}},
volume = {35},
year = {1996},
abstract = {In the present paper we are interested in properties of
forcing notions which measure in a sense the distance between the
ground model reals and the reals in the extension. We look at the ways
the ``new'' reals can be aproximated by ``old'' reals. We
consider localizations for infinite subsets of $\omega$. Though each
member of $[\omega]^\omega$ can be identified with its
increasing enumeration, the (standard) localizations of the enumeration
does not provide satisfactory information on successive points of
the set. They give us ``candidates'' for the $n$-th point of the set
but the same candidates can appear several times for distinct $n$.
That led to a suggestion that we should consider disjoint subsets
of $\omega$ as sets of ``candidates'' for successive points of
the localized set. We have two possibilities. Either we can demand
that each set from the localization contains a limited number of
members of the localized set or we can postulate that each intersection
of that kind is large. Localizations of this kind are studied
in section 1. In the second section we investigate localizations
of infinite subsets of $\omega$ by sets of integers from the
ground model. These localizations might be thought as localizations
by partitions of $\omega$ into successive intervals.},
},
@article{KoSh:502,
author = {Komjath, Peter and Shelah, Saharon},
trueauthor = {Komj\'{a}th, P\'eter and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Real Analysis Exchange},
note = { arxiv:math.LO/9308222 },
pages = {218--225},
title = {{On uniformly antisymmetric functions}},
volume = {19},
year = {1993-1994},
abstract = {We show that there is always a uniformly
antisymmetric $f:A\to\{0,1\}$ if $A\subset R$ is countable. We prove
that the continuum hypothesis is equivalent to the statement that there
is an $f:R\to\omega$ with $|S_x|\leq 1$ for every $x\in R$. If
the continuum is at least $\aleph_n$ then there exists a point $x$
such that $S_x$ has at least $2^n-1$ elements. We also show that there
is a function $f:Q\to\{0,1,2,3\}$ such that $S_x$ is always finite,
but no such function with finite range on $R$ exists},
},
@article{Sh:503,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Mathematica Japonica},
note = { arxiv:math.LO/9312212 },
pages = {1--5},
title = {{The number of independent elements in the product of interval
Boolean algebras}},
volume = {39},
year = {1994},
abstract = {We prove that in the product of $\kappa$ many
Boolean algebras we cannot find an independent set of more
than $2^\kappa$ elements solving a problem of Monk (earlier it
was known that we cannot find more than $2^{2^\kappa}$ but can
find $2^\kappa$).},
},
@incollection{KpSh:504,
author = {Koppelberg, Sabine and Shelah, Saharon},
booktitle = {Logic: from Foundations to Applications},
fromwhere = {D,IL},
note = {Proceedings of the ASL Logic Colloquium 1993 in Keele (Great
Britain); W. Hodges, M. Hyland, Ch. Steinhorn, J. Truss, editors.
arxiv:math.LO/9610227 },
pages = {261--275},
publisher = {Clarendon Press, Oxford},
series = {Oxford Science Publications},
title = {{Subalgebras of Cohen algebras need not be Cohen}},
year = {1996},
abstract = {We give an example of a regular and complete subalgebra of
a Cohen algebra which is not Cohen.},
},
@incollection{EkSh:505,
author = {Eklof, Paul C. and Shelah, Saharon},
booktitle = {Abelian group theory and related topics},
fromwhere = {1,IL},
note = {edited by R. Goebel, P. Hill and W. Liebert, Oberwolfach
proceedings. arxiv:math.LO/9403220 },
pages = {79--98},
publisher = {American Mathematical Society, Providence, RI},
series = {Contemporary Mathematics},
title = {{A Combinatorial Principle Equivalent to the Existence of
Non-free Whitehead Groups}},
volume = {171},
year = {1994},
abstract = {As a consequence of identifying the principle described
in the title, we prove that for any uncountable cardinal $\lambda $, if
there is a $\lambda $-free Whitehead group of cardinality $\lambda $
which is not free, then there are many ``nice'' Whitehead groups of
cardinality $\lambda $ which are not free.},
},
@incollection{Sh:506,
author = {Shelah, Saharon},
booktitle = {The Mathematics of Paul Erd\H{o}s, II},
fromwhere = {IL},
note = {Graham, Ne\v set\v ril, eds.. arxiv:math.LO/9502233 },
pages = {420-459},
publisher = {Springer},
series = {Algorithms and Combinatorics},
title = {{The pcf-theorem revisited}},
volume = {14},
year = {1997},
abstract = {The $\pcf$ theorem (of the possible cofinality theory) was
proved for reduced products $\prod_{i\lt \kappa} \lambda_i/I$, where
$\kappa\lt \min_{i\lt \kappa} \lambda_i$. Here we prove this theorem
under weaker assumptions such as $wsat(I)\lt \min_{i\lt \kappa}
\lambda_i$, 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 $\lt _I$ ($\lt _I$-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},
},
@article{GoSh:507,
author = {Goldstern, Martin and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9501222 },
pages = {58--73},
title = {{The Bounded Proper Forcing Axiom}},
volume = {60},
year = {1995},
abstract = {The bounded proper forcing axiom BPFA is the statement that
for any family of $\aleph_1$ many maximal antichains of a proper
forcing notion, each of size $\aleph_1$, there is a directed set
meeting all these antichains. \endgraf A regular cardinal $\kappa$ is
called ${\Sigma}_1$-reflecting, if for any regular cardinal $\chi$, for
all formulas $\varphi$, ``$H(\chi)\models `\varphi$'\,'' implies
``$\exists\delta\lt\kappa$, $H(\delta)\models `\varphi$'\,''
\endgraf 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). \endgraf 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.},
},
@article{RoSh:508,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon },
fromwhere = {PL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9606228 },
pages = {1297--1314},
title = {{Simple forcing notions and forcing axioms}},
volume = {62},
year = {1997},
abstract = {In the present paper we are interested in simple forcing
notions and Forcing Axioms. A starting point for our investigations was
the article \cite{JR1} in which several problems were posed. We
answer some of those problems here.},
},
@article{Sh:509,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/0112237 },
pages = {61--96},
title = {{Vive la diff\'erence III}},
volume = {166},
year = {2008},
abstract = {We show that, consistently, there is an ultrafilter
${\mathcal F}$ on $\omega$ such that if $N^\ell_n=(P^\ell_n\cup
Q^\ell_n, P^\ell_n,Q^\ell_n,R^\ell_n)$ (for $\ell=1,2$, $n<\omega$),
$P^\ell_n \cup Q^\ell_n \subseteq\omega$, and $\prod\limits_{n<\omega}
N^1_n/ {\mathcal F}\equiv\prod\limits_{n<\omega}N^2_n/{\mathcal F}$
are models of the canonical theory $t^{\rm ind}$ of the
strong independence property, then every isomorphism
from $\prod\limits_{n<\omega} N^1_n/{\mathcal F}$ onto
$\prod\limits_{n< \omega} N^2_n/{\mathcal F}$ is a product
isomorphism.},
},
@article{ShSr:510,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
fromwhere = {3,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9401218 },
pages = {171--180},
title = {{Decomposing Baire class 1 functions into continuous
functions}},
volume = {145},
year = {1994},
abstract = {Let $\frak dec$ 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 $cov(Meager)\leq {\frak dec}\leq {\frak d}$ 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 $\frak d$ is denoted
the least cardinal of a dominating family in $\omega^\omega$.
Steprans showed that it is consistent that $cov(Meager)\neq {\frak
dec}$. In this paper we show that the second inequality can also be
made strict. The model where ${\frak dec}$ is different from $\frak d$
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, $cov(Meager)\neq{\frak dec}$, is a slight modification of
the iteration of super-perfect forcing.},
},
@article{Sh:511,
author = {Shelah, Saharon},
fromwhere = {IL},
title = {{Building complicated index models and Boolean algebras}},
},
@article{BRSh:512,
author = {Balcerzak, Marek and Roslanowski, Andrzej and Shelah,
Saharon},
trueauthor = {Balcerzak, Marek and Ros{\l}anowski, Andrzej and Shelah,
Saharon},
fromwhere = {PL,PL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9610219 },
pages = {128--147},
title = {{Ideals without ccc}},
volume = {63},
year = {1998},
abstract = {Let $I$ be an ideal of subsets of a Polish space $X$,
containing all singletons and possessing a Borel basis. Assuming that
$I$ does not satisfy ccc, we consider the following conditions (B), (M)
and (D). Condition (B) states that there is a disjoint
family $F\subseteq P(X)$ of size ${\bf c}$, consisting of Borel sets
which are not in $I$. Condition (M) states that there is a
function $f:X\rightarrow X$ with $f^{-1}[\{x\}]\notin I$ for each $x\in
X$. Provided that $X$ is a group and $I$ is invariant, condition
(D) states that there exist a Borel set $B\notin I$ and a perfect
set $P\subseteq X$ for which the family $\{ B+x: x\in P\} $
is disjoint. The aim of the paper is to study whether the
reverse implications in the chain $(D)\Rightarrow
(M)\Rightarrow (B)\Rightarrow not-ccc$ can hold. We build a $\sigma
$-ideal on the Cantor group witnessing ``(M) and not (D)'' (Section 2).
A modified version of that $\sigma$-ideal contains the whole space
(Section 3). Some consistency results deriving (M) from (B) for
``nicely'' defined ideals are established (Section 4). We show that
both ccc and (M) can fail (Theorems 1.3 and 4.2). Finally, some
sharp versions of (M) for invariant ideals on Polish groups
are investigated (Section 5).},
},
@article{Sh:513,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9807177 },
pages = {321--359},
title = {{PCF and infinite free subsets in an algebra}},
volume = {41},
year = {2002},
abstract = {We give another proof that for every
$\lambda\geq\beth_\omega$ for every large enough regular
$\kappa<\beth_\omega$ we have $\lambda^{[\kappa]}=\lambda$, dealing
with sufficient conditions for replacing $\beth_\omega$ by
$\aleph_\omega$. In \S2 we show that large pcf$({\frak a})$ 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 \S2 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.},
},
@incollection{MgSh:514,
author = {Magidor, Menachem and Shelah, Saharon},
booktitle = {Abelian group theory and related topics},
fromwhere = {IL,IL},
note = {edited by R. Goebel, P. Hill and W. Liebert, Oberwolfach
proceedings. arxiv:math.LO/9405214 },
pages = {287--294},
publisher = {American Mathematical Society, Providence, RI},
series = {Contemporary Mathematics},
title = {{$Bext^2(G,T)$ can be nontrivial, even assuming GCH}},
volume = {171},
year = {1994},
abstract = {Using the consistency of some large cardinals we produce
a model of Set Theory in which the generalized continuum hypothesis
holds and for some torsion-free abelian group $G$ of cardinality
$\aleph_{\omega+1}$ and for some torsion group
$T$, $Bext^2(G,T)\not=0$.},
},
@incollection{Sh:515,
author = {Shelah, Saharon},
booktitle = {The Mathematics of Paul Erd\H{o}s, II},
fromwhere = {IL},
note = {Graham, Ne\v set\v ril, eds.. arxiv:math.CO/9502234 },
pages = {240-246},
publisher = {Springer},
series = {Algorithms and Combinatorics},
title = {{A finite partition theorem with double exponential bounds}},
volume = {14},
year = {1997},
abstract = {We prove that double exponentiation is an upper bound
to Ramsey theorem for colouring of pairs when we want to predetermine
the order of the differences of successive members of the homogeneous
set.},
},
@article{KoSh:516,
author = {Komjath, Peter and Shelah, Saharon},
trueauthor = {Komj\'{a}th, P\'eter and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/9505216 },
pages = {3501-3505},
title = {{Coloring finite subsets of uncountable sets}},
volume = {124},
year = {1996},
abstract = {It is consistent for every \(1\leq n\lt \omega\)
that \(2^\omega=\omega_n\) and there is a function \(F:[\omega_n]^{\lt
\omega}\to\omega \) 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.},
},
@article{Sh:517,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/0404239 },
pages = {211--245},
title = {{Zero-one laws for graphs with edge probabilities decaying with
distance. Part II}},
volume = {185},
year = {2005},
},
@article{LwSh:518,
author = {Laskowski, Michael C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0011169 },
pages = {1305--1320},
title = {{Forcing Isomorphism II}},
volume = {61},
year = {1996},
abstract = {If $T$ has only countably many complete types, yet has a
type of infinite multiplicity then there is a ccc forcing notion
$Q$ such that, in any $Q$--generic extension of the universe, there
are non-isomorphic models $M_1$ and $M_2$ of $T$ that can be
forced isomorphic by a ccc forcing. We give examples showing that
the hypothesis on the number of complete types is necessary and
what happens if ``ccc'' is replaced other
cardinal-preserving adjectives. We also give an example showing that
membership in a pseudo-elementary class can be altered by very
simple cardinal-preserving forcings.},
},
@incollection{GbSh:519,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
booktitle = {Abelian Groups and Modules},
note = { arxiv:math.LO/0104194 },
nt = {Proceedings of the Padova Conference, Padova, Italy, June 23 --
July 1, 1994. Editors: A. Facchini and C. Menini},
pages = {227--237},
publisher = {Kluwer, New York},
title = {{On the existence of rigid $\aleph_1$-free abelian groups
of cardinality $\aleph_1$}},
year = {1995},
abstract = {An abelian group is said to be $\aleph_1$--free if all
its countable subgroups are free. Our main result is: \endgraf 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 ${\rm End} G = R$. \endgraf A corollary to this
theorem is: \endgraf Indecomposable $\aleph_1$--free abelian groups of
cardinality $\aleph_1$ do exist.},
},
@article{EFSh:520,
author = {Eklof, Paul C. and Foreman, Matthew and Shelah, Saharon},
fromwhere = {1,1,IL},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.LO/9501223 },
pages = {4385--4402},
title = {{On invariants for $\omega _1$-separable groups}},
volume = {347},
year = {1995},
abstract = {We study the classification of $\omega_1$-separable
groups using Ehrenfeucht-Fra{\"\i}ss\'e games and prove a
strong classification result assuming PFA, and a strong
non-structure theorem assuming $\diamondsuit$.},
},
@article{Sh:521,
author = {Shelah, Saharon},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9503226 },
pages = {1261-1278},
title = {{If there is an exactly $\lambda$-free abelian group then there
is an exactly $\lambda$-separable one}},
volume = {61},
year = {1996},
abstract = {We give a solution stated in the title to problem 3 of part
1 of the problems listed in the book of Eklof and Mekler [EM],(p.453).
There, in pp. 241-242, this is discussed and proved in some cases. The
existence of strongly $\lambda$-free ones was proved earlier by the
criteria in [Sh:161] in [MkSh:251]. We can apply a similar proof to a
large class of other varieties in particular to the variety
of (non-commutative) groups.},
},
@article{Sh:522,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9802134 },
pages = {1--50},
title = {{Borel sets with large squares}},
volume = {159},
year = {1999},
abstract = {For a cardinal $\mu$ we give a sufficient condition
$(*)_\mu$ (involving ranks measuring existence of independent sets)
for: \endgraf [$(**)_\mu$] 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, \endgraf and also for \endgraf [$(***)_\mu$] 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.
\endgraf 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.},
},
@article{Sh:523,
author = {Shelah, Saharon},
journal = {Mathematica Japonica},
note = { arxiv:math.LO/9606229 },
pages = {1--14},
title = {{Existence of Almost Free Abelian groups and reflection of
stationary set}},
volume = {45},
year = {1997},
abstract = {\S2: 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 ${\cal S}_{\lt \aleph_2}(\lambda) = \{ a \subseteq \lambda: |a| \lt
\aleph_2 \}$]. \S3 - $NPT$ is not transitive. We
prove $NPT(\lambda,\mu) + NPT(\mu,\kappa)
\not\Rightarrow NPT(\lambda,\kappa)$},
},
@article{ShTh:524,
author = {Shelah, Saharon and Thomas, Simon},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9412230 },
pages = {902--916},
title = {{The Cofinality Spectrum of The Infinite Symmetric Group}},
volume = {62},
year = {1997},
abstract = {A group $G$ that is not finitely generated can be written
as the union of a chain of proper subgroups. The cofinality spectrum
of $G$, written $CF(S)$, is the set of regular cardinals $\lambda$
such that $G$ can be expressed as the union of a chain of
$\lambda$ proper subgroups. The cofinality of $G$, written $c(G)$, is
the least element of $CF(G)$. We show that it is consistent that
$CF(S)$ is quite a bizarre set of cardinals. For example, we
prove \endgraf Theorem (A): Let $T$ be any subset of $\omega\setminus
\{0\}$. Then it is consistent that $\aleph_n \in CF(S)$ if and only if
$n\in T$. \endgraf 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.
\endgraf Theorem (B): If $\aleph_n \in CF(S)$ for all $n\in
\omega\setminus \{0\}$, then $\aleph_{\omega+1} \in CF(S)$.},
},
@incollection{GISh:525,
author = {Gurevich, Yuri and Immerman, Neil and Shelah, Saharon},
booktitle = {Symposium on Logic in Computer Science},
fromwhere = {1,1,IL},
note = { arxiv:math.LO/9411235 },
pages = {10--19},
publisher = {IEEE Computer Society Press},
title = {{McColm Conjecture}},
year = {1994},
abstract = {Gregory McColm conjectured that positive
elementary inductions are bounded in a class $K$ of finite structures
if every $(FO + LFP)$ formula is equivalent to a first-order formula in
$K$. Here $(FO + LFP)$ is the extension of first-order logic with the
least fixed point operator. We disprove the conjecture. Our main
results are two model-theoretic constructions, one deterministic and
the other randomized, each of which refutes McColm's conjecture.},
},
@article{GuSh:526,
author = {Gurevich, Yuri and Shelah, Saharon},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9411236 },
pages = {549--562},
title = {{On Finite Rigid Structures}},
volume = {61},
year = {1996},
abstract = {The main result of this paper is a
probabilistic construction of finite rigid structures. It yields a
finitely axiomatizable class of finite rigid structures where
no $L^\omega_{\infty,\omega}$ formula with counting quantifiers defines
a linear order.},
},
@article{LeSh:527,
author = {Lifsches, Shmuel and Shelah, Saharon},
fromwhere = {Il,Il},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9701219 },
pages = {273--312},
title = {{Random Graphs in the monadic theory of order}},
volume = {38},
year = {1999},
abstract = {We continue the works of Gurevich-Shelah and Lifsches-Shelah
by showing that it is consistent with ZFC that the first-order
theory of random graphs is not interpretable in the monadic theory of
all chains. It is provable from ZFC that the theory of random graphs
is not interpretable in the monadic second order theory of short
chains (hence, in the monadic theory of the real line).},
},
@article{BlSh:528,
author = {Baldwin, John T. and Shelah, Saharon},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.LO/9607226 },
pages = {1359-1376},
title = {{Randomness and Semigenericity}},
volume = {349},
year = {1997},
abstract = {Let $L$ contain only the equality symbol and let $L^+$ be
an arbitrary finite symmetric relational language containing
$L$. Suppose probabilities are defined on finite $L^+$ structures
with ``edge probability'' $n^{-\alpha}$. By $T^\alpha$, the almost
sure theory of random $L^+$-structures we mean the collection
of $L^+$-sentences which have limit probability 1. $T_\alpha$
denotes the theory of the generic structures for $K_\alpha$, (the
collection of finite graphs $G$ with
$\delta_{\alpha}(G)=|G|-\alpha\cdot |\mbox{ edges of G }|$ hereditarily
nonnegative.) \endgraf 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.},
},
@article{HySh:529,
author = {Hyttinen, Tapani and Shelah, Saharon},
fromwhere = {SF,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9202205 },
pages = {1260--1272},
title = {{Constructing strongly equivalent nonisomorphic models for
unsuperstable theories. Part B}},
volume = {60},
year = {1995},
abstract = {We study how equivalent nonisomorphic models of
unsuperstable theories can be. We measure the equivalence by
Ehrenfeucht-Fraisse games. This paper continues [HySh:474].},
},
@article{CuSh:530,
author = {Cummings, James and Shelah, Saharon},
fromwhere = {UK,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9509227 },
pages = {992--1004},
title = {{A model in which every infinite Boolean algebra has many
subalgebras}},
volume = {60},
year = {1995},
abstract = {We show that it is consistent with ZFC (relative to
large cardinals) that every infinite Boolean algebra $B$ has
an irredundant subset $A$ such that $2^{|A|} = 2^{|B|}$. This
implies in particular that $B$ has $2^{|B|}$ subalgebras. We also
discuss some more general problems about subalgebras and free subsets
of an algebra. The result on the number of subalgebras in a
Boolean algebra solves a question of Monk. The paper is intended to
be accessible as far as possible to a general audience, in
particular we have confined the more technical material to a ``black
box'' at the end. The proof involves a variation on Foreman and
Woodin's model in which GCH fails everywhere.},
},
@article{ShSi:531,
author = {Shelah, Saharon and Spinas, Otmar},
fromwhere = {IL,CH},
journal = {Fund. Math.},
note = { arxiv:math.LO/9801151 },
pages = {81--93},
title = {{The distributivity numbers of finite products of ${\cal
P}(\omega)$/fin}},
volume = {158},
year = {1998},
abstract = {Generalizing [ShSi:494], for every $n\lt\omega $ we
construct a ZFC-model where the distributivity number of r.o.$({\cal
P}(\omega)/\hbox{fin})^{n+1}$, ${\bf h}(n+1)$, is smaller than the one
of r.o.$({\cal P}(\omega)/\hbox{fin})^{n}$. 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\lt \omega
$, hence by the first result, consistently they collapse it below ${\bf
h}(n)$},
},
@article{Sh:532,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {in preparation},
title = {{Borel rectangles}},
abstract = {We prove the consistency of the existence
of co-$\aleph_1$-Souslin equivalence relation on ${}^\omega 2$ with
any pregiven $\aleph_\alpha$ class, $\alpha < \omega_1$ but not
a perfect set of pairwise non-equivalent. We deal also
with co-$\kappa$-Souslin relations, equivalence relations,
exact characterizations and $\Pi^1_2$-equivalence relations
and rectangles. \endgraf To 666: problem on equalities of $x$'s; deal
with co-$\kappa$-Souslin deal with the $k$-notation and the
$\alpha$-notation. \endgraf 2012.8.16 In Poland - July- has written a
new try , in order to get a Borel relation with $aleph_alpha$
rectangle but no more, for any countable ordinal $alpha$. What was
written was only the forcing. Yesterday, proofread it, expand - still
has to write kappa-Delta pair proof, and thoughts about more than
$\aleph_y w_1$ But the rank in [522, \S4] seem not OK, will try to
revise},
},
@article{BGSh:533,
author = {Blass, Andreas and Gurevich, Yuri and Shelah, Saharon},
fromwhere = {1,1,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9705225 },
pages = {141--187},
title = {{Choiceless Polynomial Time}},
volume = {100},
year = {1999},
abstract = {Turing machines define polynomial time (PTime) on strings
but cannot deal with structures like graphs directly, and there is
no known, easily computable string encoding of isomorphism classes
of structures. Is there a computation model whose machines do
not distinguish between isomorphic structures and compute exactly
PTime properties? This question can be recast as follows: Does there
exist a logic that captures polynomial time (without presuming the
presence of a linear order)? Earlier, one of us conjectured the
negative answer. The problem motivated a quest for stronger and
stronger PTime logics. All these logics avoid arbitrary choice. Here we
attempt to capture the choiceless fragment of PTime. Our computation
model is a version of abstract state machines (formerly called
evolving algebras). The idea is to replace arbitrary choice with
parallel execution. The resulting logic is more expressive than other
PTime logics in the literature. A more difficult theorem shows that
the logic does not capture all PTime. },
},
@article{RoSh:534,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {PL,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9703218 },
pages = {101--151},
title = {{Cardinal invariants of ultrapoducts of Boolean algebras}},
volume = {155},
year = {1998},
abstract = {We deal with some of problems posed by Monk and related
to cardinal invariant of ultraproducts of Boolean algebras. We
also introduce and investigate some new cardinal invariants.},
},
@article{EiSh:535,
author = {Eisworth, Todd and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9808138 },
pages = {597--618},
title = {{Successors of singular cardinals and coloring theorems. I}},
volume = {44},
year = {2005},
abstract = {We investigate the existence of strong colorings on
successors of singular cardinals. This work continues Section 2
of \cite{Sh:413}, but now our emphasis is on finding colorings of
pairs of ordinals, rather than colorings of finite sets of ordinals.},
},
@inproceedings{GuSh:536,
author = {Gurevich, Yuri and Shelah, Saharon},
booktitle = {Proceedings of the 18th Annual IEEE Symposium on Logic in
Computer Science},
fromwhere = {1,IL},
note = { arxiv:math.LO/0404150 },
pages = {291--300},
title = {{Spectra of Monadic Second-Order Formulas with One Unary
Function}},
year = {2003},
abstract = {We establish the eventual periodicity of the spectrum of any
monadic second-order formula where: (i) all relation symbols, except
equality, are unary, and (ii) there is only one function symbol and
that symbol is unary.},
},
@article{AbSh:537,
author = {Abraham, Uri and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9807178 },
pages = {97--103},
title = {{Lusin sequences under CH and under Martin's Axiom}},
volume = {169},
year = {2001},
abstract = {Assuming the continuum hypothesis there is an
inseparable sequence of length $\omega_1$ that contains no Lusin
subsequence, while if Martin's Axiom and the negation of CH is assumed
then every inseparable sequence (of length $\omega_1$) is a union of
countably many Lusin subsequences.},
},
@article{Sh:538,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9607227 },
title = {{Historic iteration with $\aleph_\varepsilon$-support}},
volume = {accepted},
abstract = {One aim of this work is to get a universe in which
weak versions of Martin axioms holds for some forcing notions
of cardinality $\aleph_0,\aleph_1$ and $\aleph_2$ while on $\aleph_2$
club, the ``small'' brother of diamond, holds. As a consequence we get
the consistency of ``there is no Gross space''. Another aim is to
present a case of ``Historic iteration''.},
},
@article{LeSh:539,
author = {Lifsches, Shmuel and Shelah, Saharon},
fromwhere = {IL, IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9404227 },
pages = {1206-1227},
title = {{Uniformization, choice functions and well orders in the class
of trees}},
volume = {61},
year = {1996},
abstract = {The monadic second-order theory of trees
allows quantification over elements and over arbitrary subsets.
We classify the class of trees with respect to the question: does
a tree $T$ have a definable choice function (by a monadic formula with
parameters)? A natural dichotomy arises where the trees that fall in
the first class don't have a definable choice function and the trees in
the second class have even a definable well ordering of their elements.
This has a close connection to the uniformization problem.},
},
@article{BnSh:540,
author = {Brendle, Joerg and Shelah, Saharon},
trueauthor = {Brendle, J{\"{o}}rg and Shelah, Saharon},
fromwhere = {D, IL},
journal = {Journal of the London Mathematical Society},
note = { arxiv:math.LO/9407207 },
pages = {19--27},
title = {{Evasion and prediction II}},
volume = {53},
year = {1996},
abstract = {A subgroup $G\leq {\Bbb Z}^\omega$ exhibits the
Specker phenomenon if every homomorphism $G \to {\Bbb Z}$ maps almost
all unit vectors to $0$. We give several combinatorial
characterizations of the cardinal ${\frak se}$, the size of the
smallest $G\leq {\Bbb Z}^\omega$ exhibiting the Specker phenomenon. We
also prove the consistency of ${\bf b}\lt {\bf e}$, where ${\bf b}$
is the unbounding number and ${\bf e}$ the evasion number. Our
results answer several questions addressed by Blass.},
},
@article{CuSh:541,
author = {Cummings, James and Shelah, Saharon},
fromwhere = {UK, IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9509228 },
pages = {251--268},
title = {{Cardinal invariants above the continuum}},
volume = {75},
year = {1995},
abstract = {We prove some consistency results about ${\bf b}(\lambda)$
and ${\bf d}(\lambda)$, which are natural generalisations of
the cardinal invariants of the continuum ${\bf b}$ and ${\bf d}$.
We also define invariants ${\bf b}_{\rm cl}(\lambda)$ and ${\bf
d}_{\rm cl}(\lambda)$, and prove that almost always ${\bf b}(\lambda)
= {\bf b}_{\rm cl}(\lambda)$ and ${\bf d}(\lambda)={\bf d}_{\rm
cl}(\lambda)$},
},
@article{Sh:542,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9406219 },
pages = {341--347},
title = {{Large Normal Ideals Concentrating on a Fixed Small
Cardinality}},
volume = {35},
year = {1996},
abstract = {A property of a filter, a kind of large cardinal
property, suffices for the proof in Liu Shelah [LiSh:484] and is
proved consistent as required there. A natural property which
looks better, not only is not obtained here, but is shown to be false.
On earlier related theorems see Gitik Shelah [GiSh310].},
},
@article{FShS:543,
author = {Fuchino, Sakae and Shelah, Saharon and Soukup, Lajos},
trueauthor = {Fuchino, Saka\'e and Shelah, Saharon and Soukup, Lajos},
fromwhere = {J,IL,H},
journal = {Mathematica Japonica},
note = { arxiv:math.LO/9405215 },
pages = {199--206},
title = {{On a theorem of Shapiro}},
volume = {40},
year = {1994},
abstract = {We show that a theorem of Leonid B.\ Shapiro which
was proved under MA, is actually independent from ZFC. We also give a
direct proof of the Boolean algebra version of the theorem under
MA({\it Cohen}).},
},
@article{FShS:544,
author = {Fuchino, Sakae and Shelah, Saharon and Soukup, Lajos},
trueauthor = {Fuchino, Saka\'e and Shelah, Saharon and Soukup, Lajos},
fromwhere = {J,IL,H},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9804153 },
pages = {57--77},
title = {{Sticks and clubs}},
volume = {90},
year = {1997},
abstract = {We study combinatorial principles known as stick and club.
Several variants of these principles and cardinal invariants connected
to them are also considered. We introduce a new kind of side-by-side
product of partial orders which we call pseudo-product. Using such
products, we give several generic extensions where some of these
principles hold together with $\neg CH$ and Martin's Axiom for
countable p.o.-sets. An iterative version of the pseudo-product is used
under an inaccessible cardinal to show the consistency of the club
principle for every stationary subset of limits of $\omega_1$ together
with $\neg CH$ and Martin's Axiom for countable p.o.-sets.},
},
@article{DjSh:545,
author = {Dzamonja, Mirna and Shelah, Saharon},
trueauthor = {D\v{z}amonja, Mirna and Shelah, Saharon},
fromwhere = {BiH,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9601219 },
pages = {289--316},
title = {{Saturated filters at successors of singulars, weak reflection
and yet another weak club principle}},
volume = {79},
year = {1996},
abstract = {Suppose that $\lambda$ is the successor of a
singular cardinal $\mu$ whose cofinality is an uncountable
cardinal $\kappa$. We give a sufficient condition that the club filter
of $\lambda$ concentrating on the points of cofinality $\kappa$ is not
$\lambda^+$-saturated. The condition is phrased in terms of a notion
that we call weak reflection. We discuss various properties of weak
reflection},
},
@article{Sh:546,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9712282 },
pages = {1031--1054},
title = {{Was Sierpi\'nski right? IV}},
volume = {65},
year = {2000},
abstract = {We prove for any $\mu=\mu^{<\mu}<\theta<\lambda,\lambda$
large enough (just strongly inaccessible Mahlo) the consistency
of $2^\mu=\lambda\rightarrow [\theta]^2_3$ and even
$2^\mu=\lambda\rightarrow [\theta]^2_{\sigma,2}$ for $\sigma<\mu$. The
new point is that possibly $\theta>\mu^+$.},
},
@incollection{GbSh:547,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
booktitle = {Abelian groups, module theory, and topology (Padua, 1997)},
fromwhere = {D,IL},
note = { arxiv:math.RA/0011186 },
pages = {235--248},
publisher = {Dekker, New York},
series = {Lecture Notes in Pure and Appl. Math.},
title = {{Endomorphism Rings of Modules whose cardinality is cofinal
to $\omega$}},
volume = {201},
year = {1998},
abstract = {The main result is Theorem: Let $A$ be an $R$-algebra,
$\mu,\lambda$ be cardinals such that
$|A|\leq\mu=\mu^{\aleph_0}<\lambda\leq 2^\mu$. If $A$
is $\aleph_0$-cotorsion-free or $A$ is countably free,
respectively, then there exists an $\aleph_0$-cotorsion-free or a
separable (reduced, torsion-free) $R$-module $G$ respectively of
cardinality $|G|=\lambda$ with ${\rm End}_R G=A\oplus{\rm Fin} G$.},
},
@article{Sh:548,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Random Structures \& Algorithms},
note = { arxiv:math.LO/9606230 },
pages = {351-358},
title = {{Very weak zero one law for random graphs with order and random
binary functions}},
volume = {9},
year = {1996},
abstract = {Let $G_<(n,p)$ denote the usual random graph $G(n,p)$ on
a totally ordered set of $n$ vertices. We will fix $p=\frac{1}{2}$ for
definiteness. Let $L^<$ denote the first order language with predicates
equality $(x=y)$, adjacency $(x\sim y)$ and less than $(x 0$ then it
has probability $0(e^{-n^\varepsilon})$ for some $\varepsilon > 0$}},
volume = {82},
year = {1996},
abstract = {Shelah Spencer [ShSp:304] proved the $0-1$ law for the
random graphs $G(n,p_n)$, $p_n=n^{-\alpha}$, $\alpha\in (0,1)$
irrational (set of nodes in $[n]=\{1,\ldots,n\}$, the edges are
drawn independently, probability of edge is $p_n$). One may wonder what
can we say on sentences $\psi$ for which
Prob$(G(n,p_n)\models\psi)$ converge to zero, Lynch asked the question
and did the analysis, getting (for every $\psi$): \endgraf EITHER
[$(\alpha)$] Prob$[G(n,p_n)\models\psi]=cn^{-\beta}
+ O(n^{-\beta-\varepsilon})$ for some $\varepsilon$ such
that $\beta>\varepsilon>0$ \endgraf OR [$(\beta)$]
Prob$(G(n,p_n)\models\psi)= O(n^{-\varepsilon})$ for every
$\varepsilon>0$. \endgraf Lynch conjectured that in case $(\beta)$ we
have \endgraf [$(\beta^+)$] Prob$(G(n,p_n)\models\psi)=
O(e^{-n^\varepsilon})$ for some $\varepsilon>0$. We prove it here.},
},
@incollection{Sh:552,
author = {Shelah, Saharon},
booktitle = {Advances in Algebra and Model Theory. Editors: Manfred
Droste and Ruediger Goebel},
fromwhere = {IL},
note = { arxiv:math.LO/9609217 },
pages = {229--286},
publisher = {Gordon and Breach},
series = {Algebra, Logic and Applications},
title = {{Non-existence of universals for classes like reduced torsion
free abelian groups under embeddings which are not necessarily pure}},
volume = {9},
year = {1997},
abstract = {We consider a class $K$ of structures e.g. trees
with $\omega+1$ levels, metric spaces and mainly, classes of
Abelian groups like the one mentioned in the title and the class
of reduced separable (Abelian) $p$-groups. We say $M\in K$ is universal
for $K$ if any member $N$ of $K$ of cardinality not bigger than the
cardinality of $M$ can be embedded into $M$. This is a natural, often
raised, problem. We try to draw consequences of cardinal arithmetic to
non--existence of universal members for such natural classes.},
},
@article{SaSh:553,
author = {Shafir, Ofer and Shelah, Saharon},
trueauthor = {Shafir, Ofer and Shelah, Saharon},
fromwhere = {IL, IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9711220 },
pages = {1823--1832},
title = {{More on entangled orders}},
volume = {65},
year = {2000},
abstract = {This paper grew as a continuation of [Sh462] but in the
present form it can serve as a motivation for it as well. We deal with
the same notions, and use just one simple lemma from there.
Originally entangledness was introduced in order to get narrow Boolean
algebras and examples of the nonmultiplicativity of c.c-ness.
These applications became marginal when the hope to extract new
such objects or strong colourings were not materialized, but after
the pcf constructions which made their debut in [Sh:g] it seems
that this notion gained independence. Generally we aim at
characterizing the existence strong and weak entangled orders in
cardinal arithmetic terms. In [Sh462] necessary conditions were shown
for strong entangledness which in a previous version was
erroneously proved to be equivalent to plain entangledness. In section
1 we give a forcing counterexample to this equivalence and in section 2
we get those results for entangledness (certainly the most
interesting case). In \S3 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.},
},
@article{GoSh:554,
author = {Goldstern, Martin and Shelah, Saharon},
trueauthor = {Goldstern, Martin and Shelah, Saharon},
fromwhere = {A, IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9707202 },
pages = {255--265},
title = {{A Partial Order Where All Monotone Maps Are Definable}},
volume = {152},
year = {1997},
abstract = {We show the consistency of ``There is a p.o.\ of
size continuum on which all monotone maps are first order definable''.
The continuum can be $aleph_1$ or larger, and we may even have Martin's
axiom.},
},
@article{SmSh:555,
author = {Scheepers, Marion and Shelah, Saharon},
trueauthor = {Scheepers, Marion and Shelah, Saharon},
fromwhere = {1, IL},
journal = {in preparation},
title = {{Embeddings of partial orders into $\omega^\omega$}},
},
@article{FKSh:556,
author = {Fuchino, Sakae and Koppelberg, Sabine and Shelah, Saharon},
trueauthor = {Fuchino, Saka\'e and Koppelberg, Sabine and Shelah,
Saharon},
fromwhere = {D,D,IL},
journal = {Topology and its Applications},
note = {A special issue: Proceedings of Matsuyama Topological
Conference. arxiv:math.LO/9505212 },
pages = {141-148},
title = {{A game on partial orderings}},
volume = {74},
year = {1996},
abstract = {We study the determinacy of the game
$G_\kappa(A)$ introduced in [FKSh:549] for uncountable regular $\kappa$
and several classes of partial orderings $A$. Among trees or
Boolean algebras, we can always find an $A$ such that $G_\kappa(A)$
is undetermined. For the class of linear orders, the existence of such
$A$ depends on the size of $\kappa^{\lt \kappa}$. In particular we
obtain a characterization of $\kappa^{\lt \kappa}=\kappa$ in terms of
determinacy of the game $G_\kappa(L)$ for linear orders $L$.},
},
@article{NShS:557,
author = {Niedermeyer, Frank and Shelah, Saharon and Steffens, Karsten},
trueauthor = {Niedermeyer, Frank and Shelah, Saharon and Steffens,
Karsten},
fromwhere = {D, IL, D},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.XX/012454 },
pages = {665--672},
title = {{The $f$-Factor Problem for Graphs and the Hereditary
Property}},
volume = {45},
year = {2006},
abstract = {If $P$ is a hereditary property then we show that, for
the existence of a perfect $f$-factor, $P$ is a sufficient condition
for countable graphs and yields a sufficient condition for graphs
of size $\aleph_1$. Further we give two examples of a
hereditary property which is even necessary for the existence of a
perfect $f$-factor. We also discuss the $\aleph_2$-case.},
},
@article{GeSh:558,
author = {Geschke, Stefan and Shelah, Saharon},
trueauthor = {Geschke, Stefan and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0702600 },
pages = {151--164},
title = {{The number of openly generated Boolean algebras}},
volume = {73},
year = {2008},
abstract = {This article is devoted to two different generalizations of
projective Boolean algebras: openly generated Boolean algebras and
tightly $\sigma$-filtered Boolean algebras. \endgraf 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. \endgraf 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. \endgraf 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 $\kappa$.},
},
@incollection{EkSh:559,
author = {Eklof, Paul C. and Shelah, Saharon},
trueauthor = {Eklof, Paul C. and Shelah, Saharon},
booktitle = {Abelian Groups and Modules},
fromwhere = {1,IL},
note = {ed. by Arnold \& Rangaswamy},
pages = {15--22},
publisher = {Marcel Dekker},
title = {{New non-free Whitehead groups by coloring}},
volume = {Mistake in proof --- corrected version in [EkSh 559a]},
year = {1996},
abstract = {We show that it is consistent that there is a
strongly $\aleph_{1}$-free $\aleph_{1}$-coseparable group of
cardinality $\aleph_{1}$ which is not $\aleph_{1}$-separable.},
},
@article{EkSh:559a,
author = {Eklof, Paul C. and Shelah, Saharon},
trueauthor = {Eklof, Paul C. and Shelah, Saharon},
fromwhere = {1, IL},
journal = {preprint},
note = { arxiv:math.LO/9711221 },
title = {{New non-free Whitehead groups (corrected version)}},
abstract = {We show that it is consistent that there is a
strongly $\aleph_1$-free $\aleph_1$-coseparable group of
cardinality $\aleph_1$ which is not $\aleph_1$-separable.},
},
@article{LwSh:560,
author = {Laskowski, Michael C. and Shelah, Saharon},
trueauthor = {Laskowski, Michael C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0011167 },
pages = {69--88},
title = {{The Karp complexity of unstable classes}},
volume = {40},
year = {2001},
abstract = {A class ${\bf K}$ of structures is controlled if, for
all cardinals $\lambda$, the relation
of $L_{\infty,\lambda}$-equivalence partitions ${\bf K}$ into a set
of equivalence classes (as opposed to a proper class). We prove
that the class of doubly transitive linear orders is controlled,
while any pseudo-elementary class with the $\omega$-independence
property is not controlled.},
},
@article{ShZa:561,
author = {Shelah, Saharon and Zapletal, Jindrich},
trueauthor = {Shelah, Saharon and Zapletal, Jind\v{r}ich},
fromwhere = {IL,1},
journal = {Advances in Mathematics},
note = { arxiv:math.LO/9502230 },
pages = {93--118},
title = {{Embeddings of Cohen algebras}},
volume = {126},
year = {1997},
abstract = {Complete Boolean algebras proved to be an important tool
in topology and set theory. Two of the most prominent examples
are $B(\kappa)$, the algebra of Borel sets modulo measure zero ideal
in the generalized Cantor space $\{0,1\}^\kappa$ equipped with
product measure, and $C(\kappa),$ the algebra of regular open sets in
the space $\{0,1\}^\kappa$, for $\kappa$ an infinite cardinal.
$C(\kappa)$ is much easier to analyse than $B(\kappa)$:
$C(\kappa)$ has a dense subset of size $\kappa$, while the density
of $B(\kappa)$ depends on the cardinal characteristics of the
real line; and the definition of $C(\kappa)$ is simpler.
Indeed, $C(\kappa)$ seems to have the simplest definition among all
algebras of its size. In the Main Theorem of this paper we show that in
a certain precise sense, $C(\aleph_1)$ has the simplest structure among
all algebras of its size, too. \endgraf 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 $B$''.},
},
@article{DjSh:562,
author = {Dzamonja, Mirna and Shelah, Saharon},
trueauthor = {D\v{z}amonja, Mirna and Shelah, Saharon},
fromwhere = {BiH,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9510216 },
pages = {165--198},
title = {{On squares, outside guessing of clubs and $I_{0$)) for them we have NDOP when one side comes from this
type, then use a decomposion theorem with zero and two successors},
},
@article{BGSh:570,
author = {Baldwin, John and Grossberg, Rami and Shelah, Saharon},
trueauthor = {Baldwin, John and Grossberg, Rami and Shelah, Saharon},
fromwhere = {1,1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9511205 },
pages = {678--684},
title = {{Transfering saturation, the finite cover property,
and stability}},
volume = {64},
year = {2000},
abstract = {Saturation is $(\mu,\kappa)$-transferable in $T$ if and only
if there is an expansion $T_1$ of $T$ with $|T_1|=|T|$ such that if
$M$ is a $\mu$-saturated model of $T_1$ and $|M|\geq\kappa$ then
the reduct $M\restriction L(T)$ is $\kappa$-saturated. We
characterize theories which are superstable without f.c.p., or without
f.c.p. as, respectively those where saturation
is $(\aleph_0,\lambda)$-transferable
or $(\kappa(T),\lambda)$-transferable for all $\lambda$. Further if
for some $\mu\geq |T|$, $2^\mu>\mu^+$, stability is equivalent to
for all $\mu\geq |T|$, saturation is $(\mu,2^\mu)$-transferable.},
},
@article{CDSh:571,
author = {Cummings, James and Dzamonja, Mirna and Shelah, Saharon},
trueauthor = {Cummings, James and D\v{z}amonja, Mirna and Shelah,
Saharon},
fromwhere = {UK,1,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9504221 },
pages = {91--100},
title = {{A consistency result on weak reflection}},
volume = {148},
year = {1995},
abstract = {In this paper we study the notion of strong non-reflection,
and its contrapositive weak reflection. We say $\theta$
strongly non-reflects at $\lambda$ iff there is a
function $F:\theta\longrightarrow\lambda$ such that for all
$\alpha<\theta$ with $cf(\alpha)=\lambda$ there is $C$ club in $\alpha$
such that $F\restriction C$ is strictly increasing. We prove that it
is consistent to have a cardinal $\theta$ such that
strong non-reflection and weak reflection each hold on an unbounded set
of cardinals less than $\theta$.},
},
@article{Sh:572,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9609218 },
pages = {153-174},
title = {{Colouring and non-productivity of $\aleph_2$-cc}},
volume = {84},
year = {1997},
abstract = {We prove that colouring of pairs from $\aleph_2$ with strong
properties exists. The easiest to state (and quite a well known
problem) it solves: there are two topological spaces with cellularity
$\aleph_1$ whose product has cellularity $\aleph_2$; equivalently we
can speak on cellularity of Boolean algebras or on Boolean algebras
satisfying the $\aleph_2$-c.c. whose product fails the $\aleph_2$-c.c.
We also deal more with guessing of clubs.},
},
@article{LeSh:573,
author = {Lifsches, Shmuel and Shelah, Saharon},
trueauthor = {Lifsches, Shmuel and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9412231 },
pages = {103--127},
title = {{Uniformization and Skolem Functions in the Class of Trees}},
volume = {63},
year = {1998},
abstract = {The monadic second-order theory of trees
allows quantification over elements and over arbitrary subsets.
We classify the class of trees with respect to the question: does a
tree T have definable Skolem functions (by a monadic formula with
parameters)? This continues [LiSh539] where the question was asked
only with respect to choice functions. Here we define a subclass of the
class of tame trees (trees with a definable choice function) and prove
that this is exactly the class (actually set) of trees with definable
Skolem functions.},
},
@article{DjSh:574,
author = {Dzamonja, Mirna and Shelah, Saharon},
trueauthor = {D\v{z}amonja, Mirna and Shelah, Saharon},
fromwhere = {BiH,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9710215 },
pages = {180--198},
title = {{Similar but not the same: various versions of $\clubsuit$ do
not coincide}},
volume = {64},
year = {1999},
abstract = {We consider various versions of the $\clubsuit$ principle.
This principle is a known consequence of $\diamondsuit$. It is well
known that $\diamondsuit$ is not sensitive to minor changes in
its definition, e.g. changing the guessing requirement form
``guessing exactly'' to ``guessing modulo a finite set''. We show
however, that this is not true for $\clubsuit$. We consider some
other variants of $\clubsuit$ as well.},
},
@article{Sh:575,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematica},
note = { arxiv:math.LO/9508221 },
pages = {153--208},
title = {{Cellularity of free products of Boolean algebras (or
topologies)}},
volume = {166},
year = {2000},
abstract = {We answer Problem 1 of Monk if there are Boolean
algebras $B_1,B_2$ such that $c(B_i)\leq\lambda_i$ but $c(B_1\times
B_2)> \lambda_1+\lambda_2$ where $\lambda_1=\mu$ is singular and
$\mu>\lambda_2=\theta>cf(\mu)$},
},
@article{Sh:576,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9805146 },
pages = {29--128},
title = {{Categoricity of an abstract elementary class in two
successive cardinals}},
volume = {126},
year = {2001},
abstract = {We investigate categoricity of abstract elementary
classes without any remnants of compactness (like non-definability of
well ordering, existence of E.M. models or existence of large
cardinals). We prove (assuming a weak version of GCH around $\lambda$)
that if ${\frak K}$ is categorical in $\lambda,\lambda^+$,
$LS({\frak K}) \le\lambda$ and $1\le I(\lambda^{++},{\frak K})\lt
2^{\lambda^{++}}$ then ${\frak K}$ has a model in $\lambda^{+++}$.},
},
@article{GiSh:577,
author = {Gitik, Moti and Shelah, Saharon},
trueauthor = {Gitik, Moti and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/9503203 },
pages = {1523--1530},
title = {{Less saturated ideals}},
volume = {125},
year = {1997},
abstract = {We prove the following: \endgraf (1) If $\kappa$ is weakly
inaccessible then $NS_\kappa$ is not $\kappa^+$-saturated.
\endgraf (2) If $\kappa$ is weakly inaccessible and $\theta<\kappa$
is regular then $NS^\theta_\kappa$ is not $\kappa^+$-saturated.
\endgraf (3) If $\kappa$ is singular then $NS^{cf(\kappa)}_{\kappa^+}$
is not $\kappa^{++}$-saturated. \endgraf 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.},
},
@incollection{MlSh:578,
author = {Milner, Eric C. and Shelah, Saharon},
trueauthor = {Milner, Eric C. and Shelah, Saharon},
booktitle = {Set Theory: Techniques and Applications, (J. Bagaria, C.
Di Prisco, J. Larson, A.R.D. Mathias, eds.)},
fromwhere = {3,IL},
note = { arxiv:math.LO/9708210 },
pages = {175--182},
publisher = {Kluwer Acad. Publ.},
title = {{A tree--arrowing graph}},
year = {1998},
abstract = {We answer a variant of a question of R{\"{o}}dl 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.},
},
@incollection{GbSh:579,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
booktitle = {Proceedings of the Conference on Abelian Groups in Colorado
Springs, 1995},
fromwhere = {D,IL},
note = { arxiv:math.GR/0011185 },
pages = {253--271},
publisher = {Marcel Dekker},
series = {Lecture Notes in Pure and Applied Math.},
title = {{G.C.H. implies existence of many rigid almost free abelian
groups}},
volume = {182},
year = {1996},
abstract = {We begin with the existence of groups with trivial duals
for cardinals $\aleph_n$ ($n\in \omega$). Then we derive results
about strongly $\aleph_n$-free abelian groups of cardinality
$\aleph_n$ ($n\in\omega$) with prescribed free, countable
endomorphism ring. Finally we use combinatorial results of [Sh:108],
[Sh:141] to give similar answers for cardinals $>\aleph_\omega$. As in
Magidor and Shelah [MgSh:204], a paper concerned with the existence
of $\kappa $-free, non-free abelian groups of cardinality $\kappa$, the
induction argument breaks down at $\aleph_\omega$. Recall
that $\aleph_\omega$ is the first singular cardinal and such groups
of cardinality $\aleph_\omega$ do not exist by the well-known
Singular Compactness Theorem (see [Sh:52]).},
},
@article{Sh:580,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9604243 },
pages = {87--107},
title = {{Strong covering without squares}},
volume = {166},
year = {2000},
abstract = {We continue \cite[Ch XIII]{Sh:b} and \cite{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 ${\cal M} \in V$ 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 $V\models |Y|<\kappa$.
\endgraf 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).
},
},
@article{Sh:581,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL,IL},
journal = {in preparation},
title = {{When 0--1 law hold for $G_{n,\bar{p}}$, $\bar{p}$ monotonic}},
},
@article{GiSh:582,
author = {Gitik, Moti and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9507208 },
pages = {221--242},
title = {{More on real-valued measurable cardinals and forcing with
ideals}},
volume = {124},
year = {2001},
abstract = {(1) It is shown that if $c$ is real-valued measurable then
the Maharam type of $(c, {\cal P}(c),\sigma)$ is $2^c$. This answers
a question of D. Fremlin. \endgraf (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. \endgraf (3) The
forcing with a $\kappa$-complete ideal over a set $X$, $|X|\geq\kappa$
cannot be isomorphic to Random$\times$Cohen or Cohen$\times$Random. 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 $X$.},
},
@article{GcSh:583,
author = {Gilchrist, Martin and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9603219 },
pages = {1151--1160},
title = {{The Consistency of ${\rm
ZFC}+2^{\aleph_{0}}>\aleph_{\omega}+ {\cal I}(\aleph_2)={\cal
I}(\aleph_{\omega})$}},
volume = {62},
year = {1997},
abstract = {An $\omega$-coloring is a pair $\langle f,B\rangle$
where $f:[B]^{2}\longrightarrow\omega$. The set $B$ is the field of
$f$ and denoted $Fld(f)$. Let $f,g$ be $\omega$-colorings. We say
that $f$ realizes the coloring $g$ if there is a one-one
function $k:Fld(g)\longrightarrow Fld(f)$ such that for all
$\{x,y\}$, $\{u,v\}\in dom(g)$ we have $f(\{k(x),k(y)\})\neq
f(\{k(u),k(v)\}) \Rightarrow g(\{x,y\})\neq g(\{u,v\})$. We write
$f\sim g$ if $f$ realizes $g$ and $g$ realizes $f$. We call the
$\sim$-classes of $\omega$-colorings with finite fields identities. We
say that an identity $I$ is of size $r$ if $|Fld(f)|=r$ for some/all
$f\in I$. For a cardinal $\kappa$ and
$f:[\kappa]^2\longrightarrow\omega$ we define ${\cal I}(f)$ to be the
collection of identities realized by $f$ and ${\cal I }(\kappa)$ to be
$\bigcap\{{\cal I}(f)| f:[\kappa]^2\longrightarrow\omega\}$.
\endgraf We show that, if ZFC is consistent then ZFC +
$2^{\aleph_0}>\aleph_\omega + {\cal I}(\aleph_2)={\cal
I}(\aleph_\omega)$ is consistent.},
},
@article{ShST:584,
author = {Shelah, Saharon and Saxl, Jan and Thomas, Simon},
fromwhere = {IL,1},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.IG/9605202 },
pages = {4611-4641},
title = {{Infinite products of finite simple groups}},
volume = {348},
year = {1996},
abstract = {We classify the sequences $\langle S_{n}: n\in N\rangle$
of finite simple nonabelian groups such that $\prod_n S_n$
has uncountable cofinality.},
},
@article{RbSh:585,
author = {Rabus, Mariusz and Shelah, Saharon},
fromwhere = {3,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9706223 },
pages = {229--240},
title = {{Covering a function on the plane by two continuous
functions on an uncountable square - the consistency}},
volume = {103},
year = {2000},
abstract = {It is consistent that for every function $f:{\Bbb R}\times
{\Bbb R}\rightarrow {\Bbb R}$ there is an uncountable set $A\subseteq
{\Bbb R}$ and two continuous functions $f_0,f_1:D(A)\rightarrow {\Bbb
R}$ such that
$f(\alpha,\beta)\in \{f_0(\alpha,\beta),f_1(\alpha,\beta)\}$ for every
$(\alpha,\beta) \in A^2$, $\alpha\not =\beta$.},
},
@article{Sh:586,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9706224 },
pages = {153-160},
title = {{A polarized partition relation and failure of GCH }},
volume = {155},
year = {1998},
abstract = {The main result is that for $\lambda$ strong limit
singular failing the continuum hypothesis (i.e. $2^\lambda>
\lambda^+$), a polarized partition theorem holds.},
},
@article{Sh:587,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9707225 },
pages = {29--115},
title = {{Not collapsing cardinals $\leq\kappa$ in $(<\kappa)$--support
iterations}},
volume = {136},
year = {2003},
abstract = {We deal with the problem of preserving various versions
of completeness in $(<\kappa)$--support iterations of forcing
notions, generalizing the case ``$S$--complete proper is preserved by
CS iterations for a stationary co-stationary $S\subseteq\omega_1$''.
We give applications to Uniformization and the Whitehead problem.
In particular, for a strongly inaccessible cardinal $\kappa$ and
a stationary set $S\subseteq\kappa$ with fat complement we can
have uniformization for $\langle A_\delta:\delta\in S'\rangle$,
$A_\delta \subseteq\delta=\sup A_\delta$, $cf(\delta)=otp(A_\delta)$
and a stationary non-reflecting set $S'\subseteq S$.},
},
@article{Sh:588,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Periodica Mathematica Hungarica},
pages = {131--137},
title = {{Large weight does not yield an irreducible base}},
volume = {66},
year = {2013},
abstract = {Answering a question of Juh\'asz, Soukup and SzentmiklÃ³ssy
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.},
},
@article{Sh:589,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9804155 },
pages = {1624--1674},
title = {{Applications of PCF theory}},
volume = {65},
year = {2000},
abstract = {We deal with several pcf problems; we characterize
another version of exponentiation: number of $\kappa$-branches in a
tree with $\lambda$ nodes, deal with existence of independent sets
in stable theories, possible cardinality of ultraproduct, the depth
of ultraproducts of Boolean Algebras. Also we give cardinal
invariant for each $\lambda$ with pcf restriction and investigate
further $T_D(f)$.},
},
@article{Sh:590,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9705226 },
pages = {137--151},
title = {{On a problem of Steve Kalikow}},
volume = {166},
year = {2000},
abstract = {The Kalikow problem for a pair $(\lambda,\kappa)$ of
cardinal numbers, $\lambda >\kappa$ (in particular $\kappa=2$) is
whether we can map the family of $\omega$--sequences from $\lambda$ to
the family of $\omega$--sequences from $\kappa$ in a very
continuous manner. Namely, we demand that for
$\eta,\nu\in\lambda^\omega$ we have: \endgraf $\eta,\nu$ are almost
equal if and only if their images are. \endgraf 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.},
},
@article{GbSh:591,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Canadian Journal of Mathematics},
note = { arxiv:math.RA/0011182 },
pages = {719--738},
title = {{Indecomposable almost free modules - the local case}},
volume = {50},
year = {1998},
abstract = {Let $R$ be a countable, principal ideal domain which is not
a field and $A$ be a countable $R$-algebra which is free as
an $R$-module. Then we will construct an $\aleph_1$-free $R$-module
$G$ of rank $\aleph_1$ with endomorphism algebra End$_RG=A$. Clearly
the result does not hold for fields. Recall that an $R$-module
is $\aleph_1$-free if all its countable submodules are free,
a condition closely related to Pontryagin's theorem. This result
has many consequences, depending on the algebra $A$ in use.
For instance, if we choose $A=R$, then clearly $G$ is an
indecomposable `almost free' module. The existence of such modules was
unknown for rings with only finitely many primes like $R={\mathbb
Z}_{(p)}$, the integers localized at some prime $p$. The result
complements a classical realization theorem of Corner's showing that
any such algebra is an endomorphism algebra of some torsion-free,
reduced $R$-module $G$ of countable rank. Its proof is based on
new combinatorial-algebraic techniques related with what we call
rigid tree-elements coming from a module generated over a forest
of trees.},
},
@article{Sh:592,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9810181 },
pages = {109--136},
title = {{Covering of the null ideal may have countable cofinality}},
volume = {166},
year = {2000},
abstract = {We prove that it is consistent that the covering of the
ideal of measure zero sets has countable cofinality.},
},
@article{FMShV:593,
author = {Fuchino, Sakae and Mildenberger, Heike and Shelah, Saharon and
Vojtas, Peter},
trueauthor = {Fuchino, Saka\'e and Mildenberger, Heike and Shelah,
Saharon and Peter Vojt\'a\v{s}},
fromwhere = {J,D,IL,SL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9903114 },
pages = {255--268},
title = {{On absolutely divergent series}},
volume = {160},
year = {1999},
abstract = {We show that in the $\aleph_2$-stage countable
support iteration of Mathias forcing over a model of CH the
complete Boolean algebra generated by absolutely divergent series
under eventual dominance is not isomorphic to the completion
of $P(\omega)/$fin. This complements Vojtas' result, that
under $cf(c)=p$ the two algebras are isomorphic.},
},
@incollection{Sh:594,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
booktitle = {Proceedings of the Logic Colloquium Haifa'95},
fromwhere = {IL},
note = { arxiv:math.LO/9611221 },
pages = {305--324},
publisher = {Springer},
series = {Lecture Notes in Logic},
title = {{There may be no nowhere dense ultrafilter}},
volume = {11},
year = {1998},
abstract = {We show the consistency of ZFC + ``there is no
NWD-ultrafilter on $\omega$'', which means: for every non principle
ultrafilter $D$ on the set of natural numbers, there is a function $f$
from the set of natural numbers to the reals, such that for some
nowhere dense set $A$ of reals, the set $\{n: f(n)\in A\}$ is not in
$D$. This answers a question of van Douwen, which was put in more
general context by Baumgartner},
},
@article{Sh:595,
author = {Shelah, Saharon},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9508201 },
pages = {83--86},
title = {{Embedding Cohen algebras using pcf theory}},
volume = {166},
year = {2000},
abstract = {Using a theorem from pcf theory, we show that for any
singular cardinal $\nu$, the product of the Cohen forcing notions
on $\kappa$, $\kappa<\nu$ adds a generic for the Cohen forcing
notion on $\nu^+$. This solves Problem 5.1 in Miller's list (attributed
to Rene David and Sy Friedman).},
},
@article{CuSh:596,
author = {Cummings, James and Shelah, Saharon},
trueauthor = {Cummings, James and Shelah, Saharon},
fromwhere = {UK,IL},
journal = {Journal of the London Mathematical Society},
note = { arxiv:math.LO/9703219 },
pages = {37--49},
title = {{Some independence results on reflection}},
volume = {59},
year = {1999},
abstract = {We prove that there is a certain degree of
independence between stationary reflection phenomena at
different cofinalities; e.g. it is consistent that every stationary
subset of $S_1^3$ reflects at a point of cofinality $\aleph_2$
while every stationary subset of $S^3_0$ has a
non-reflecting stationary subset},
},
@article{GiSh:597,
author = {Gitik, Moti and Shelah, Saharon},
trueauthor = {Gitik, Moti and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Topology and its Applications},
note = { arxiv:math.LO/9603206 },
pages = {219--237},
title = {{On densities of box products}},
volume = {88},
year = {1998},
abstract = {We construct two universes $V_1, V_2$ satisfying the
following: GCH below $\aleph_\omega$,
$2^{\aleph_\omega}=\aleph_{\omega+2}$ and the topological density of
the space ${}^{\aleph_\omega} 2$ with $\aleph_0$ box product topology
$d_{<\aleph_1}(\aleph_\omega)$ is $\aleph_{\omega+1}$ in $V_1$ and
$\aleph_{\omega+2}$ in $V_2$. Further related results are discussed as
well.},
},
@article{AbSh:598,
author = {Abraham, Uri and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0404151 },
pages = {518--532},
title = {{Ladder gaps over stationary sets}},
volume = {69, 2},
year = {2004},
abstract = {For a stationary set $S\subseteq \omega_1$ and a ladder
system $C$ over $S$, a new type of gaps called $C$-Hausdorff is
introduced and investigated. We describe a forcing model of ZFC in
which, for some stationary set $S$, for every ladder $C$ over $S$,
every gap contains a subgap that is $C$-Hausdorff. But for every ladder
$E$ over $\omega_1\setminus S$ there exists a gap with no subgap that
is $E$-Hausdorff. A new type of chain condition, called polarized
chain condition, is introduced. We prove that the iteration with
finite support of polarized c.c.c posets is again a polarized
c.c.c poset.},
},
@article{RoSh:599,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {PL,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9808056 },
pages = {1--37},
title = {{More on cardinal invariants of Boolean algebras}},
volume = {103},
year = {2000},
abstract = {We address several questions of Donald Monk related
to irredundance and spread of Boolean algebras, gaining both some ZFC
knowledge and consistency results. We show in ZFC that $irr(B_0\times
B_1)=\max\{irr(B_0),irr(B_1)\}$. We prove consistency of the statement
``there is a Boolean algebra $B$ such that $irr(B)\lt s(B\otimes B)$''
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 $irr(B_0)\lt inc(B_0)$ 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 $\lambda\notin{\rm
Depth}_{\rm Hs}(B)$.},
},
@inbook{Sh:600,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
booktitle = {Classification Theory for Abstract Elementary Classes},
fromwhere = {IL},
note = { arxiv:math.LO/0011215 },
title = {{Categoricity in abstract elementary classes: going up
inductively}},
abstract = { We deal with beginning stability theory for ``reasonable''
non-elementary classes without any remnants of compactness like
dealing with models above Hanf number or by the class being definable
by $\Bbb L_{\omega_1,\omega}$. We introduce and investigate good
$\lambda$-frame, show that they can be found under reasonable
assumptions and prove we can advance from $\lambda$ to $\lambda^+$ when
non-structure fail. That is, assume $2^{\lambda^{+n}} <
2^{\lambda^{+n+1}}$ for $n < \omega$. So if an a.e.c. is cateogorical
in $\lambda,\lambda^+$ and has intermediate number of models
in $\lambda^{++}$ and $2^\lambda < 2^{\lambda^+} <
2^{\lambda^{++}}$, LS$({\frak K}) \le \lambda)$. Then there is a good
$\lambda$-frame ${\frak s}$ and if ${\frak s}$ fails non-structure in
$\lambda^{++}$ then ${\frak s}$ has a successor ${\frak s}^+$, a
good $\lambda^+$-frame hence $K^{\frak s}_{\lambda^{+3}} \ne
\emptyset$, and we can continue.},
},
@article{KKSh:601,
author = {Kuhlmann, Franz--Viktor and Kuhlmann, Salma and Shelah,
Saharon},
trueauthor = {Kuhlmann, Franz--Viktor and Kuhlmann, Salma and Shelah,
Saharon},
fromwhere = {D,D,IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.RA/9608214 },
pages = {3177--3183},
title = {{Exponentiation in power series fields}},
volume = {125},
year = {1997},
abstract = {We prove that for no nontrivial ordered abelian group $G$,
the ordered power series field $R((G))$ admits an exponential, i.e.
an isomorphism between its ordered additive group and its
ordered multiplicative group of positive elements, but that there is
a non-surjective logarithm. For an arbitrary ordered field $k$,
no exponential on $k((G))$ is compatible, that is, induces
an exponential on $k$ through the residue map. This is proved
by showing that certain functional equations for lexicographic
powers of ordered sets are not solvable.},
},
@article{HySh:602,
author = {Hyttinen, Tapani and Shelah, Saharon},
fromwhere = {SF,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9709229 },
pages = {634--642},
title = {{Constructing strongly equivalent nonisomorphic models
for unsuperstable theories, Part C}},
volume = {64},
year = {1999},
abstract = {In this paper we prove a strong nonstructure theorem
for $\kappa(T)$-saturated models of a stable theory $T$ with dop.},
},
@incollection{Sh:603,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
booktitle = {Proceedings of the 11 International Congress of
Logic, Methodology and Philosophy of Science, Krakow August'99; In the
Scope of Logic, Methodology and Philosophy of Science},
fromwhere = {IL},
note = { arxiv:math.LO/9906023 },
pages = {29--53},
publisher = {Kluwer Academic Publishers},
title = {{Few non minimal types and non-structure}},
volume = {1},
year = {2002},
abstract = {We pay two debts from \cite{Sh:576}. The main demands little
knowledge from \cite{Sh:576}, just quoting a model theoretic
consequence of the weak diamond. We assume that ${\frak K}$ 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 \cite{Sh:576} of $\lambda \ne \aleph_0$, it is minor as for this
case by \cite{Sh:88} we ``usually'' know more.},
},
@incollection{Sh:604,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
booktitle = {Proceedings of LC'2001},
fromwhere = {IL},
note = { arxiv:math.LO/0404240 },
pages = {402--433},
publisher = {ASL},
series = {Lecture Notes in Logic},
title = {{The pair $(\aleph_n,\aleph_0)$ may fail
$\aleph_0$--compactness}},
volume = {20},
year = {2005},
abstract = {Let $P$ be a distinguished unary predicate and $K=\{M:$ $M$
a model of cardinality $\aleph_n$ with $P^M$ of
cardinality $\aleph_0\}$. We prove that consistently for $n=4$, for
some countable first order theory $T$ we have: $T$ has no model in
$K$ whereas every finite subset of $T$ has a model in $K$. We then
show how we prove it also for $n=2$, too.},
},
@article{ShTr:605,
author = {Shelah, Saharon and Truss, John},
trueauthor = {Shelah, Saharon and Truss, John},
fromwhere = {UK,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9805147 },
pages = {47--83},
title = {{On distinguishing quotients of symmetric groups}},
volume = {97},
year = {1999},
abstract = {A study is carried out of the elementary theory of quotients
of symmetric groups in a similar spirit to [Sh:24]. Apart from
the trivial and alternating subgroups, the normal subgroups of the
full symmetric group $S(\mu)$ on an infinite cardinal $\mu$ are all
of the form $S_\kappa(\mu)=$ the subgroup consisting of elements
whose support has cardinality $<\kappa$ for some $\kappa\le\mu^+$.
A many-sorted structure ${\cal M}_{\kappa\lambda\mu}$ 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 ${\cal
M}_{\kappa\lambda\mu}$ 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 $cf(\kappa) > 2^{\aleph_0},
cf(\kappa)\le 2^{\aleph_0}\lt\kappa, \aleph_0\lt\kappa\lt
2^{\aleph_0}$, 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 ${\cal N}^2_{\kappa \lambda \mu}$,
all of whose sorts have cardinality at most $2^{\aleph_0}$.},
},
@article{Sh:606,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Periodica Math. Hungarica},
note = { arxiv:math.LO/9811177 },
pages = {87--98},
title = {{On $T_3$--topological space omitting many cardinals}},
volume = {38},
year = {1999},
abstract = {We prove that for every (infinite cardinal) $\lambda$ there
is a $T_3$-space $X$ with clopen basis, $2^\mu$ points where $\mu =
2^\lambda$, such that every closed subspace of cardinality $<|X|$ has
cardinality $<\lambda$.},
},
@article{BrSh:607,
author = {Bartoszynski, Tomek and Shelah, Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Mathematical Logic},
note = { arxiv:math.LO/9805148 },
pages = {1--34},
title = {{Strongly meager sets do not form an ideal}},
volume = {1},
year = {2001},
abstract = {A set $X \subseteq {\bf R}$ is strongly meager if for
every measure zero set $H$, $X+H \neq {\bf R}$. Let SM denote
the collection of strongly meager sets. We show that assuming CH, SM
is not an ideal.},
},
@article{ShSt:608,
author = {Shelah, Saharon and Stanley, Lee},
fromwhere = {IL, 1},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9710216 },
pages = {1359--1370},
title = {{Forcing Many Positive Polarized Partition Relations Between a
Cardinal and its Powerset}},
volume = {66},
year = {2001},
abstract = {Corrected in [918]. We present a forcing for blowing up
$2^\lambda$ and making ``many positive polarized partition relations''
(in a sense made precise in (c) of our main theorem) hold in the
interval $[\lambda,\ 2^\lambda]$. This generalizes results of
\cite{276}, Section 1, and the forcing is a \lq\lq many cardinals''
version of the forcing there.},
},
@article{KjSh:609,
author = {Kojman, Menachem and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/9512202 },
pages = {2459--2465},
title = {{A ZFC Dowker space in $\aleph_{\omega+1}$: an application of
pcf theory to topology.}},
volume = {126},
year = {1998},
abstract = {A Dowker space is a normal Hausdorff topological space whose
product with the unit interval is not normal. Using pcf theory we
construct a Dowker space of cardinality $\aleph_{\omega+1}$.},
},
@article{ShZa:610,
author = {Shelah, Saharon and Zapletal, Jindrich},
fromwhere = {IL,1},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9806166 },
pages = {217--259},
title = {{Canonical models for $\aleph_1$ combinatorics}},
volume = {98},
year = {1999},
abstract = {We define the property of $\Pi_2$-compactness of a
statement $\phi$ of set theory, meaning roughly that the hard core of
the impact of $\phi$ on combinatorics of $\aleph_1$ can be isolated in
a canonical model for the statement $\phi$. We show that the
following statements are $\Pi_2$-compact: ``dominating
number$=\aleph_1,$'' ``cofinality of the meager ideal$=\aleph_1$'',
``cofinality of the null ideal$=\aleph_1$'', existence of various types
of Souslin trees and variations on uniformity of measure and category
$=\aleph_1$. Several important new metamathematical patterns among
classical statements of set theory are pointed out.},
},
@incollection{RShW:611,
author = {Rosen, Eric and Shelah, Saharon and Weinstein, Scott},
booktitle = {Logic and Random Structures: DIMACS Workshop, November 5-7,
1995},
fromwhere = {IL, IL, 1},
note = { arxiv:math.LO/9604244 },
pages = {65-77},
publisher = {American Mathematical Society},
series = {DIMACS Series in Discrete Mathematics and Theoretical Computer
Science},
title = {{$k$--Universal Finite Graphs}},
volume = {33},
year = {1997},
abstract = {This paper investigates the class of $k$-universal
finite graphs, a local analog of the class of universal graphs,
which arises naturally in the study of finite variable logics. The
main results of the paper, which are due to Shelah, establish that
the class of $k$-universal graphs is not definable by an
infinite disjunction of first-order existential sentences with a
finite number of variables and that there exist $k$-universal graphs
with no $k$-extendible induced subgraphs.},
},
@article{JuSh:612,
author = {Juhasz, Istvan and Shelah, Saharon},
trueauthor = {Juh\'asz, Istv\'an and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9703220 },
pages = {91--94},
title = {{On the cardinality and weight spectra of compact spaces, II}},
volume = {155},
year = {1998},
abstract = {Let $B(\kappa,\lambda)$ be the subalgebra of
${\cal P}(\kappa)$ generated by $[\kappa]^{\le\lambda}$. It is
shown that if $B$ is any homomorphic image of $B(\kappa,\lambda)$
then either $|B|\lt 2^\lambda$ 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.},
},
@article{JiSh:613,
author = {Jin, Renling and Shelah, Saharon},
trueauthor = {Jin, Renling and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9604211 },
pages = {1371--1392},
title = {{Compactness of Loeb Spaces}},
volume = {63},
year = {1998},
abstract = {In this paper we show that the compactness of a Loeb space
depends on its cardinality, the nonstandard universe it belongs to and
the underlying model of set theory we live in. In \S 1 we prove that
Loeb spaces are compact under various assumptions, and in \S 2 we prove
that Loeb spaces are not compact under various other assumptions. The
results in \S 1 and \S 2 give a quite complete answer to a question of
D. Ross.},
},
@article{DjSh:614,
author = {Dzamonja, Mirna and Shelah, Saharon},
trueauthor = {D\v{z}amonja, Mirna and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9805149 },
pages = {901--936},
title = {{On the existence of universal models}},
volume = {43},
year = {2004},
abstract = {Suppose that $\lambda=\lambda^{<\lambda}\ge\aleph_0$, and we
are considering a theory $T$. We give a criterion on $T$ which
is sufficient for the consistent existence of $\lambda^{++}$
universal models of $T$ of size $\lambda^+$ for models of $T$ of
size $\le\lambda^+$, and is meaningful
when $2^{\lambda^+}>\lambda^{++}$. In fact, we work more generally
with abstract elementary classes. The criterion for the
consistent existence of universals applies to various well known
theories, such as triangle-free graphs and simple theories.
\endgraf 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
$x$.},
},
@article{KKSh:615,
author = {Kuhlmann, Franz--Viktor and Kuhlmann, Salma and Shelah,
Saharon},
trueauthor = {Kuhlmann, Franz--Viktor and Kuhlmann, Salma and Shelah,
Saharon},
fromwhere = {2,2,IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/0107206 },
pages = {2969--2976},
title = {{Functorial Equations for Lexicographic Products}},
volume = {131},
year = {2003},
abstract = {We generalize the main result of [KKSh:601] concerning
the convex embeddings of a chain $\Gamma$ in a lexicographic
power $\Delta^\Gamma$. For a fixed nonempty chain $\Delta$, we
derive necessary and sufficient conditions for the existence of
nonempty solutions $\Gamma$ to each of the lexicographic functional
equations $(\Delta^\Gamma)^{\leq 0}\simeq \Gamma$,
$(\Delta^\Gamma)\simeq \Gamma$, and
$(\Delta^\Gamma)^{<0}\simeq\Gamma$.},
},
@article{BRSh:616,
author = {Bartoszynski, Tomek and Roslanowski, Andrzej and Shelah,
Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Ros{\l}anowski, Andrzej and
Shelah, Saharon },
fromwhere = {1,PL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9711222 },
pages = {803--816},
title = {{After all, there are some inequalities which are provable in
ZFC}},
volume = {65},
year = {2000},
abstract = {We address ZFC inequalities between some cardinal invariants
of the continuum, which turned to be true in spite of
strong expectations given by \cite{RoSh:470}.},
},
@article{EHSh:617,
author = {Eklof, Paul C. and Huisgen--Zimmermann, Birge and Shelah,
Saharon},
fromwhere = {1,IL,1},
journal = {Bulletin of the London Mathematical Society},
note = { arxiv:math.LO/9703221 },
pages = {547--555},
title = {{Torsion modules, lattices and $p$-points}},
volume = {29},
year = {1997},
abstract = {Answering a long-standing question in the theory of
torsion modules, we show that weakly productively bounded domains
are necessarily productively bounded. Moreover, we prove a twin
result for the ideal lattice $L$ of a domain equating weak and
strong global intersection conditions for families $(X_i)_{i\in I}$
of subsets of $L$ with the property that $\bigcap_{i\in I} A_i\ne 0$
whenever $A_i\in X_i$. Finally, we show that, for domains with
Krull dimension (and countably generated extensions thereof),
these lattice-theoretic conditions are equivalent to
productive boundedness.},
},
@article{HmSh:618,
author = {Hamkins, Joel David and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9612227 },
pages = {549--554},
title = {{Superdestructibility: A Dual to Laver Indestructibility}},
volume = {63},
year = {1998},
abstract = {After small forcing, any $<\kappa$-closed forcing
will destroy the supercompactness, even the strong compactness,
of $\kappa$.},
},
@article{Sh:619,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9705213 },
pages = {97--129},
title = {{The null ideal restricted to some non-null set may
be $\aleph_1$-saturated}},
volume = {179},
year = {2003},
abstract = {Our main result is that possibly some non-null set of
reals cannot be divided to uncountably many non-null sets. We deal
also with a non-null set of reals, the graph of any function from it
is null and deal with our iterations somewhat more generally.},
},
@article{Sh:620,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Topology and its Applications},
note = {8th Prague Topological Symposium on General Topology and its
Relations to Modern Analysis and Algebra, Part II (1996).
arxiv:math.LO/9804156 },
pages = {135--235},
title = {{Special Subsets of ${}^{{\rm cf}(\mu)}\mu$, Boolean
Algebras and Maharam measure Algebras}},
volume = {99},
year = {1999},
abstract = {The original theme of the paper is the existence proof of
``there is $\bar{\eta}=\langle\eta_\alpha:\alpha\lt\lambda\rangle$
which is a $(\lambda,J)$-sequence for $\bar{I}=\langle I_i:i\lt
\delta\rangle$, a sequence of ideals. This can be thought of as in a
generalization to Luzin sets and Sierpinski sets, but for the product
$\prod_{i\lt \delta} Dom(I_i)$, 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 $\mu=\mu^{\lt\kappa}\lt\lambda\lt 2^\mu$;
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. },
},
@article{EkSh:621,
author = {Eklof, Paul C. and Shelah, Saharon},
trueauthor = {Eklof, Paul C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Pure and Applied Algebra},
note = { arxiv:math.LO/9908157 },
pages = {199--214},
title = {{A non-reflexive Whitehead group}},
volume = {156},
year = {2001},
abstract = {It is proved, via an iterated forcing construction with
finite support, that it is consistent that there is a
strongly $\aleph_1$-free Whitehead group of cardinality $\aleph_1$
which is strongly non-reflexive. It is also proved consistent that
there is a group $A$ satisfying Ext$(A, Z)$ is torsion and
Hom$(A,Z)=0$.},
},
@article{Sh:622,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Group Theory},
note = { arxiv:math.LO/9808139 },
pages = {169--191},
title = {{Non-existence of universal members in classes of Abelian
groups}},
volume = {4},
year = {2001},
abstract = {We prove that if
$\mu^+<\lambda=cf(\lambda)<\mu^{\aleph_0}$, then there is no universal
reduced torsion free abelian group. Similarly if $\aleph_0\lt\lambda\lt
2^{\aleph_0}$. We also prove that if
$2^{\aleph_0}\lt\mu^+\lt\lambda=cf(\lambda)\lt \mu^{\aleph_0}$, 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$(\mu)=\aleph_0\lt\mu$).},
},
@article{BlSh:623,
author = {Baldwin, John and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Theoretical Computer Science},
note = { arxiv:math.LO/9801152 },
pages = {117--129},
title = {{On the classifiability of cellular automata}},
volume = {230},
year = {2000},
abstract = {Based on computer simulations Wolfram presented in
several papers conjectured classifications of cellular automata into
4 types. He distinguishes the 4 classes of cellular automata by
the evolution of the pattern generated by applying a cellular
automaton to a finite input. Wolfram's qualitative classification is
based on the examination of a large number of simulations. In addition
to this classification based on the rate of growth, he conjectured
a similar classification according to the eventual pattern.
We consider here one formalization of his rate of growth suggestion.
After completing our major results (based only on Wolfram's work), we
investigated other contributions to the area and we report the relation
of some them to our discoveries.},
},
@article{Sh:624,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Colloquium Mathematicum},
note = { arxiv:math.LO/9608215 },
pages = {1--7},
title = {{On full Suslin trees}},
volume = {79},
year = {1999},
abstract = {In this note we answer a question of Kunen (15.13
[Mi91]) showing that it is consistent that there are full Souslin
trees},
},
@article{EkSh:625,
author = {Eklof, Paul C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/9709230 },
pages = {1901--1907},
title = {{The Kaplansky test problems for $\aleph_1$-separable groups}},
volume = {126},
year = {1998},
abstract = {We answer a long-standing open question by proving in
ordinary set theory, ZFC, that the Kaplansky test problems have
negative answers for $\aleph_1$-separable abelian groups of
cardinality $\aleph_1$. In fact, there is an $\aleph_1$-separable
abelian group $M$ such that $M$ is isomorphic to $M\oplus M\oplus M$
but not to $M\oplus M$.},
},
@article{JiSh:626,
author = {Jin, Renling and Shelah, Saharon},
fromwhere = {1, IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9801153 },
pages = {61--77},
title = {{Possible Size of an ultrapower of $\omega$}},
volume = {38},
year = {1999},
abstract = {Let $\omega$ be the first infinite ordinal (or the set
of all natural numbers) with the usual order $<$. In \S 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 \cite{CK}, modulo the assumption
of supercompactness. In \S 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 \cite{KS},
modulo the assumption of measurability.},
},
@article{Sh:627,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Combinatorial Theory. Ser. A},
note = { arxiv:math.CO/9707226 },
pages = {179--185},
title = {{Erdos and Renyi Conjecture}},
volume = {82},
year = {1998},
abstract = {Affirming a conjecture of Erd{\"{o}}s and R\'enyi 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.},
},
@article{RoSh:628,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {PL,IL},
journal = {Journal of Applied Analysis},
note = { arxiv:math.LO/9703222 },
pages = {103--127},
title = {{Norms on possibilities II: More ccc ideals on
$2^{\textstyle\omega}$}},
volume = {3},
year = {1997},
abstract = {We use the method of {\it 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.},
},
@article{HySh:629,
author = {Hyttinen, Tapani and Shelah, Saharon},
fromwhere = {SF,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9911229 },
pages = {201--228},
title = {{Strong splitting in stable homogeneous models}},
volume = {103},
year = {2000},
abstract = {In this paper we study elementary submodels of a
stable homogeneous structure. We improve the independence relation
defined in [Hy1]. We apply this to prove a structure theorem. We also
solve a question from [Hy3]: We show that dop and sdop are
essentially equivalent, where the negation of dop is a property used to
get structure theorems and sdop implies nonstructure.},
},
@article{Sh:630,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Applied Analysis},
note = { arxiv:math.LO/9712283 },
pages = {168--289},
title = {{Properness Without Elementaricity}},
volume = {10},
year = {2004},
abstract = {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 $({\cal H}(\chi),\in)$. 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. },
},
@article{RbSh:631,
author = {Rabus, Mariusz and Shelah, Saharon},
trueauthor = {Rabus, Mariusz and Shelah, Saharon},
fromwhere = {IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/9709231 },
pages = {2573--2581},
title = {{Topological density of ccc Boolean algebras---every
cardinality occurs.}},
volume = {127},
year = {1999},
abstract = {For every uncountable cardinal $\mu$ there is a ccc
Boolean algebra whose topological density is $\mu$.},
},
@article{HySh:632,
author = {Hyttinen, Tapani and Shelah, Saharon},
trueauthor = {Hyttinen, Tapani and Shelah, Saharon},
fromwhere = {SF,IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:math.LO/9702228 },
pages = {354--358},
title = {{On the Number of Elementary Submodels of an Unsuperstable
Homogeneous Structure}},
volume = {44},
year = {1998},
abstract = {We show that if $M$ is a stable unsuperstable
homogeneous structure, then for most $\kappa\lt |M|$, the number of
elementary submodels of $ M$ of power $\kappa$ is $2^\kappa$.},
},
@article{GoSh:633,
author = {Goldstern, Martin and Shelah, Saharon},
fromwhere = {A, IL},
journal = {Algebra Universalis},
note = { arxiv:math.LO/9707203 },
pages = {197--209},
title = {{Order-polynomially complete lattices must be LARGE}},
volume = {39},
year = {1998},
abstract = {If $L$ is an order-polynomially complete lattice, then
the cardinality of $L$ must be a strongly inaccessible cardinal},
},
@incollection{Sh:634,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
booktitle = {Computer Science Logic, 14th International Workshop,
CSL 2000, Annual Conference of the EACSL, Fischbachau, Germany, August
21--26, 2000, Proceedings},
fromwhere = {IL},
note = { arxiv:math.LO/9807179 },
pages = {72--125},
publisher = {Springer},
series = {Lecture Notes in Computer Science},
title = {{Choiceless Polynomial Time Logic: Inability to express}},
volume = {1862},
year = {2000},
abstract = {Here we deal with the logic of [GuSh 533], which tries
to capture polynomial time (for finite models). There it is proved that
the logic cannot say much on models with equality only. Here we prove
that it cannot say much on models for which we expect it cannot say
much, like random enough graphs. This is the result of having a general
criterion.},
},
@article{ShVi:635,
author = {Shelah, Saharon and Villaveces, Andres},
trueauthor = {Shelah, Saharon and Villaveces, Andr\'es},
fromwhere = {IL, COL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9707227 },
pages = {1--25},
title = {{Toward Categoricity for Classes with no Maximal Models}},
volume = {97},
year = {1999},
abstract = {We provide here the first steps toward Classification Theory
of Abstract Elementary Classes with no maximal models, plus some
mild set theoretical assumptions, when the class is categorical in
some $\lambda$ greater than its L{\"{o}}wenheim-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.},
},
@article{Sh:636,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Applied Analysis},
note = { arxiv:math.LO/9712284 },
pages = {1--17},
title = {{The lifting problem with the full ideal}},
volume = {4},
year = {1998},
abstract = {We prove in ZFC that for $\mu\geq\aleph_2$ there is
a $\sigma$--ideal $I$ on $\mu$ and a Boolean $\sigma$--subalgebra $B$
of the family of subsets of $\mu$ which includes $I$ such that the
natural homomorphism from $B$ onto $B/I$ cannot be lifted.},
},
@article{Sh:637,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {in preparation},
title = {{0.1 Laws: Putting together two contexts randomly }},
},
@article{Sh:638,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {The East-West Journal of Mathematics},
note = { arxiv:math.LO/9807180 },
title = {{More on Weak Diamond}},
volume = {accepted},
abstract = {We deal with the combinatorial principle Weak
Diamond, showing that we always either a local version is not saturated
or we can increase the number of colours. Then we point out a
model theoretic consequence of Weak Diamond.},
},
@article{Sh:639,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9809201 },
pages = {1055--1075},
title = {{On quantification with a finite universe}},
volume = {65},
year = {2000},
abstract = {We consider a finite universe ${\Cal U}$ (more exactly -
a family ${\frak U}$ of them). Can second order quantifier
$Q_K$, where for each ${\Cal U}$ this means quantifying over a family
of $n(K)$-place relations closed under permuting ${\Cal U}$. We
define some natural orders and shed some light on the
classification problem of those quantifiers.},
},
@article{BzSh:640,
author = {Blaszczyk, Aleksander and Shelah, Saharon},
trueauthor = {B{\l}aszczyk, Aleksander and Shelah, Saharon},
fromwhere = {PL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9712285 },
pages = {792--800},
title = {{Regular subalgebras of complete Boolean algebras}},
volume = {66},
year = {2001},
abstract = {It is shown that there exists a complete,
atomless, $\sigma$-centered Boolean algebra, which does not contain
any regular countable subalgebra if and only if there exist a
nowhere dense ultrafilter. Therefore the existence of such algebras
is undecidable in ZFC.},
},
@article{Sh:641,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Algebra Universalis},
note = { arxiv:math.LO/9712286 },
pages = {353--373},
title = {{Constructing Boolean algebras for cardinal invariants}},
volume = {45},
year = {2001},
abstract = {We construct Boolean Algebras answering questions of Monk
on cardinal invariants. The results are proved in ZFC (rather
than giving consistency results). We deal with the existence
of superatomic Boolean Algebras with ``few automorphisms'',
with entangled sequences of linear orders, and with semi-ZFC examples
of the non-attainment of the spread (and hL, hd).},
},
@article{BnSh:642,
author = {Brendle, Joerg and Shelah, Saharon},
trueauthor = {Brendle, J{\"{o}}rg and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.LO/9710217 },
pages = {2643--2674},
title = {{Ultrafilters on $\omega$ --- their ideals and their cardinal
characteristics}},
volume = {351},
year = {1999},
abstract = {For a free ultrafilter $\cal U$ on $\omega$ we study several
cardinal characteristics which describe part of the combinatorial
structure of $\cal U$. 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 $\cal U$ and calculate cardinal
coefficients of these ideals in terms of cardinal characteristics of
the underlying ultrafilter.},
},
@article{ShSj:643,
author = {Shelah, Saharon and Spasojevic, Zoran},
trueauthor = {Shelah, Saharon and Spasojevi{\' c}, Zoran},
fromwhere = {1,IL},
journal = {Publications de L'Institute Math\'ematique - Beograd,
Nouvelle S\'erie},
note = { arxiv:math.LO/0003141 },
pages = {1--9},
title = {{Cardinal invariants $\frak{b}_\kappa$ and $\frak{t}_\kappa$}},
volume = {72},
year = {2002},
abstract = {This paper studies cardinal invariants ${\frak b}_\kappa$
and ${\frak t}_\kappa$, the natural generalizations of the
invariants ${\frak b}$ and ${\frak t}$ to a regular cardinal
$\kappa$.},
},
@article{ShVs:644,
author = {Shelah, Saharon and Vaisanen, Pauli},
trueauthor = {Shelah, Saharon and V{\"{a}}is{\"{a}}nen, Pauli},
fromwhere = {IL, SF},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9807181 },
pages = {272--284},
title = {{On inverse $\gamma$-systems and the number
of $L_{\infty,\lambda}$-equivalent, non-isomorphic models for
$\lambda$ singular}},
volume = {65},
year = {2000},
abstract = {Suppose $\lambda$ is a singular cardinal of
uncountable cofinality $\kappa$. For a model $M$ of cardinality
$\lambda$, let $No(M)$ denote the number of isomorphism types of models
$N$ of cardinality $\lambda$ which are $L_{\infty\lambda}$-equivalent
to $M$. In \cite{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 \cite{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
$\theta<\lambda$.},
},
@article{KoSh:645,
author = {Komjath, Peter and Shelah, Saharon},
trueauthor = {Komj\'{a}th, P\'eter and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9807182 },
pages = {333--338},
title = {{Two consistency results on set mappings}},
volume = {65},
year = {2000},
abstract = {It is consistent that there is a set mapping from
the four-tuples of $\omega_n$ into the finite subsets with no
free subsets of size $t_n$ for some natural number $t_n$. For
any $n<\omega$ it is consistent that there is a set mapping from
the pairs of $\omega_n$ into the finite subsets with no infinite
free sets.},
},
@article{ShVs:646,
author = {Shelah, Saharon and Vaisanen, Pauli},
trueauthor = {Shelah, Saharon and V{\"{a}}is{\"{a}}nen, Pauli},
fromwhere = {IL,SF},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.LO/9908160 },
pages = {1781--1817},
title = {{On the number of
$L_{\infty,\omega_1}$-equivalent non-isomorphic models}},
volume = {353},
year = {2001},
abstract = {We prove that if ZF is consistent then ZFC + GCH
is consistent with the following statement: There is for every $k \lt
\omega$ 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.},
},
@article{GbSh:647,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.LO/9910159 },
pages = {5357--5379},
title = {{Cotorsion theories and splitters}},
volume = {352},
year = {2000},
abstract = {Let $R$ be a subring of the rationals. We want to
investigate self splitting $R$-modules $G$ that is ${\rm Ext}_R(G,G)=0$
holds and follow Schultz to call such modules splitters. Free modules
and torsion-free cotorsion modules are classical examples
for splitters. Are there others? Answering an open problem by Schultz
we will show that there are more splitters, in fact we are able
to prescribe their endomorphism $R$-algebras with a free
$R$-module structure. As a byproduct we are able to answer a problem of
Salce showing that all rational cotorsion theories have enough
injectives and enough projectives.},
},
@article{ShVi:648,
author = {Shelah, Saharon and Villaveces, Andres},
trueauthor = {Shelah, Saharon and Villaveces, Andr\'es},
fromwhere = {IL,COL},
journal = {Revista de la Academia Colombiana de Ciencias Exactas,
Fisicas y Naturales (Journal of the Colombia Academy of Sciences)},
note = { arxiv:math.LO/0404258 },
title = {{Dimension blocking categoricity in higher logics}},
volume = {submitted},
abstract = {We build an example that generalizes [HaSh:323]
to uncountable cases. In particular, our example yields a
sentence $\psi\in {\mathcal L}_{(2^\lambda)^+,\omega}$ that is
categorical in $\lambda,\lambda^+,\dots,\lambda^{+k}$ but not
in $\beth_{k+1}(\lambda)^+$. This is connected with the \L
o\'s Conjecture and with Shelah's own conjecture and construction
of excellent classes for the $\psi\in {\mathcal L}_{\omega_1,\omega}$
case.},
},
@article{KjSh:649,
author = {Kojman, Menachem and Shelah, Saharon},
trueauthor = {Kojman, Menachem and Shelah, Saharon},
fromwhere = {IL, IL},
journal = {Journal of Combinatorial Theory, Series A},
note = { arxiv:math.CO/9805150 },
pages = {177--181},
title = {{Regressive Ramsey numbers are Ackermannian}},
volume = {86},
year = {1999},
abstract = {We give an elementary proof of the fact that regressive
Ramsey numbers are Ackermannian. This fact was first proved by Kanamori
and McAloon with mathematical logic techniques.},
},
@incollection{GbSh:650,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
booktitle = {Proceedings of the Algebra Conference at Venice, June 2002;
in the series: Lecture Notes in Pure and Appl. Math.},
fromwhere = {D,IL},
note = { arxiv:math.LO/0404259 },
pages = {271--290},
title = {{Uniquely Transitive Torsion-free Abelian Groups}},
volume = {236},
year = {2004},
abstract = {We will answer a question raised by Farjoun concerning
the existence of torsion-free abelian groups $G$ such that for
any ordered pair of pure elements there is a unique automorphism
mapping the first element onto the second one. We will show the
existence of many such groups in the constructible universe L.},
},
@article{RoSh:651,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {PL,IL},
journal = {Colloquium Mathematicum},
note = { arxiv:math.LO/9808104 },
pages = {273--310},
title = {{Forcing for hL and hd}},
volume = {88},
year = {2001},
abstract = {The present paper addresses the problem of attainment of
the supremums in various equivalent definitions of hereditary
density ${\rm hd}$ and hereditary Lindel{\"{o}}f 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.},
},
@article{Sh:652,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9605235 },
pages = {401--441},
title = {{More constructions for Boolean algebras}},
volume = {41},
year = {2002},
abstract = {We address a number of problems on Boolean Algebras.
For example, we construct, in ZFC, for any BA $B$, and cardinal
$\kappa$ BAs $B_1,B_2$ extending $B$ such that the depth of the free
product of $B_1,B_2$ over $B$ is strictly larger than the depths of
$B_1$ and of $B_2$ than $\kappa$. We give a condition (for
$\lambda$, $\mu$ and $\theta$) which implies that for some BA
$A_\theta$ there are $B_1=B^1_{\lambda,\mu,\theta}$ and
$B_2B^2_{\lambda,\mu,\theta}$ such that Depth$(B_t)\leq\mu$ and
Depth$(B_1\oplus_{A_\theta} B_1) \geq \lambda$. We then investigate for
a fixed $A$, the existence of such $B_1,B_2$ giving sufficient and
necessary conditions, involving consistency results. Further we prove
that e.g. if $B$ is a BA of cardinality $\lambda$, $\lambda\ge\mu$ and
$\lambda,\mu$ are strong limit singular of the same cofinality, then
$B$ has a homomorphic image of cardinality $\mu$ (and with $\mu$
ultrafilters). Next we show that for a BA $B$, if $d(B)^\kappa<|B|$
then ind$(B)>\kappa$ or Depth$(B)\geq\log(|B|)$. Finally we prove that
if $\square_\lambda$ holds and $\lambda=\lambda^{\aleph_0}$ then for
some BAs $B_n$, Depth$(B_n)\leq\lambda$ but for any uniform ultrafilter
$D$ on $\omega$, $\prod_{n<\omega} B_n/D$ has depth $\ge\lambda^+$.},
},
@article{CiSh:653,
author = {Ciesielski, Krzysztof and Shelah, Saharon},
trueauthor = {Ciesielski, Krzysztof and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9801154 },
pages = {1467--1490},
title = {{A model with no magic sets}},
volume = {64},
year = {1999},
abstract = {We will prove that there exists a model of ZFC+``${\frak
c}=\omega_2$'' in which every $M\subseteq R$ of cardinality less than
continuum ${\frak c}$ is meager, and such that for every $X\subseteq R$
of cardinality ${\frak c}$ 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 $f,g:R\to
R$.},
},
@article{JShT:654,
author = {Just, Winfried and Shelah, Saharon and Thomas, Simon},
trueauthor = {Just, Winfried and Shelah, Saharon and Thomas, Simon},
fromwhere = {1,IL,1},
journal = {Advances in Mathematics},
note = { arxiv:math.LO/0003120 },
pages = {243--265},
title = {{The automorphism tower problem revisited}},
volume = {148},
year = {1999},
abstract = {It is known that the automorphism towers of
infinite centreless groups of cardinality $\kappa$ terminate in less
than $\left( 2^{\kappa} \right)^{+}$ steps. In this paper, we show that
$ZFC$ cannot settle the question of whether such automorphism towers
actually terminate in less than $2^{\kappa}$ steps.},
},
@article{RoSh:655,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {International Journal of Mathematics and Mathematical
Sciences},
note = { arxiv:math.LO/9906024 },
pages = {63--82},
title = {{Iteration of $\lambda$-complete forcing notions not collapsing
$\lambda^+$.}},
volume = {28},
year = {2001},
abstract = {We look for a parallel to the notion of ``proper forcing''
among $\lambda$-complete forcing notions not collapsing $\lambda^+$.
We suggest such a definition and prove that it is preserved by
suitable iterations.},
},
@article{Sh:656,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Preprint},
note = { arxiv:math.LO/0003115 },
title = {{NNR Revisited}},
abstract = {We are interested in proving that if we use CS iterations of
forcing notions not adding reals and additional conditions then the
limit forcing does not add reals. As a result we prove that we can
amalgamate two earlier methods and so can prove the consistency with
ZFC + G.C.H. of two statements gotten separately earlier: SH and
non-club guessing. We also prove the consistency of further cases of
``non-club guessing''.},
},
@article{ShVa:657,
author = {Shelah, Saharon and Vaananen, Jouko},
trueauthor = {Shelah, Saharon and V{\"{a}}{\"{a}}n{\"{a}}nen, Jouko},
fromwhere = {IL,SF},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9706225 },
pages = {1311--1320},
title = {{Stationary Sets and Infinitary Logic}},
volume = {65},
year = {2000},
abstract = {Let $K^0_\lambda$ be the class of
structures $\langle\lambda,\lt ,A\rangle$, where $A\subseteq\lambda$
is disjoint from a club, and let $K^1_\lambda$ be the class
of structures $\langle\lambda,\lt ,A\rangle$, where $A\subseteq\lambda$
contains a club. We prove that if $\lambda=\lambda^{\lt \kappa}$ 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 $\lambda=\mu^+$, $\mu=\mu^{\lt \mu}$, 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 $K^1_\lambda$.},
},
@article{BrSh:658,
author = {Bartoszynski, Tomek and Shelah, Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9907137 },
pages = {245--250},
title = {{Strongly meager and strong measure zero sets}},
volume = {41},
year = {2002},
abstract = {In this paper we present several consistency
results concerning the existence of large strong measure zero
and strongly meager sets.},
},
@article{DjSh:659,
author = {Dzamonja, Mirna and Shelah, Saharon},
trueauthor = {D\v{z}amonja, Mirna and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0102043 },
pages = {366--388},
title = {{Universal graphs at the successor of a singular cardinal}},
volume = {68},
year = {2003},
abstract = {The paper is concerned with the existence of a universal
graph at the successor of a strong limit singular $\mu$ of
cofinality $\aleph_0$. Starting from the assumption of the existence of
a supercompact cardinal, a model is built in which for some such
$\mu$ there are $\mu^{++}$ graphs on $\mu^+$ that taken jointly
are universal for the graphs on $\mu^+$, while $2^{\mu^+}>>\mu^{++}$.
The paper also addresses the general problem of obtaining a framework
for consistency results at the successor of a singular strong limit
starting from the assumption that a supercompact cardinal $\kappa$
exists. The result on the existence of universal graphs is obtained as
a specific application of a more general method.},
},
@article{Sh:660,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Real Analysis Exchange},
note = { arxiv:math.LO/9711223 },
pages = {205--213},
title = {{Covering Numbers Associated with Trees Branching into a
Countably Generated Set of Possibilities}},
volume = {24},
year = {1998/99},
abstract = {This paper is concerned with certain generalizations
of meagreness and their combinatorial equivalents. The
simplest example, and the one which motivated further study in this
area, comes about by considering the following definition: a
set $X\subseteq R$ is said to be $Q$-nowhere dense if and only if
for every rational $q$ there exists and integer $k$ such that
the interval whose endpoints are $q$ and $q+1/k$ is disjoint from $X$.
A set which is the union of countably many $Q$-nowhere dense sets
will be called $Q$-very meagre. \endgraf 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.},
},
@article{KlSh:661,
author = {Kolman, Oren and Shelah, Saharon},
fromwhere = {IR,IL},
journal = {Journal of Applied Analysis},
note = { arxiv:math.LO/9712287 },
pages = {161--165},
title = {{A result related to the problem CN of Fremlin}},
volume = {4},
year = {1998},
abstract = {We show that the set of injective functions from
any uncountable cardinal less than the continuum into the real
numbers is of second category in the box product topology.},
},
@article{HkSh:662,
author = {Halko, Aapo and Shelah, Saharon},
trueauthor = {Halko, Aapo and Shelah, Saharon},
fromwhere = {SF,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9710218 },
pages = {219--229},
title = {{On strong measure zero subsets of ${}^\kappa 2$.}},
volume = {170},
year = {2001},
abstract = {This paper answers three questions posed by the
first author. In Theorem 2.6 we show that the family of strong
measure zero subsets of ${}^{\omega_1}2$ is $2^{\aleph_1}$-additive
under GMA and CH. In Theorem 3.1 we prove that the generalized
Borel conjecture is false in ${}^{\omega_1}2$ assuming ZFC+CH. Next,
in Theorem 4.2, we show that the family of subsets of
${}^{\omega_1}2$ with the property of Baire is not closed under the
Souslin operation.},
},
@article{ShSi:663,
author = {Shelah, Saharon and Spinas, Otmar},
trueauthor = {Shelah, Saharon and Spinas, Otmar},
fromwhere = {IL,CH},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/9802135 },
pages = {3475--3480},
title = {{On tightness and depth in superatomic Boolean algebras}},
volume = {127},
year = {1999},
abstract = {We introduce a large cardinal property which is consistent
with $L$ and show that for every superatomic Boolean algebra $B$
and every cardinal $\lambda$ with the large cardinal property,
if tightness$^+(B)\geq\lambda^+$, then depth$(B)\geq\lambda$.
This improves a theorem of Dow and Monk.},
},
@article{Sh:664,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Results in Mathematics},
note = { arxiv:math.LO/9807183 },
pages = {131--154},
title = {{Strong dichotomy of cardinality}},
volume = {39},
year = {2001},
abstract = {A usual dichotomy is that in many cases, reasonably
definable sets, satisfy the CH, i.e. if they are uncountable they
have cardinality continuum. A strong dichotomy is when: if
the cardinality is infinite it is continuum as in \cite{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 $\aleph_0$.},
},
@article{ShSr:665,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
fromwhere = {IL, 3},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/9712288 },
pages = {707--718},
title = {{The covering numbers of Mycielski ideals are all equal}},
volume = {66},
year = {2001},
abstract = {The Mycielski ideal $M_k$ is defined to consist of all
sets $A\subseteq k^\omega$ such that $\{f\restriction X: f\in
A\}\neq k^X$ for all $X\in [\omega]^{\aleph_0}$. It will be shown that
the covering numbers for these ideals are all equal. However,
the covering numbers of the closely associated Roslanowski ideals
will be shown to be consistently different.},
},
@article{Sh:666,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9906113 },
pages = {1--82},
title = {{On what I do not understand (and have something to say:) Part
I}},
volume = {166},
year = {2000},
abstract = {This is a non-standard paper, containing some problems
I have in various degrees been interested in, sometimes with discussion
on what I have to say; why they seem interesting, sometimes how I have
tried to solve them, failed tries, anecdote and opinion, so the
discussion is quite personal, in other words, egocentric and somewhat
accidental. As we discuss many problems, history and side references
are erratic, usually kept at a minimum (see $\ldots$ means: see the
references there and possibly the paper itself). \endgraf The base
were lectures in Rutgers Fall '97.},
},
@article{Sh:667,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9808140 },
pages = {127--155},
title = {{Successor of singulars: combinatorics and not collapsing
cardinals $\leq\kappa$ in $(<\kappa)$-support iterations}},
volume = {134},
year = {2003},
abstract = {We deal with $(< \kappa)$-supported iterated forcing
notions which are $(E_0,E_1)$-complete, have in mind problems on
Whitehead groups, uniformizations and the general problem. We deal
mainly with the successor of a singular case. This continues [Sh:587].
We also deal with complimentary combinatorial results.},
},
@article{Sh:668,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Scientiae Mathematicae Japonicae},
note = { arxiv:math.LO/9906025 },
pages = {203--255},
title = {{Anti--homogeneous Partitions of a Topological Space}},
volume = {59, No. 2; (special issue:e9, 449--501)},
year = {2004},
abstract = {We prove the consistency (modulo supercompact) of a negative
answer to Arhangelskii's problem (some Hausdorff compact space cannot
be partitioned to two sets not containing a closed copy of Cantor
discontinuum). In this model we have CH. Without CH we get consistency
results using a pcf assumption, close relatives of which are necessary
for such results.},
},
@article{Sh:669,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Applied Analysis},
note = { arxiv:math.LO/0303294 },
pages = {1--17},
title = {{Non-Cohen Oracle c.c.c.}},
volume = {12},
year = {2006},
abstract = {The oracle c.c.c. is closed related to Cohen forcing. During
an iteration we can ``omit a type''; i.e. preserve ``the intersection
of a given family of Borel sets of reals is empty'' provided that
Cohen forcing satisfies it. We generalize this to other cases. In \S1
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.},
},
@article{RoSh:670,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1, IL},
journal = {in preparation},
title = {{Norms on possibilities III: strange subsets of the real
line}},
},
@article{JeSh:671,
author = {Jech, Thomas and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.LO/9801078 },
pages = {2507--2515},
title = {{On reflection of stationary sets in ${\cal
P}_\kappa\lambda$}},
volume = {352},
year = {2000},
abstract = {Let $\kappa$ be an inaccessible cardinal, and
let $E_0=\{x\in{\Cal P}_\kappa\kappa^+:cf(\lambda_x)=cf(\kappa_x)\}$
and $E_1=\{x\in{\Cal P}_\kappa\kappa^+:\kappa_x$ is regular
and $\lambda_x =\kappa_x^+\}$. \endgraf It is consistent that the
set $E_1$ is stationary and that every stationary subset of $E_0$
reflects at almost every $a\in E_1$.},
},
@article{RoSh:672,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9909115 },
pages = {583--663},
title = {{Sweet {\&} Sour and other flavours of ccc forcing notions}},
volume = {43},
year = {2004},
abstract = {We continue developing the general theory of forcing notions
built with the use of norms on possibilities, this time concentrating
on ccc forcing notions and classifying them.},
},
@article{KjSh:673,
author = {Kojman, Menachem and Shelah, Saharon},
trueauthor = {Kojman, Menachem and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9712289 },
pages = {213--218},
title = {{The PCF trichotomy theorem does not hold for short
sequences}},
volume = {39},
year = {2000},
abstract = {We show that the assumption $\lambda\gt \kappa^+$ in
the Trichotomy Theorem cannot be relaxed to $\lambda\gt \kappa$.},
},
@article{BDJShS:674,
author = {Balogh, Z.T. and Davis, S.W. and Just, W. and Shelah, S.
and Szeptycki, P.J.},
trueauthor = {Balogh, Z.T. and Davis, S.W. and Just, W. and Shelah, S.
and Szeptycki, P.J},
fromwhere = {1,1,1,IL,1},
journal = {Transactions of the AMS},
note = { arxiv:math.LO/9803167 },
pages = {4971--4987},
title = {{Strongly almost disjoint sets and weakly uniform bases}},
volume = {352},
year = {2000},
abstract = {A combinatorial principle CECA is formulated and
its equivalence with GCH+ certain weakenings of $\Box_\lambda$
for singular $\lambda$ is proved. CECA is used to show that certain
``almost point-$< \tau$'' families can be refined to point-$< \tau$
families by removing a small set from each member of the family. This
theorem in turn is used to show the consistency of ``every first
countable $T_1$-space with a weakly uniform base has a point-countable
base.''},
},
@article{Sh:675,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Applied Analysis},
note = { arxiv:math.LO/9801155 },
pages = {191-209},
title = {{On Ciesielski's Problems}},
volume = {3},
year = {1997},
abstract = {We discuss some problems posed by Ciesielski. For example we
show that, consistently, $d_c$ is a singular cardinal and $e_c0$, 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).},
},
@article{CiSh:680,
author = {Ciesielski, Krzysztof and Shelah, Saharon},
trueauthor = {Ciesielski, Krzysztof and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Real Analysis Exchange},
note = { arxiv:math.LO/9805151 },
pages = {615--619},
title = {{Uniformly antisymmetric function with bounded range}},
volume = {24},
year = {1998--99},
abstract = {The goal of this note is to construct a uniformly
antisymmetric function $f:R\to R$ with a bounded countable range. This
answers Problem 1(b) of Ciesielski and Larson. (See also list of
problems in Thomson and Problem 2(b) from Ciesielski's survey.) A
problem of existence of uniformly antisymmetric function $f:R\to R$
with finite range remains open.},
},
@incollection{GShS:681,
author = {Goebel, Rudiger and Shelah, Saharon and Struengmann, Lutz},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon and
Str{\"{u}}ngmann, Lutz},
booktitle = {Proceedings of the Venice Conference on Rings,
modules, algebras, and abelian groups (2002); in the series: Lecture
Notes in Pure and Appl. Math.},
fromwhere = {D,IL,D},
note = { arxiv:math.LO/0404271 },
pages = {291--306},
title = {{Generalized $E$-Rings}},
volume = {236},
year = {2004},
abstract = {A ring $R$ is called an $E$-ring if the canonical
homomorphism from $R$ to the endomorphism ring $End(R_{\mathbb Z})$ of
the additive group $R_{\mathbb Z}$, taking any $r \in R$ to
the endomorphism left multiplication by $r$ turns out to be
an isomorphism of rings. In this case $R_{\mathbb Z}$ is called
an $E$-group. Obvious examples of $E$-rings are subrings of ${\mathbb
Q}$. However there is a proper class of examples constructed recently.
$E$-rings come up naturally in various topics of algebra. This also led
to a generalization: an abelian group $G$ is an ${\mathbb E}$-group if
there is an epimorphism from $G$ onto the additive group of $End(G)$.
If $G$ is torsion-free of finite rank, then $G$ is an $E$-group if and
only if it is an ${\mathbb E}$-group. The obvious question was raised a
few years ago which we will answer by showing that the two notions do
not coincide. We will apply combinatorial machinery to non-commutative
rings to produce an abelian group $G$ with (non-commutative) $End(G)$
and the desired epimorphism with prescribed kernel $H$. Hence, if we
let $H=0$, we obtain a non-commutative ring $R$ such that
$End(R_{{\mathbb Z}}) \cong R$ but $R$ is not an $E$-ring.},
},
@article{GbSh:682,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Colloquium Mathematicum},
note = {The paper contains an error, see [GbSh:E22].
arxiv:math.LO/9910161 },
pages = {193-221},
title = {{Almost free splitters}},
volume = {81},
year = {1999},
abstract = {Let $R$ be a subring of the rationals. We want to
investigate self splitting $R$-modules $G$ that is ${\rm
Ext}_R(G,G)=0$ holds. For simplicity we will call such modules
splitters. Our investigation continues [GbSh:647]. In [GbSh:647] we
answered an open problem by constructing a large class of splitters.
Classical splitters are free modules and torsion-free, algebraically
compact ones. In [GbSh:647] we concentrated on splitters which are
larger then the continuum and such that countable submodules are
not necessarily free. The `opposite' case of $\aleph_1$-free
splitters of cardinality less or equal to $\aleph_1$ was singled out
because of basically different techniques. This is the target of the
present paper. If the splitter is countable, then it must be free over
some subring of the rationals by a result of Hausen. We can show that
all $\aleph_1$-free splitters of cardinality $\aleph_1$ are
free indeed.},
},
@incollection{KlSh:683,
author = {Kolman, Oren and Shelah, Saharon},
trueauthor = {Kolman, Oren and Shelah, Saharon},
booktitle = {Abelian groups and modules (Dublin, 1998)},
fromwhere = {UK, IL},
note = { arxiv:math.LO/0102057 },
pages = {225--230},
publisher = {Birkh{\"{a}}user, Basel},
series = {Trends in Mathematics},
title = {{Almost disjoint pure subgroups of the Baer-Specker group}},
year = {1999},
abstract = {We prove in ZFC that the Baer-Specker group ${\bf
Z}^\omega$ has $2^{\aleph_1}$ non-free pure subgroups of cardinality
$\aleph_1$ which are almost disjoint: there is no non-free subgroup
embeddable in any pair.},
},
@article{MdSh:684,
author = {Mildenberger, Heike and Shelah, Saharon},
trueauthor = {Mildenberger, Heike and Shelah, Saharon},
fromwhere = {D, IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9901096 },
pages = {207--261},
title = {{Changing cardinal characteristics without changing
$\omega$--sequences or cofinalities}},
volume = {106},
year = {2000},
abstract = {We show: There are pairs of universes $V_1\subseteq V_2$ and
there is a notion of forcing $P\in V_1$ such that the change mentioned
in the title occurs when going from $V_1[G]$ to $V_2[G]$ for a
$P$--generic filter $G$ over $V_2$. We use forcing iterations with
partial memories. Moreover, we implement highly transitive automorphism
groups into the forcing orders.},
},
@article{DjSh:685,
author = {Dzamonja, Mirna and Shelah, Saharon},
trueauthor = {D\v{z}amonja, Mirna and Shelah, Saharon},
fromwhere = {UK,IL},
journal = {Mathematica Japonica},
note = { arxiv:math.LO/9911228 },
pages = {53--61},
title = {{On versions of $\clubsuit$ on cardinals larger than
$\aleph_1$}},
volume = {51},
year = {2000},
abstract = {We give two results on guessing unbounded subsets
of $\lambda^+$. The first is a positive result and applies to
the situation of $\lambda$ regular and at least equal to
$\aleph_3$, while the second is a negative consistency result which
applies to the situation of $\lambda$ a singular strong limit
with $2^\lambda>\lambda^+$. The first result shows that in ZFC there is
a guessing of unbounded subsets of $S^{\lambda^+}_\lambda$. The
second result is a consistency result (assuming a supercompact
cardinal exists) showing that a natural guessing fails. A result of
Shelah in [Sh:667] shows that if $2^\lambda=\lambda^+$ and $\lambda$ is
a strong limit singular, then the corresponding guessing holds.
Both results are also connected to an earlier result of
D{\v z}amonja-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 D{\v z}amonja-Shelah result. },
},
@article{RoSh:686,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/9810179 },
pages = {279--291},
title = {{The Yellow Cake}},
volume = {129},
year = {2001},
abstract = {We consider the following property: \endgraf $(*)$ 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 $(\forall
x,y\in R)(f(x,y)=\sum_{n<\omega}g^0_n(x)g^1_n(y))$. \endgraf 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 $(*)$.},
},
@article{LwSh:687,
author = {Laskowski, Michael C. and Shelah, Saharon},
trueauthor = {Laskowski, Michael C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/0303345 },
pages = {263--283},
title = {{Karp complexity and classes with the independence property}},
volume = {120},
year = {2003},
abstract = {A class ${\bf K}$ of structures is {\em 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.},
},
@article{GoSh:688,
author = {Goldstern, Martin and Shelah, Saharon},
trueauthor = {Goldstern, Martin and Shelah, Saharon},
fromwhere = {AT,IL},
journal = {Algebra Universalis},
note = { arxiv:math.LO/9810050 },
pages = {49--57},
title = {{There are no infinite order polynomially complete
lattices, after all}},
volume = {42},
year = {1999},
abstract = {If $L$ is a lattice with the interpolation property whose
cardinality is a strong limit cardinal of uncountable cofinality, then
some finite power $L^n$ has an antichain of size~$ \kappa $. Hence
there are no infinite opc lattices (i.e., lattices on which every
$n$-ary monotone function is a polynomial). \endgraf However, the
existence of strongly amorphous sets implies (in ZF) the existence of
infinite opc lattices.},
},
@article{CShS:689,
author = {Cherlin, Gregory and Shelah, Saharon and Shi, Niandong},
trueauthor = {Cherlin, Gregory and Shelah, Saharon and Shi, Niandon},
fromwhere = {1, IL, 1},
journal = {Advances in Applied Mathematics},
note = { arxiv:math.LO/9809202 },
pages = {454--491},
title = {{Universal graphs with forbidden subgraphs and algebraic
closure}},
volume = {22},
year = {1999},
abstract = {We apply model theoretic methods to the problem of existence
of countable universal graphs with finitely many forbidden connected
subgraphs. We show that to a large extent the question reduces to one
of local finiteness of an associated ``algebraic closure'' operator.
The main applications are new examples of universal graphs with
forbidden subgraphs and simplified treatments of some previously known
cases.},
},
@article{ENSh:690,
author = {Eisworth, Todd and Nyikos, Peter and Shelah, Saharon},
trueauthor = {Eisworth, Todd and Nyikos, Peter and Shelah, Saharon},
fromwhere = {1, 1, IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/9812133 },
pages = {189--220},
title = {{Gently Killing S--spaces}},
volume = {136},
year = {2003},
abstract = {We produce a model of ZFC in which there are no
locally compact first countable S--spaces, and in
which $2^{\aleph_0}<2^{\aleph_1}$. A consequence of this is that
in this model there are no locally compact, separable, hereditarily
normal spaces of size $\aleph_1$, answering a question of the second
author},
},
@article{DjSh:691,
author = {Dzamonja, Mirna and Shelah, Saharon},
trueauthor = {D\v{z}amonja, Mirna and Shelah, Saharon},
fromwhere = {UK,IL},
journal = {Journal of the London Mathematical Society},
note = { arxiv:math.LO/0003118 },
pages = {1--15},
title = {{Weak reflection at the successor of a singular cardinal}},
volume = {67},
year = {2003},
abstract = {The notion of stationary reflection is one of the most
important notions of combinatorial set theory. We investigate weak
reflection, which is, as its name suggests, a weak version of
stationary reflection. Our main result is that modulo a large
cardinal assumption close to 2-hugeness, there can be a regular
cardinal $\kappa$ such that the first cardinal weakly reflecting at
$\kappa$ is the successor of a singular cardinal. This answers a
question of Cummings, D\v zamonja and Shelah.},
},
@article{DjSh:692,
author = {Dzamonja, Mirna and Shelah, Saharon},
trueauthor = {D\v{z}amonja, Mirna and Shelah, Saharon},
fromwhere = {UK, IL},
note = { arxiv:math.LO/0009087 },
pages = {119--158},
title = {{{On $\mathrel{<\!\vrule height 5pt depth
0pt}^\ast$-maximality}}, *journal = {Annals of Pure and Applied
Logic}},
volume = {125},
year = {2004},
abstract = {This paper investigates a connection between the
ordering $\triangleleft^\ast$ among theories in model theory and
the (N)SOP${}_n$ hierarchy of Shelah. It introduces two properties
which are natural extensions of this hierarchy, called SOP${}_2$
and SOP${}_1$, and gives a strong connection between SOP${}_1$ and
the maximality in Keisler ordering. Together with the known
results about the connection between the (N)SOP${}_n$ hierarchy and
the existence of universal models in the absence of $GCH$, the
paper provides a step toward the classification of unstable
theories without the strict order property.},
},
@article{ShTl:693,
author = {Shelah, Saharon and Trlifaj, Jan},
trueauthor = {Shelah, Saharon and Trlifaj, Jan},
fromwhere = {IL,CZ},
journal = {Journal of Pure and Applied Algebra},
note = { arxiv:math.LO/0009060 },
pages = {367--379},
title = {{Spectra of the $\Gamma$-invariant of uniform modules}},
volume = {162},
year = {2001},
abstract = {For a ring $R$, denote by ${\rm Spec}^R_\kappa (\Gamma)$
the $\kappa$-spectrum of the $\Gamma$-invariant of strongly uniform
right $R$-modules. Recent realization techniques of Goodearl and
Wehrung show that ${\rm Spec}^R_{\aleph_1} (\Gamma)$ is full for
suitable von Neumann regular algebras $R$, but the techniques do not
extend to cardinals $\kappa > \aleph_1$. By a direct construction, we
prove that for any field $F$ and any regular uncountable cardinal
$\kappa$ there is an $F$-algebra $R$ such that ${\rm Spec}^R_\kappa
(\Gamma)$ is full. We also derive some consequences for the complexity
of Ziegler spectra of infinite dimensional algebras.},
},
@article{JeSh:694,
author = {Jech, Thomas and Shelah, Saharon},
trueauthor = {Jech, Thomas and Shelah, Saharon},
fromwhere = {1, IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/0406438 },
pages = {543--549},
title = {{Simple Complete Boolean Algebras}},
volume = {129},
year = {2001},
abstract = {For every regular cardinal $\kappa$ there exists a
simple complete Boolean algebra with $\kappa$ generators.},
},
@article{CiSh:695,
author = {Ciesielski, Krzysztof and Shelah, Saharon},
trueauthor = {Ciesielski, Krzysztof and Shelah, Saharon},
fromwhere = {1, IL},
journal = {Journal of Applied Analysis},
note = { arxiv:math.LO/9905147 },
pages = {159--172},
title = {{Category analogue of sup-measurability problem}},
volume = {6},
year = {2000},
abstract = {A function $F\colon{\mathbb R}^2\to{\mathbb R}$
is sup-measurable if $F_f\colon{\mathbb R}\to{\mathbb R}$ given
by $F_f(x)=F(x,f(x))$, $x\in{\mathbb R}$, is measurable for
each measurable function $f\colon{\mathbb R}\to{\mathbb R}$. It is
known that under different set theoretical assumptions, including
CH, there are sup-measurable non-measurable functions, as well as
their category analog. In this paper we will show that the existence
of category analog of sup-measurable non-measurable functions
is independent of ZFC. A problem whether the similar is true for
the original measurable case remains open.},
},
@article{GoSh:696,
author = {Goldstern, Martin and Shelah, Saharon},
trueauthor = {Goldstern, Martin and Shelah, Saharon},
fromwhere = {AT, IL},
journal = {Order},
note = { arxiv:math.LO/9902054 },
pages = {213--222},
title = {{Antichains in products of linear orders}},
volume = {19},
year = {2002},
abstract = {We show that: For many cardinals $ \lambda$, for all $n\in
\{2,3,4,\ldots\}$ There is a linear order $L$ such that $L^n$ has no
(incomparability-)antichain of cardinality $\lambda$, while $L^{n+1}$
has an antichain of cardinality $\lambda$. For any nondecreasing
sequence $(\lambda_n: n \in \{2,3,4,\ldots\})$ of infinite cardinals it
is consistent that there is a linear order $L$ such that $L^n$ has an
antichain of cardinality $\lambda_n$, but not one of
cardinality $\lambda_n^+$.},
},
@article{HJSh:697,
author = {Hajnal, Andras and Juhasz, Istvan and Shelah, Saharon},
trueauthor = {Hajnal, Andras and Juh\'asz, Istv\'an and Shelah,
Saharon},
ams-subject = {(03E05); (03E35); (03E55); (04A20); (04A30)},
fromwhere = {1, H, IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9812114 },
pages = {13--23},
title = {{Strongly almost disjoint families, {I}{I}}},
volume = {163},
year = {2000},
abstract = {The relations $M(\kappa,\lambda,\mu)\to B$ (resp.
$B(\sigma)$) meaning that if ${\mathcal A}\subset [\kappa]^\lambda$
with $|{\mathcal A}|=\kappa$ is $\mu$-almost disjoint then ${\mathcal
A}$ has property $B$ (resp. has a $\sigma$-transversal) had
been introduced and studied under GCH by Erdos and Hajnal in 1961. Our
two main results here say the following: \endgraf 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 $\lambda\le\varrho^{(+\varrho+1)}$). },
},
@article{Sh:698,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9908159 },
pages = {207--213},
title = {{On the existence of large subsets of $[\lambda]^{<\kappa}$
which contain no unbounded non--stationary subsets}},
volume = {41},
year = {2002},
abstract = {Here we deal with some problems of Mate, written as
it developed. The first section deals with the existence of stationary
subsets of $[\lambda]^{<\kappa}$ with no unbounded subsets which are
not stationary, where $\kappa$ is regular uncountable $\leq\lambda$. In
the section section we deal with the existence of such clubs.},
},
@article{HlSh:699,
author = {Halbeisen, Lorenz and Shelah, Saharon},
trueauthor = {Halbeisen, Lorenz and Shelah, Saharon},
fromwhere = {CH, IL},
journal = {The Bulletin of Symbolic Logic},
note = { arxiv:math.LO/0010268 },
pages = {237-261},
title = {{Relations between some cardinals in the absence of the
Axiom of Choice}},
volume = {7},
year = {2001},
abstract = {If we assume the axiom of choice, then every two
cardinal numbers are comparable. In the absence of the axiom of choice,
this is no longer so. For a few cardinalities related to an
arbitrary infinite set, we will give all the possible relationships
between them, where possible means that the relationship is consistent
with the axioms of set theory. Further we investigate the
relationships between some other cardinal numbers in specific
permutation models and give some results provable without using the
axiom of choice.},
},
@article{Sh:700,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Acta Mathematica},
note = {Also known under the title ``Are $\mathfrak a$ and $\mathfrak d$
your cup of tea?''. arxiv:math.LO/0012170 },
pages = {187--223},
title = {{Two cardinal invariants of the continuum
(${\mathfrak d}<{\mathfrak a}$) and FS linearly ordered iterated
forcing}},
volume = {192},
year = {2004},
abstract = {We show that consistently, every MAD family has
cardinality strictly bigger than the dominating number, that is ${\frak
a} > {\frak d}$, 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.},
},
@article{GRSh:701,
author = {Goebel, Ruediger and Rodriguez, Jose and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Rodr\'{\i}guez, Jos\'e and
Shelah, Saharon},
fromwhere = {D,S,IL},
journal = {Journal f{\"{u}}r die Reine und Angew. Mathematik
(Crelle Journal)},
note = { arxiv:math.LO/9912191 },
pages = {1--24},
title = {{Large localizations of finite simple groups}},
volume = {550},
year = {2002},
abstract = {We answer a question of Emanuel Farjoun on homotopy
groups using cancellation theory},
},
@article{Sh:702,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Mathematica Japonica},
note = { arxiv:math.LO/9910158 },
pages = {329--377},
title = {{On what I do not understand (and have something to say), model
theory}},
volume = {51},
year = {2000},
abstract = {This is a non-standard paper, containing some problems,
mainly in model theory, which I have, in various degrees, been
interested in. Sometimes with a discussion on what I have to say;
sometimes, of what makes them interesting to me, sometimes the problems
are presented with a discussion of how I have tried to solve them,
and sometimes with failed tries, anecdote and opinion. So
the discussion is quite personal, in other words, egocentric
and somewhat accidental. As we discuss many problems, history and
side references are erratic, usually kept at a minimum (``See...''
means: see the references there and possibly the paper itself). The
base were lectures in Rutgers Fall '97 and reflect my knowledge then.
The other half, concentrating on set theory, is in print [Sh:666], but
the two halves are independent. We thank A. Blass, G. Cherlin and R.
Grossberg for some corrections.},
},
@article{Sh:703,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0012171 },
pages = {569--581},
title = {{On ultraproducts of Boolean Algebras and irr}},
volume = {42},
year = {2003},
abstract = {We prove the consistency of ${\rm irr}(\prod\limits_{i<
\kappa} B_i/D)<\prod\limits_{i<\kappa}{\rm irr}(B_i)/D$, where $D$ is
an ultrafilter on $\kappa$ and each $B_i$ is a Boolean Algebra.
This solves the last problem of this form from the Monk's list
of problems, that is number 35. The solution applies to many
other properties, e.g., Souslinity. Next, we get similar results with
$\kappa=\aleph_1$ (easily we cannot have it for $\kappa = \aleph_0$)
and Boolean Algebras $B_i$ ($i<\kappa$) of cardinality
$<\beth_{\omega_1}$.},
},
@incollection{Sh:704,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
booktitle = {Set Theory, The Hajnal Conference},
fromwhere = {IL},
note = { arxiv:math.LO/0009075 },
pages = {107--128},
publisher = {DIMACS Ser. Discrete Math. Theoret. Comput. Sci.},
series = {Proceedings from MAMLS Conference in honor of Andras Hajnal at
DIMACS Center, Rutgers, Oct. 15--17, 1999, S. Thomas, Ed.},
title = {{Superatomic Boolean Algebras: maximal rigidity}},
volume = {58},
year = {2002},
abstract = {We prove that for any superatomic Boolean Algebra
of cardinality $>\beth_\omega$ there is an automorphism
moving uncountably many atoms. Similarly for larger cardinals any of
those results are essentially best possible.},
},
@inbook{Sh:705,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
booktitle = {Classification Theory for Abstract Elementary Classes},
fromwhere = {IL},
note = { arxiv:math.LO/0404272 },
title = {{Toward classification theory of good $\lambda$ frames and
abstract elementary classes}},
abstract = { Our main aim is to investigate a good
$\lambda$-frame ${\frak s}$ which is as in the end of {600}, i.e.
${\frak s}$ is $n$-successful for every $n$ (i.e. we can define a good
$\lambda^{+n}$-frame ${\frak s}^{+n}$ such that ${\frak s}^{+0}
={\frak s},{\frak s}^{+(n+1)} = ({\frak s}^{+n})^+$). We would like
to prove then $K^{\frak s}$ has model in every cardinal $> \lambda$,
and it is categorical in one of them iff it is categorical in every one
of them. For this we shall show that $K_{{\frak s}^{+n}}$'s are
similar to superstable elementary classse with prime existence.
(Actually also $K^{\frak s}_{\ge \lambda^{+ \omega}}$, but the
full proof are delayed).},
},
@article{Sh:706,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Combinatorica},
note = { arxiv:math.LO/0102058 },
pages = {325--362},
title = {{Universality among graphs omitting a complete bipartite
graph}},
volume = {32},
year = {2012},
abstract = {For cardinals $\lambda,\kappa,\theta$ we consider the class
of graphs of cardinality $\lambda$ which has no subgraph which
is $(\kappa,\theta)$-complete bipartite graph. The question is
whether in such a class there is a universal one under (weak)
embedding. We solve this problem completely under GCH. Under various
assumptions mostly related to cardinal arithmetic we prove nonexistence
of universals for this problem and some related ones.},
},
@article{Sh:707,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0112238 },
title = {{Long iterations for the continuum}},
volume = {submitted},
abstract = {We deal with an iteration theorem of forcing notion with a
kind of countable support of nice enough forcing notion which is proper
$\aleph_2$-c.c. forcing notions. We then look at some special cases
(${\mathbb Q}_D$'s preceded by random forcing).},
},
@article{GiSh:708,
author = {Gitik, Moti and Shelah, Saharon},
trueauthor = {Gitik, Moti and Shelah, Saharon},
fromwhere = {IL, IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/9909087 },
pages = {639--650},
title = {{On some configurations related to the Shelah weak
hypothesis}},
volume = {40},
year = {2001},
abstract = {We show that some cardinal arithmetic configurations related
to the negation of the Shelah Weak Hypothesis and natural from
the forcing point of view are impossible.},
},
@article{KlSh:709,
author = {Kolman, Oren and Shelah, Saharon},
trueauthor = {Kolman, Oren and Shelah, Saharon},
fromwhere = {UK, IL},
journal = {Bull. Belg. Math. Soc.},
note = { arxiv:math.LO/9910162 },
pages = {623--629},
title = {{Infinitary Axiomatizability of Slender and Cotorsion-Free
Groups}},
volume = {7},
year = {2000},
abstract = {The classes of slender and cotorsion-free abelian groups
are axiomatizable in the infinitary logics $L_{\infty\omega_1}$
and $L_{\infty\omega}$ respectively. The Baer-Specker group ${\mathbb
Z}^\omega$ is not $L_{\infty\omega_1}$-equivalent to a slender group.},
},
@article{DjSh:710,
author = {Dzamonja, Mirna and Shelah, Saharon},
trueauthor = {D\v{z}amonja, Mirna and Shelah, Saharon},
fromwhere = {UK,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/0009078 },
pages = {280--302},
title = {{On properties of theories which preclude the existence
of universal models}},
volume = {139},
year = {2006},
abstract = {In this paper we investigate some properties of first
order theories which prevent them from having universal models
under certain cardinal assumptions. Our results give a new
syntactical condition, oak property, which is a sufficient condition
for a theory not to have universal models in cardinality $\lambda$
when certain cardinal arithmetic assumptions implying the failure
of $GCH$ (and close to the failure of $SCH$) hold.},
},
@article{Sh:711,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Applied Analysis},
note = { arxiv:math.LO/0303293 },
pages = {1--17},
title = {{On nicely definable forcing notions}},
volume = {11, No.1},
year = {2005},
abstract = {We prove that if $\Bbb Q$ is a nw-nep forcing then it
cannot add a dominating real. We also prove that Amoeba forcing
cannot be ${\Cal P}(X)/I$ if $I$ is an $\aleph_1$-complete ideal.},
},
@article{FGShS:712,
author = {Fuchino, Sakae and Geschke, Stefan and Shelah, Saharon
and Soukup, Lajos},
trueauthor = {Fuchino, Saka\'e and Geschke, Stefan and Shelah, Saharon
and Soukup, Lajos},
fromwhere = {J,D,IL,H},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/9911230 },
pages = {89--105},
title = {{On the weak Freese-Nation property of complete Boolean
algebras}},
volume = {110},
year = {2001},
abstract = {The following results are proved: (a) In a Cohen model,
there is always a ccc complete Boolean algebras without the weak
Freese-Nation property. (b) Modulo the consistency strength of a
supercompact cardinal, the existence of a ccc complete Boolean algebras
without the weak Freese-Nation property consistent with GCH. (c)
Under some consequences of $\neg0^\#$, the weak Freese-Nation property
of $({\cal P}(\omega),{\subseteq})$ is equivalent to the weak
Freese-Nation property of any of ${\Bbb C}(\kappa)$ or
${\Bbb R}(\kappa)$ 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 $({\cal P}(\omega),{\subseteq})$.},
},
@article{MPSh:713,
author = {Matet, Pierre and Pean, Cedric and Shelah, Saharon},
trueauthor = {Matet, Pierre and P\'ean, C\'edric and Shelah, Saharon},
fromwhere = {F,F,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0404318 },
title = {{Cofinality of normal ideals on $P_\kappa(\lambda)$, I}},
volume = {appeared online},
abstract = {Given an ordinal $\delta\leq\lambda$ and a
cardinal $\theta\leq\kappa$, an ideal $J$ on $P_{\kappa}(\lambda)$ is
said to be $\lbrack\delta\rbrack^{<\theta}$-normal if given $B_e\in J$
for $e\in P_\theta(\delta)$, the set of all $a\in
P_{\kappa}(\lambda)$ such that $a\in B_e$ for some $e\in
P_{|a\cap\theta|}(a\cap\delta)$ lies in $J$. We give necessary and
sufficient conditions for the existence of such ideals and we describe
the least one and we compute its cofinality.},
},
@article{JShSS:714,
author = {Juhasz, Istvan and Shelah, Saharon and Soukup, Lajos and
Szentmiklossy, Zoltan},
trueauthor = {Juh\'asz, Istv\'an and Shelah, Saharon and Soukup, Lajos
and Szentmikl\'{o}ssy, Zolt\'{a}n},
fromwhere = {H,IL,H,H},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/0104198 },
pages = {1907--1916},
title = {{A tall space with a small bottom}},
volume = {131},
year = {2003},
abstract = {We introduce a general method of constructing locally
compact scattered spaces from certain families of sets and then, with
the help of this method, we prove that if $\kappa^{<\kappa} =
\kappa$ then there is such a space of height $\kappa^+$ with only
$\kappa$ many isolated points. This implies that there is a locally
compact scattered space of height ${\omega}_2$ with $\omega_1$
isolated points in ZFC, solving an old problem of the first author.},
},
@article{Sh:715,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Scientiae Mathematicae Japonicae},
note = { arxiv:math.LO/0009056 },
pages = {265--316},
title = {{Classification theory for elementary classes with the
dependence property - a modest beginning}},
volume = {59, No. 2; (special issue: e9, 503--544)},
year = {2004},
abstract = {Our thesis is that for the family of classes of the
form EC$(T),T$ a complete first order theory with the dependence
property (which is just the negation of the independence property)
there is a substantial theory which means: a substantial body of basic
results for all such classes and some complimentary results for the
first order theories with the independence property, as for the family
of stable (and the family of simple) first order theories. We
examine some properties.},
},
@article{GbSh:716,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Archiv der Mathematik},
note = { arxiv:math.LO/0003165 },
pages = {166--181},
title = {{Decompositions of reflexive modules}},
volume = {76},
year = {2001},
abstract = {We continue [GbSh:568], proving a stronger result under
the special continuum hypothesis (CH). The original question of
Eklof and Mekler related to dual abelian groups. We want to find
a particular example of a dual group, which will provide a
negative answer to the question. In order to derive a stronger and also
more general result we will concentrate on reflexive modules
over countable principal ideal domains $R$. Following H.~Bass,
an $R$-module $G$ is reflexive if the evaluation
map $\sigma:G\longrightarrow G^{**}$ is an isomorphism. Here
$G^*={\rm Hom} (G,R)$ denotes the dual group of $G$. Guided by
classical results the question about the existence of a reflexive
$R$-module $G$ of infinite rank with $G\not\cong G\oplus R$ is natural.
We will use a theory of bilinear forms on free $R$-modules which
strengthens our algebraic results in [GbSh:568]. Moreover we want to
apply a model theoretic combinatorial theorem from [Sh:e] which allows
us to avoid the weak diamond principle. This has the great advantage
that the used prediction principle is still similar to the diamond,
but holds under CH.},
},
@article{EkSh:717,
author = {Eklof, Paul C. and Shelah, Saharon},
trueauthor = {Eklof, Paul C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Mathematische Zeitschrift},
note = { arxiv:math.LO/0303344 },
pages = {143--157},
title = {{The structure of ${\rm Ext}(A,{\mathbb Z})$ and GCH:
possible co-Moore spaces}},
volume = {239},
year = {2002},
abstract = {We consider what ${\rm Ext}(A,{\mathbb{Z}})$ can be when $A$
is torsion-free and ${\rm Hom}(A,{\mathbb{Z}})=0$. We thereby give
an answer to a question of Golasi\'{n}ski and Gon\c{c}alves which
asks for the divisible Abelian groups which can be the type of a
co-Moore space. },
},
@article{ShVs:718,
author = {Shelah, Saharon and Vaisanen, Pauli},
trueauthor = {Shelah, Saharon and V{\"{a}}is{\"{a}}nen, Pauli},
fromwhere = {IL,SF},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9911232 },
pages = {97--126},
title = {{The number of $L_{\infty\kappa}$--equivalent
nonisomorphic models for $\kappa$ weakly compact}},
volume = {174},
year = {2002},
abstract = {For a cardinal $\kappa$ and a model ${\mathcal M}$
of cardinality $\kappa$ let ${\rm No}({\mathcal M})$ denote the
number of non-isomorphic models of cardinality $\kappa$ which
are $L_{\infty\kappa}$--equivalent to ${\mathcal M}$. In [Sh:133]
Shelah established that when $\kappa$ is a weakly compact cardinal and
$\mu \leq \kappa$ is a nonzero cardinal, there exists a model
${\mathcal M}$ of cardinality $\kappa$ with ${\rm No}({\mathcal
M})=\mu$. We prove here that if $\kappa$ is a weakly compact cardinal,
the question of the possible values of ${\rm No}({\mathcal M})$
for models ${\mathcal M}$ of cardinality $\kappa$ is equivalent to
the question of the possible numbers of equivalence classes
of equivalence relations which are $\Sigma^1_1$-definable
over $V_\kappa$. In [ShVa:719] we prove that, consistent wise,
the possible numbers of equivalence classes of
$\Sigma^1_1$-equivalence relations can be completely controlled under
the singular cardinal hypothesis. These results settle the problem of
the possible values of ${\rm No}({\mathcal M})$ for models of weakly
compact cardinality, provided that the singular cardinal hypothesis
holds.},
},
@article{ShVs:719,
author = {Shelah, Saharon and Vaisanen, Pauli},
trueauthor = {Shelah, Saharon and V{\"{a}}is{\"{a}}nen, Pauli},
fromwhere = {IL,SF},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/9911231 },
pages = {1--21},
title = {{On equivalence relations second order definable over
$H(\kappa)$}},
volume = {174},
year = {2002},
abstract = {Let $\kappa$ be an uncountable regular cardinal. Call
an equivalence relation on functions from $\kappa$ into
2 $\Sigma_1^1$-definable over $H(\kappa)$ if there is a first
order sentence $\phi$ and a parameter $R\subseteq H(\kappa)$ such
that functions $f ,g \in {}^\kappa 2$ are equivalent iff for some
$h\in {}^\kappa 2$, the structure $(H(\kappa),\in,R,f,g,h)$
satisfies $\phi$, where $\in$, $R$, $f$, $g$, and $h$ are
interpretations of the symbols appearing in $\phi$. All the values
$\mu$, $1\leq\mu \leq\kappa^+$ or $\mu=2^\kappa$, are possible numbers
of equivalence classes for such a $\Sigma_1^1$-equivalence relation.
Additionally, the possibilities are closed under unions of
$\leq\kappa$-many cardinals and products of $<\kappa$-many cardinals.
We prove that, consistent wise, these are the only restrictions under
the singular cardinal hypothesis. The result is that the possible
numbers of equivalence classes of $\Sigma_1^1$-equivalence relations
might consistent wise be exactly those cardinals which are in
a prearranged set, provided that the singular cardinal hypothesis holds
and that some necessary conditions are fulfilled.},
},
@article{KjSh:720,
author = {Kojman, Menachem and Shelah, Saharon},
trueauthor = {Kojman, Menachem and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/0009079 },
pages = {117--129},
title = {{Fallen Cardinals}},
volume = {109},
year = {2001},
abstract = {We prove that for every singular cardinal $\mu$ of
cofinality $\omega$, the complete Boolean algebra ${\rm comp}{\Cal
P}_\mu(\mu)$ 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
${\Cal P}_\mu(\mu)={\Cal P}(\mu)/[\mu]^{<\mu}$ 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. \endgraf The proof uses pcf theory.},
},
@article{GShW:721,
author = {Goebel, Ruediger and Shelah, Saharon and Wallutis, Simone },
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon and
Wallutis, Simone},
fromwhere = {D,D,IL},
journal = {Journal of Algebra},
note = { arxiv:math.LO/0103154 },
pages = {292--313},
title = {{On the Lattice of Cotorsion Theories}},
volume = {238},
year = {2001},
abstract = {We discuss the lattice of cotorsion theories for
abelian groups. First we show that the sublattice of the
well--studied rational cotorsion theories can be identified with the
well--known lattice of types. Using a recently developed method for
making Ext vanish we also prove that any power set together with the
ordinary set inclusion (and thus any poset) can be embedded into the
lattice of all cotorsion theories.},
},
@article{BrSh:722,
author = {Bartoszynski, Tomek and Shelah, Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Topology and its Applications},
note = { arxiv:math.LO/0001051 },
pages = {243--253},
title = {{Continuous images of sets of reals}},
volume = {116},
year = {2001},
abstract = {We show that, consistently, every uncountable set can
be continuously mapped onto a non measure zero set, while there
exists an uncountable set whose all continuous images into a Polish
space are meager.},
},
@article{Sh:723,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Combinatorica},
note = { arxiv:math.LO/0003139 },
pages = {309--319},
title = {{Consistently there is no non trivial ccc forcing notion with
the Sacks or Laver property }},
volume = {21},
year = {2001},
abstract = {The result in the title answers a problem of
Boban Velickovic. A definable version of it (that is for Souslin
forcing notions) has been answered in [Sh 480], and our proof
follows it. Independently Velickovic proved this consistency, following
[Sh 480] and some of his works, proving it from PFA and from OCA.
We prove that moreover, consistently there is no ccc forcing with
the Laver property. Note that if cov(meagre)=continuum (which
follows e.g. from PFA) then there is a (non principal) Ramsey
ultrafilter on $\omega$ hence a forcing notion with the Laver property.
So the results are incomparable.},
},
@article{Sh:724,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0009064 },
pages = {31--64},
title = {{On nice equivalence relations on ${}^\lambda 2$}},
volume = {43},
year = {2004},
abstract = {The main question here is the possible generalization of
the following theorem on ``simple'' equivalence relation on ${}^\omega
2$ to higher cardinals.\endgraf 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.\endgraf (2) Instead of ``$E$ is
Borel'', ``$E$ is analytic (or even a Borel combination of analytic
relations)'' is enough.\endgraf (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.},
},
@article{MdSh:725,
author = {Mildenberger, Heike and Shelah, Saharon},
trueauthor = {Mildenberger, Heike and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/0104276 },
pages = {1--37},
title = {{On needed reals}},
volume = {141},
year = {2004},
abstract = {Following Blass, we call a real $a$ ``needed'' for a
binary relation $R$ on the reals if in every $R$-adequate set we find
an element from which $a$ is Turing computable. We show that every
real needed for ${\bf Cof}({\mathcal N})$ is hyperarithmetic.
Replacing ``$R$-adequate'' by ``$R$-adequate with minimal cardinality''
we get related notion of being ``weakly needed''. We show that is
is consistent that the two notions do not coincide for the
reaping relation. (They coincide in many models.) We show that not
all hyperarithmetical reals are needed for the reaping relation.
This answers some questions asked by Blass at the Oberwolfach
conference in December 1999.},
},
@article{ShVa:726,
author = {Shelah, Saharon and Vaananen, Jouko},
trueauthor = {Shelah, Saharon and V{\"{a}}{\"{a}}n{\"{a}}nen, Jouko},
fromwhere = {IL,SF},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0009080 },
pages = {63--69},
title = {{A Note on Extensions of Infinitary Logic}},
volume = {44},
year = {2005},
abstract = {We show that a strong form of the so called
Lindstr{\"{o}}m'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
L{\"{o}}wenheim-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 L{\"{o}}wenheim-Skolem theorem down to $<\kappa$.},
},
@incollection{GbSh:727,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
booktitle = {Proceedings of the Perth Conference 2000 ``AGRAM''},
fromwhere = {D,IL},
note = { arxiv:math.LO/0009062 },
pages = {145--158},
series = {Contemporary Mathematics},
title = {{Reflexive subgroups of the Baer-Specker group and
Martin's axiom}},
volume = {273},
year = {2001},
abstract = {In two recent papers we answered a question raised in the
book by Eklof and Mekler (p. 455, Problem 12) under the set
theoretical hypothesis of $\diamondsuit_{\aleph_1}$ which holds in many
models of set theory, respectively of the special continuum
hypothesis (CH). The objects are reflexive modules over countable
principal ideal domains $R$, which are not fields. Following H. Bass,
an $R$-module $G$ is reflexive if the evaluation map
$\sigma: G\longrightarrow G^{**}$ is an isomorphism. Here $G^*={\rm
Hom}(G, R)$ denotes the dual module of $G$. We proved the existence
of reflexive $R$-modules $G$ of infinite rank with $G \not\cong
G \oplus R$, which provide (even essentially indecomposable)
counter examples to the question mentioned above. Is CH a
necessary condition to find `nasty' reflexive modules? In the last part
of this paper we will show (assuming the existence of
supercompact cardinals) that large reflexive modules always have
large summands. So at least being essentially indecomposable needs
an additional set theoretic assumption. However the assumption need
not be CH as shown in the first part of this paper. We will use
Martin's axiom to find reflexive modules with the above decomposition
which are submodules of the Baer-Specker module $R^\omega$.},
},
@article{KeSh:728,
author = {Kennedy, Juliette and Shelah, Saharon},
trueauthor = {Kennedy, Juliette and Shelah, Saharon},
fromwhere = {F, IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/0105134 },
pages = {17--24},
title = {{On embedding models of arithmetic of cardinality $\aleph_1$
into reduced powers}},
volume = {176},
year = {2003},
abstract = {In the early 1970's S.Tennenbaum proved that all
countable models of $PA^- + \forall_1 -Th({\mathbb N})$ are embeddable
into the reduced product ${\mathbb N}^\omega/{\mathcal F}$,
where ${\mathcal F}$ is the cofinite filter. In this paper we show that
if $M$ is a model of $PA^- + \forall_1 -Th({\mathbb N})$,
and $|M|=\aleph_1$, then $M$ is embeddable into ${\mathbb
N}^\omega/D$, where $D$ is any regular filter on $\omega$.},
},
@article{ShSm:729,
author = {Shelah, Saharon and Struengmann, Lutz},
trueauthor = {Shelah, Saharon and Str{\"{u}}ngmann, Lutz},
fromwhere = {IL,D},
journal = {The Journal of Group Theory},
note = { arxiv:math.LO/0009045 },
pages = {417--426},
title = {{The failure of the uncountable non-commutative
Specker Phenomenon}},
volume = {4},
year = {2001},
abstract = {Higman proved in 1952 that every free group
is non-commutatively slender, this is to say that if $G$ is a
free group and $h$ is a homomorphism from the countable complete
free product $\bigotimes_\omega{\mathbb Z}$ to $G$, then there exists
a finite subset $F\subseteq\omega$ and a homomorphism $\bar{h}: *_{i\in
F} {\mathbb Z}\longrightarrow G$ such that $h=\bar{h}\rho_F$, where
$\rho_F$ is the natural map from $\bigotimes_{i\in\omega}{\mathbb Z}$
to $*_{i\in F}{\mathbb Z}$. Corresponding to the abelian case this
phenomenon was called the non-commutative Specker Phenomenon. In this
paper we show that Higman's result fails if one passes from countable
to uncountable. In particular, we show that for non-trivial
groups $G_\alpha$ ($\alpha\in\lambda$) and uncountable cardinal
$\lambda$ there are $2^{2^\lambda}$ homomorphisms from the complete
free product of the $G_\alpha$'s to the ring of integers.},
},
@article{Sh:730,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Periodica Mathematica Hungarica},
note = { arxiv:math.LO/0009047 },
pages = {81--84},
title = {{A space with only Borel subsets}},
volume = {40},
year = {2000},
abstract = {Mikl\'os 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. },
},
@article{MdSh:731,
author = {Mildenberger, Heike and Shelah, Saharon},
trueauthor = {Mildenberger, Heike and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0009077 },
pages = {297--314},
title = {{The relative consistency of ${\frak g}<{\rm
cf}({\rm Sym}(\omega))$}},
volume = {67},
year = {2002},
abstract = {We prove the consistency result from the title. By forcing
we construct a model of ${\frak g}=\aleph_1$, ${\frak b}={\rm cf}({\rm
Sym}(\omega))=\aleph_2$. },
},
@article{BrSh:732,
author = {Bartoszynski, Tomek and Shelah, Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/0102011 },
pages = {3701--3711},
title = {{Perfectly meager sets and universally null sets}},
volume = {130},
year = {2002},
abstract = {For a set of reals $X$: (a) $X$ is perfectly meager (PM) if
for every perfect set $P\subseteq{\mathbb R}$, $P\cap X$ is meager
in $P$. (b) $X$ is universal null (UN) if every Borel isomorphic
image of $X$ has Lebesgue measure zero. \endgraf We show that it is
consistent with ZFC that PM is a subset of UN.},
},
@article{RoSh:733,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Colloquium Mathematicum},
note = { arxiv:math.LO/0006219 },
pages = {99--115},
title = {{Historic forcing for {\rm Depth}}},
volume = {89},
year = {2001},
abstract = {We show that, consistently, for some regular
cardinals $\theta<\lambda$, there exist a Boolean algebra $B$ such that
$|B|= \lambda^+$ and for every subalgebra $B'\subseteq B$ of
size $\lambda^+$ we have ${\rm Depth}(B')=\theta$.},
},
@inbook{Sh:734,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
booktitle = {Classification Theory for Abstract Elementary Classes},
fromwhere = {IL},
note = { arxiv:0808.3023 },
title = {{Categoricity and solvability of A.E.C., quite highly}},
abstract = { We investigate in ZFC what can be the family of
large enough cardinals $\mu$ in which an a.e.c. ${\frak K}$ is
categorical or even just solvable. We show that for not few cardinals
$\lambda < \mu$ there is a superlimit model in ${\frak K}_\lambda$.
Moreover, our main result is that we can find a good $\lambda$-frame
${\frak s}$ categorical in $\lambda$ such that ${\frak K}_{\frak s}
\subseteq {\frak K}_\lambda$. We then show how to use {705} to
get categoricity in every large enough cardinality if ${\frak K}$
has cases of $\mu$-amalgamation for enough $\mu$ and $2^\mu <
2^{\mu^{+1}} < \ldots < 2^{\mu^{+n}} \ldots$ for enough $\mu$.},
},
@article{ShSr:735,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
fromwhere = {IL, 2},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/0011166 },
pages = {2097--2106},
title = {{Martin's axiom is consistent with the existence of
nowhere trivial automorphisms}},
volume = {130},
year = {2002},
abstract = {Martin's~Axiom does not imply that all automorphisms
of ${\mathcal P}({\mathbb N})/[{\mathbb N}]^{<\aleph_0}$ are somewhere
trivial.},
},
@article{RoSh:736,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/0010070 },
pages = {61--110},
title = {{Measured creatures}},
volume = {151},
year = {2006},
abstract = {Using forcing with measured creatures we build a universe
of set theory in which (a) every sup-measurable function
$f:{\mathbb R}^2\longrightarrow{\mathbb R}$ is measurable, and (b)
every function $f:{\mathbb R}\longrightarrow{\mathbb R}$ is continuous
on a non-measurable set. This answers von Weizs{\"{a}}cker's problem
(see Fremlin's list of problems) and a question of Balcerzak,
Ciesielski and Kharazishvili.},
},
@article{GoSh:737,
author = {Goldstern, Martin and Shelah, Saharon},
trueauthor = {Goldstern, Martin and Shelah, Saharon},
fromwhere = {AT, IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.RA/0005273 },
pages = {1-20},
title = {{Clones on regular cardinals}},
volume = {173},
year = {2002},
abstract = {We investigate the structure of the lattice of clones on
an infinite set $X$. We first observe that ultrafilters
naturally induce clones; this yields a simple proof of Rosenberg's
theorem: there are $2^{2^{\lambda}}$ many maximal (= ``precomplete'')
clones on a set of size~$\lambda$. The clones we construct do not
contain all unary functions. We then investigate clones that do
contain all unary functions. Using a strong negative partition theorem
we show that for many cardinals $\lambda$ (in particular, for all
successors of regulars) there are $2^{2^\lambda }$ many such clones on
a set of size $\lambda$. Finally, we show that on a weakly compact
cardinal there are exactly 2 maximal clones which contain all unary
functions.},
},
@article{GbSh:738,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Math. Proc. Camb. Phil. Soc.},
note = { arxiv:math.GR/0009091 },
pages = {23--31},
title = {{Philip Hall's problem on non-Abelian splitters}},
volume = {134},
year = {2003},
abstract = {Philip Hall raised around 1965 the following question which
is stated in the Kourovka Notebook: {\it 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.},
},
@article{GbSh:739,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Communication in Algebra},
note = { arxiv:math.GR/0009089 },
pages = {809--837},
title = {{Constructing Simple Groups For Localizations}},
volume = {30},
year = {2002},
abstract = {A group homomorphism $\eta: A\to H$ is called a
{\it 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. },
},
@article{GPSh:740,
author = {Goebel, Ruediger and Paras, Agnes T. and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Paras, Agnes T. and Shelah,
Saharon},
fromwhere = {D,D,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.GR/0009088 },
pages = {21--27},
title = {{Groups isomorphic to all their non--trivial normal
subgroups}},
volume = {129},
year = {2002},
abstract = {In answer to a question of P. Hall, we supply
another construction of a group which is isomorphic to each of
its non--trivial normal subgroups.},
},
@article{GbSh:741,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.GR/0010303 },
pages = {673--674},
title = {{Radicals and Plotkin's problem concerning geometrically
equivalent groups}},
volume = {130},
year = {2002},
abstract = {If $G$ and $X$ are groups and $N$ is a normal subgroup of
$X$, then the $G-$closure of $N$ in $X$ is the normal
subgroup ${\overline X}^G=\bigcap\{{\rm
ker}\varphi|\varphi:X\rightarrow G, \mbox{ with } N \subseteq{\rm
ker}\varphi\}$ 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.},
},
@article{GShW:742,
author = {Goebel, Ruediger and Shelah, Saharon and Wallutis, Simone},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon and
Wallutis, Simone},
fromwhere = {D,IL,?},
journal = {Illinois Journal of Mathematics},
note = { arxiv:math.LO/0112252 },
pages = {223--236},
title = {{On universal and epi-universal locally nilpotent groups}},
volume = {47},
year = {2003},
abstract = {In this paper we mainly consider the class $LN$ of all
locally nilpotent groups. We first show that there is no universal
group in $LN_\lambda$ if $\lambda$ is a cardinal such
that $\lambda=\lambda^{\aleph_0}$; here we call a group $G$
{\em 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 {\em 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.},
},
@article{DrSh:743,
author = {Droste, Manfred and Shelah, Saharon},
trueauthor = {Droste, Manfred and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Forum Mathematicum},
note = { arxiv:math.GR/0010304 },
pages = {605--621},
title = {{Outer automorphism groups of ordered permutation groups}},
volume = {14},
year = {2002},
abstract = {An infinite linearly ordered set $(S,\leq)$ is called
doubly homogeneous if its automorphism group $A(S)$ acts
$2$-transitively on it. We show that any group $G$ arises as outer
automorphism group $G\cong{\rm Out}(A(S))$ of the automorphism group
$A(S)$, for some doubly homogeneous chain $(S,\leq)$.},
},
@article{Sh:744,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Bulletin of the London Mathematical Society},
note = { arxiv:math.LO/0010305 },
pages = {1--7},
title = {{A countable structure does not have a free
uncountable automorphism group}},
volume = {35},
year = {2003},
abstract = {Solecki proved that the group of automorphisms of a
countable structure cannot be an uncountable free abelian group. See
more in Just, Shelah and Thomas [JShT:654] where as a by product we can
say something on on uncountable structures. We prove here the following
Theorem: If ${\mathbb A}$ is a countable model, then ${\rm
Aut}(M)$ cannot be a free uncountable group.},
},
@article{NeSh:745,
author = {Nesetril, Jaroslav and Shelah, Saharon},
trueauthor = {Ne\v{s}et\v{r}il, Jaroslav and Shelah, Saharon},
fromwhere = {Cz,IL},
journal = {European Journal of Combinatorics},
note = { arxiv:math.LO/0404319 },
pages = {649--663},
title = {{On the order of countable graphs}},
volume = {24},
year = {2003},
abstract = {A set of graphs is said to be {\em 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 {\em 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.},
},
@article{LrSh:746,
author = {Larson, Paul and Shelah, Saharon},
trueauthor = {Larson, Paul and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Mathematical Logic},
note = { arxiv:math.LO/0011187 },
pages = {193--215},
title = {{Bounding by canonical functions, with CH}},
volume = {3, No.2},
year = {2003},
abstract = {We show that that a certain class of semi-proper
iterations does not add $\omega$-sequences. As a result, starting
from suitable large cardinals one can obtain a model in which
the Continuum Hypothesis holds and every function from $\omega_{1}$
to $\omega_{1}$ is bounded by a canonical function on a club, and
so $\omega_{1}$ is the $\omega_{2}$-nd canonical function.},
},
@article{GoSh:747,
author = {Goldstern, Martin and Shelah, Saharon},
trueauthor = {Goldstern, Martin and Shelah, Saharon},
fromwhere = {AT, IL},
journal = {Algebra Universalis},
note = { arxiv:math.RA/0208066 },
pages = {367--374},
title = {{Large Intervals in the Clone Lattice}},
volume = {62},
year = {2010},
abstract = {We give three examples of large intervals in the lattice
of (local) clones on an infinite set $X$, by exhibiting
clones ${\mathcal C}_1$, ${\mathcal C}_2$, ${\mathcal C}_3$ such
that: \endgraf (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, \endgraf (2) the
interval $[{\mathcal C}_2, {\mathcal O}]$ in the lattice of local
clones is isomorphic to the congruence lattice of an arbitrary
semilattice, \endgraf (3) the interval $[{\mathcal C}_3,{\mathcal O}]$
in the lattice of all clones is isomorphic to the lattice of
all filters on $X$. \endgraf 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.},
},
@article{KkSh:748,
author = {Kikyo, Hirotaka and Shelah, Saharon},
trueauthor = {Kikyo, Hirotaka and Shelah, Saharon},
fromwhere = {J,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0010306 },
pages = {214--216},
title = {{The strict order property and generic automorphisms}},
volume = {67},
year = {2002},
abstract = {If $T$ is an model complete theory with the strict
order property, then the theory of the models of $T$ with an
automorphism has no model companion.},
},
@article{EkSh:749,
author = {Eklof, Paul C. and Shelah, Saharon},
trueauthor = {Eklof, Paul C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Illinois Journal of Mathematics},
note = {A special volume dedicated to Baer. arxiv:math.LO/0011228 },
pages = {173--188},
title = {{On the existence of precovers}},
volume = {47},
year = {2003},
abstract = {It is proved undecidable in ZFC + GCH whether every ${\Bbb
Z}$-module has a $^{\perp}\{{\Bbb Z}\}$-precover.},
},
@article{Sh:750,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {CUBO, A Mathematical Journal},
note = { arxiv:0902.0439 },
pages = {59--72},
title = {{On $\lambda$ strony homogeneity existence for cofinality
logic}},
volume = {13},
year = {2011},
},
@article{EdSh:751,
author = {Eda, Katsuya and Shelah, Saharon},
trueauthor = {Eda, Katsuya and Shelah, Saharon},
fromwhere = {J, IL},
journal = {Journal of Algebra},
note = { arxiv:math.LO/0011231 },
pages = {22--26},
title = {{The non-commutative Specker phenomenon in the uncountable
case}},
volume = {252},
year = {2002},
abstract = {An infinitary version of the notion of free products has
been introduced and investigated by G.Higman. Let $G_i$ (for $i\in I$)
be groups and $\ast_{i\in X} G_i$ the free product of $G_i$ ($i\in
X$) for $X \Subset I$ and $ p _{XY}:\ast_{i\in Y}
G_{i}\rightarrow \ast_{i\in X} G_{i}$ the canonical homomorphism for
$X\subseteq Y \Subset I.$ ( $X\Subset I$ denotes that $X$ is a finite
subset of $I$.) Then, the unrestricted free product is the inverse
limit $\lim (\ast_{i\in X} G_i, p_{XY}: X\subseteq Y\Subset I).$
\endgraf We remark $\ast_{i\in\emptyset} G_i=\{e\}$. We
prove: \endgraf 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 $U_0\in u_0,\cdots ,U_m\in u_m$.
\endgraf 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.},
},
@article{EkSh:752,
author = {Eklof, Paul C. and Shelah, Saharon},
trueauthor = {Eklof, Paul C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Forum Mathematicum},
note = { arxiv:math.LO/0011230 },
pages = {477--482},
title = {{Whitehead modules over large principal ideal domains}},
volume = {14},
year = {2002},
abstract = {We consider the Whitehead problem for principal ideal
domains of large size. It is proved, in ZFC, that some p.i.d.'s of
size $\geq\aleph_{2}$ have non-free Whitehead modules even though
they are not complete discrete valuation rings.},
},
@article{MdSh:753,
author = {Mildenberger, Heike and Shelah, Saharon},
trueauthor = {Mildenberger, Heike and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/0011188 },
pages = {167--176},
title = {{The splitting number can be smaller than the matrix chaos
number}},
volume = {171},
year = {2002},
abstract = {Let $\chi$ be the minimum cardinal of a subset of
$2^\omega$ that cannot be made convergent by multiplication with a
single Toeplitz matrix. By an application of creature forcing we show
that ${\mathfrak s}<\chi$ is consistent. We thus answer a question
by Vojt\'a\v{s}. 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.},
},
@article{ShSm:754,
author = {Shelah, Saharon and Struengmann, Lutz},
trueauthor = {Shelah, Saharon and Str{\"{u}}ngmann, Lutz},
fromwhere = {IL,D},
journal = {Forum Mathematicum},
note = { arxiv:math.LO/0012172 },
pages = {507--524},
title = {{It is consistent with ZFC that $B_1$-groups are not
$B_2$-groups}},
volume = {15},
year = {2003},
abstract = {A torsion-free abelian group $B$ of arbitrary rank is called
a $B_1$-group if ${\rm Bext}^1(B,T)=0$ for every torsion abelian
group $T$, where ${\rm Bext}^1$ denotes the group of equivalence
classes of all balanced exact extensions of $T$ by $B$. It is
a long-standing problem whether or not the class of
$B_1$-groups coincides with the class of $B_2$-groups. A torsion-free
abelian group $B$ is called a $B_2$-group if there exists a
continuous well-ordered ascending chain of pure subgroups, $0=B_0
\subset B_1 \subset\cdots\subset B_\alpha\subset\cdots\subset
B_\lambda=B= \bigcup\limits_{\alpha\in\lambda} B_\alpha$ such that
$B_{\alpha+1} =B_\alpha+G_\alpha$ for every $\alpha\in\lambda$ for some
finite rank Butler group $G_\alpha.$ Both, $B_1$-groups and
$B_2$-groups are natural generalizations of finite rank Butler groups
to the infinite rank case and it is known that every $B_2$-group is
a $B_1$-group. Moreover, assuming $V=L$ it was proven that the
two classes coincide. Here we demonstrate that it is undecidable in
ZFC whether or not all $B_1$-groups are $B_2$-groups. Using
Cohen forcing we prove that there is a model of ZFC in which there
exists a $B_1$-group that is not a $B_2$-group.},
},
@article{Sh:755,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Mathematica Japonica},
note = { arxiv:math.LO/0107207 },
pages = {531--538},
title = {{Weak Diamond}},
volume = {55},
year = {2002},
abstract = {Under some cardinal arithmetic assumptions, we prove that
every stationary of $\lambda$ of a right cofinality has the weak
diamond. This is a strong negation of uniformization. We then deal with
a weaker version of the weak diamond- colouring restrictions. We
then deal with semi- saturated (normal) filters.},
},
@article{HySh:756,
author = {Hyttinen, Tapani and Shelah, Saharon},
trueauthor = {Hyttinen, Tapani and Shelah, Saharon},
fromwhere = {F, IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/0102044 },
pages = {2837--2843},
title = {{Forcing a Boolean Algebra with predesigned automorphism
group}},
volume = {130},
year = {2002},
abstract = {For suitable groups $G$ we will show that one can add a
Boolean algebra $B$ by forcing in such a way that $Aut(B)$ is
almost isomorphic to $G$. In particular, we will give a positive answer
to the following question due to J. Roitman: Is $\aleph_{\omega}$
a possible number of automorphisms of a rich Boolean algebra?},
},
@article{Sh:757,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/0112212 },
pages = {261--272},
title = {{Quite Complete Real Closed Fields}},
volume = {142},
year = {2004},
abstract = {We prove that any ordered field can be extended to one
for which every decreasing sequence, of closed intervals has a non
empty intersection.},
},
@article{MsSh:758,
author = {Matsubara, Yo and Shelah, Saharon},
trueauthor = {Matsubara, Yo and Shelah, Saharon},
fromwhere = {J,IL},
journal = {Journal of Mathematical Logic},
note = { arxiv:math.LO/0102045 },
pages = {81--89},
title = {{Nowhere precipitousness of the non-stationary ideal
over ${\mathcal P}_{\kappa}\lambda$}},
volume = {2},
year = {2002},
abstract = {We prove that if $\lambda$ is a strong limit singular
cardinal and $\kappa$ a regular uncountable cardinal $<\lambda$,
then $NS_{\kappa\lambda}$, the non-stationary ideal over
${\mathcal P}_{\kappa}\lambda$, is nowhere precipitous. We also show
that under the same hypothesis every stationary subset of
${\mathcal P}_{\kappa}\lambda$ can be partitioned into $\lambda^{<
\kappa}$ disjoint stationary sets.},
},
@article{BlSh:759,
author = {Baldwin, John and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Notre Dame Journal of Formal Logic},
note = { arxiv:math.LO/0105136 },
pages = {129--142},
title = {{Model Companions of $T_{\rm Aut}$ for stable $T$}},
volume = {42},
year = {2001},
abstract = {Let $T$ be a complete first order theory in a
countable relational language $L$. We assume relation symbols have been
added to make each formula equivalent to a predicate. Adjoin a new
unary function symbol $\sigma$ to obtain the language
$L_\sigma$; $T_\sigma$ is obtained by adding axioms asserting that
$\sigma$ is an $L$-automorphism. We provide necessary and sufficient
conditions for $T_{\rm Aut}$ to have a model companion when $T$ is
stable. Namely, we introduce a new condition: $T$ admits obstructions,
and show that $T_{\rm Aut}$ has a model companion iff and only if $T$
does not admit obstructions. This condition is weakening of the finite
cover property: if a stable theory $T$ has the finite cover property
then $T$ admits obstructions.},
},
@article{BGSh:760,
author = {Blass, Andreas and Gurevich, Yuri and Shelah, Saharon},
fromwhere = {1,1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0102059 },
pages = {1093--1125},
title = {{On polynomial time computation over unordered structures}},
volume = {67},
year = {2002},
abstract = {This paper is motivated by the question whether there exists
a logic capturing polynomial time computation over
unordered structures. We consider several algorithmic problems near
the border of the known, logically defined complexity classes
contained in polynomial time. We show that fixpoint logic plus
counting is stronger than might be expected, in that it can express
the existence of a complete matching in a bipartite graph. We
revisit the known examples that separate polynomial time from fixpoint
plus counting. We show that the examples in a paper of Cai,
F{\"{u}}rer, 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.},
},
@article{Sh:761,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/0103155 },
pages = {2585--2592},
title = {{A partition relation using strongly compact cardinals}},
volume = {131},
year = {2003},
abstract = {If $\kappa$ is strongly compact, $\lambda>\kappa$ is
regular, then $(2^{<\lambda})^+\to (\lambda+\eta)^2_\theta$ holds for
$\eta$, $\theta<\kappa$.},
},
@article{BnSh:762,
author = {Brendle, Joerg and Shelah, Saharon},
trueauthor = {Brendle, J{\"{o}}rg and Shelah, Saharon},
fromwhere = {J,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0103153 },
pages = {349--360},
title = {{Evasion and prediction IV: Strong forms of constant
prediction}},
volume = {42},
year = {2003},
abstract = {Say that a function $\pi:n^{<\omega}\to n$ (henceforth
called a predictor) $k$--constantly predicts a real $x\in n^\omega$ if
for almost all intervals $I$ of length $k$, there is $i\in I$ such
that $x(i)=\pi(x\restriction i)$. We study the $k$--constant
prediction number ${\mathfrak v}_n^{\rm const}(k)$, that is, the size
of the least family of predictors needed to $k$--constantly predict
all reals, for different values of $n$ and $k$, and investigate
their relationship.},
},
@article{FGSh:763,
author = {Fuchino, Sakae and Greenberg, Noam and Shelah, Saharon},
trueauthor = {Fuchino, Saka\'e and Greenberg, Noam and Shelah, Saharon},
fromwhere = {J,1,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/0601087 },
pages = {380--397},
title = {{Models of Real-Valued Measurability}},
volume = {142},
year = {2006},
abstract = {Solovay's random-real forcing is the standard way
of producing real-valued measurable cardinals. Following questions of
Fremlin, by giving a new construction, we show that there are
combinatorial, measure-theoretic properties of Solovay's model that do
not follow from the existence of real-valued measurability.},
},
@article{ShSy:764,
author = {Shelah, Saharon and Shioya, Masahiro},
trueauthor = {Shelah, Saharon and Shioya, Masahiro},
fromwhere = {IL,J},
journal = {Advances in Mathematics},
note = { arxiv:math.LO/0405013 },
pages = {185--191},
title = {{Nonreflecting stationary sets in ${\mathcal
P}_\kappa\lambda$}},
volume = {199},
year = {2006},
abstract = {Let $\kappa$ be a regular uncountable cardinal and
$\lambda\geq \kappa^+$. The principle of stationary reflection for
${\mathcal P}_\kappa\lambda$ has been successful in settling problems
of infinite combinatorics in the case $\kappa=\omega_1$. For a
greater $\kappa$ the principle is known to fail at some $\lambda$. This
note shows that it fails at every $\lambda$ if $\kappa$ is the
successor of a regular uncountable cardinal or $\kappa$ is countably
closed.},
},
@article{JShSS:765,
author = {Juhasz, Istvan and Shelah, Saharon and Soukup, Lajos and
Szentmiklossy, Zoltan},
trueauthor = {Juh\'asz, Istv\'an and Shelah, Saharon and Soukup, Lajos
and Szentmikl\'{o}ssy, Zolt\'{a}n},
fromwhere = {H,IL,H,H},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/0404322 },
pages = {75--88},
title = {{Cardinal sequences and Cohen real extensions}},
volume = {181},
year = {2004},
abstract = {We show that if we add any number of Cohen reals to the
ground model then, in the generic extension, a locally compact
scattered space has at most $(2^{\aleph_0})^V$ many levels of size
$\omega$. We also give a complete $ZFC$ characterization of the
cardinal sequences of regular scattered spaces. Although the classes of
the regular and of the $0$-dimensional scattered spaces are
different, we prove that they have the same cardinal sequences.},
},
@article{FuSh:766,
author = {Fuchs, Laszlo and Shelah, Saharon},
trueauthor = {Fuchs, Laszlo and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Rend. Sem. Mat. Univ. Padova},
note = { arxiv:math.LO/0405015 },
pages = {235--239},
title = {{On a non-vanishing Ext}},
volume = {109},
year = {2003},
abstract = {The existence of valuation domains admitting
non-standard uniserial modules for which certain Exts do not vanish was
proved under Jensen's Diamond Principle. In this note, the same is
verified using the ZFC axioms alone.},
},
@article{ShTs:767,
author = {Shelah, Saharon and Tsuboi, Akito},
trueauthor = {Shelah, Saharon and Tsuboi, Akito},
fromwhere = {IL,J},
journal = {Notre Dame Journal of Formal Logic},
note = { arxiv:math.LO/0104277 },
pages = {65--73},
title = {{Definability of initial segments}},
volume = {43(2002), no.2},
year = {2003},
abstract = {We consider implicit definability of the standard
part $\{0,1,...\}$ in nonstandard models of Peano arithmetic (PA),
and we ask whether there is a model of PA in which the standard part
is implicitly definable. In \S 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 \S 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.},
},
@article{ShTb:768,
author = {Shelah, Saharon and Tsaban, Boaz},
trueauthor = {Shelah, Saharon and Tsaban, Boaz},
fromwhere = {IL,IL},
journal = {Journal of Applied Analysis},
note = { arxiv:math.LO/0304019 },
pages = {149--162},
title = {{Critical Cardinalities and Additivity Properties
of Combinatorial Notions of Smallness}},
volume = {9},
year = {2003},
abstract = {Motivated by the minimal tower problem, an earlier work
studied diagonalizations of covers where the covers are related to
linear quasiorders ($\tau$-covers). We deal with two types of
combinatorial questions which arise from this study. \endgraf (a) Two
new cardinals introduced in the topological study are expressed in
terms of well known cardinals characteristics of
the continuum. \endgraf (b) We study the additivity numbers of the
combinatorial notions corresponding to the topological diagonalization
notions. \endgraf 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.},
},
@article{KeSh:769,
author = {Kennedy, Juliette and Shelah, Saharon},
trueauthor = {Kennedy, Juliette and Shelah, Saharon},
fromwhere = {F, IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0105135 },
pages = {1169--1177},
title = {{On regular reduced products}},
volume = {67},
year = {2002},
abstract = {Assume
$\langle\aleph_0,\aleph_1\rangle\rightarrow\langle \lambda,\lambda^+\ra
ngle$. 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 $\prod_i
M_i/D\cong\prod_i N_i/D$.},
},
@article{HHSh:770,
author = {Hellsten, Alex and Hyttinen, Tapani and Shelah, Saharon},
trueauthor = {Hellsten, Alex and Hyttinen, Tapani and Shelah, Saharon},
fromwhere = {F,F,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/0112288 },
pages = {127--142},
title = {{Potential isomorphism and semi--proper trees}},
volume = {175},
year = {2002},
abstract = {We study a notion of potential isomorphism, where
two structures are said to be potentially isomorphic if they
are isomorphic in some generic extension that preserves stationary
sets and does not add new sets of cardinality less than the
cardinality of the models. We introduce the notions of semi-proper and
weakly semi-proper trees, and note that there is a strong
connection between the existence of potentially isomorphic models for a
given complete theory and the existence of weakly semi-proper trees.
We prove the existence of semi-proper trees under certain
cardinal arithmetic assumptions. We also show the consistency of
the non-existence of weakly semi-proper trees assuming the
consistency of some large cardinals.},
},
@article{Sh:771,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/0212250 },
pages = {477--507},
title = {{Polish Algebras, shy from freedom}},
volume = {181},
year = {2011},
abstract = {Every Polish group is not free whereas some $F_\sigma$ group
may be free. Also every automorphism group of a structure of
cardinality, e.g. $\beth_\omega$ is not free.},
},
@article{ShSm:772,
author = {Shelah, Saharon and Struengmann, Lutz},
trueauthor = {Shelah, Saharon and Str{\"{u}}ngmann, Lutz},
fromwhere = {IL,D},
journal = {Journal of the London Mathematical Society},
note = { arxiv:math.LO/0112253 },
pages = {626--642},
title = {{Kulikov's problem on universal torsion-free abelian groups}},
volume = {67},
year = {2003},
abstract = {Let $T$ be an abelian group and $\lambda$ an
uncountable regular cardinal. We consider the question of whether there
is a $\lambda$-universal group $G^*$ among all torsion-free
abelian groups $G$ of cardinality less than or equal to $\lambda$
satisfying ${\rm Ext}(G,T)=0$. Here $G^*$ is said to be
$\lambda$-universal for $T$ if, whenever a torsion-free abelian group
$G$ of cardinality less than or equal to $\lambda$ satisfies ${\rm
Ext}(G,T)=0$, then there is an embedding of $G$ into $G^*$. For large
classes of abelian groups $T$ and cardinals $\lambda$ it is shown that
the answer is consistently no. In particular, for $T$ torsion,
this solves a problem of Kulikov.},
},
@article{ShSm:773,
author = {Shelah, Saharon and Struengmann, Lutz},
trueauthor = {Shelah, Saharon and Str{\"{u}}ngmann, Lutz},
fromwhere = {IL,D},
journal = {Rocky Mountain Journal of Mathematics},
note = {Proceedings of the Second Honolulu Conf. on Abelian Groups and
Modules (Honolulu, HI, 2001). arxiv:math.LO/0107208 },
pages = {1617--1626},
title = {{Cotorsion theories cogenerated by $\aleph_1$ free
Abelian groups}},
volume = {32},
year = {2002},
abstract = {Given an $\aleph_1$-free abelian group $G$ we characterize
the class ${\mathfrak C_G}$ of all torsion abelian groups $T$
satisfying ${\rm Ext}(G,T)=0$ assuming the continuum hypothesis CH.
Moreover, in G{\"{o}}del'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.},
},
@article{BShT:774,
author = {Bartoszynski, Tomek and Shelah, Saharon and Tsaban, Boaz},
trueauthor = {Bartoszy\'nski, Tomek and Shelah, Saharon and Tsaban,
Boaz},
fromwhere = {1,IL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0112262 },
title = {{Additivity Properties of Topological Diagonalizations}},
volume = {68},
year = {2003},
abstract = {In a work of Just, Miller, Scheepers and Szeptycki it was
asked whether certain diagonalization properties for sequences of
open covers are provably closed under taking finite or
countable unions. In a recent work, Scheepers proved that one of the
classes in question is closed under taking countable unions. In this
paper we show that none of the remaining classes is provably closed
under taking finite unions, and thus settle the problem. We also show
that one of these properties is consistently (but not provably)
closed under taking unions of size less than the continuum, by relating
a combinatorial version of this problem to the Near Coherence
of Filters (NCF) axiom, which asserts that the Rudin-Keisler
ordering is downward directed.},
},
@article{Sh:775,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0212249 },
pages = {527--560},
title = {{Middle Diamond}},
volume = {44},
year = {2005},
abstract = {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.},
},
@article{HShV:776,
author = {Hyttinen, Tapani and Shelah, Saharon and Vaananen, Jouko},
trueauthor = {Hyttinen, Tapani and Shelah, Saharon and
V{\"{a}}{\"{a}}n{\"{a}}nen, Jouko},
fromwhere = {SF,IL,SF},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/0212234 },
pages = {79--96},
title = {{More on the Ehrenfeucht-Fra{\"\i}ss\'e game of length
$\omega_1$}},
volume = {175},
year = {2002},
abstract = { Let $A$ and $B$ be two first order structures of the
same relational vocabulary $L$. The Ehrenfeucht-Fra{\"\i}ss\'e-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 $2^\omega\leq\omega_3$.},
},
@article{RoSh:777,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/0210205 },
pages = {109--174},
title = {{Sheva-Sheva-Sheva: Large Creatures}},
volume = {159},
year = {2007},
abstract = {We develop the theory of the forcing with trees and
creatures for an inaccessible $\lambda$ continuing Ros{\l}anowski 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 Ros{\l}anowski and Shelah [RoSh:655].)},
},
@article{MdSh:778,
author = {Mildenberger, Heike and Shelah, Saharon},
trueauthor = {Mildenberger, Heike and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0112287 },
pages = {627--647},
title = {{Specialising Aronszajn trees by countable approximations}},
volume = {42},
year = {2003},
abstract = {We show that there are proper forcings based upon countable
trees of creatures that specialize a given Aronszajn tree.},
},
@article{LrSh:779,
author = {Larson, Paul and Shelah, Saharon},
trueauthor = {Larson, Paul and Shelah, Saharon},
fromwhere = {1, IL},
journal = {Colloquium Mathematicum},
pages = {1--13},
title = {{Consistency of a strong uniformization principle}},
volume = {146},
year = {2017},
abstract = {We prove the consistency of a strong uniformization for
some $\aleph_1$ branches ${}^{\omega>}\omega$. As a consequence we
get the consistency of a relative speaking on groups.},
},
@article{GbSh:780,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Bulletin of the London Mathematical Society},
note = { arxiv:math.LO/0112264 },
pages = {289--292},
title = {{Characterizing automorphism groups of ordered abelian
groups}},
volume = {35},
year = {2003},
abstract = {We want to characterize the groups isomorphic to
full automorphism groups of ordered abelian groups. The result
will follow from classical theorems on ordered groups adding an
argument from proofs used to realize rings as endomorphism rings of
abelian groups.},
},
@article{KjSh:781,
author = {Kojman, Menachem and Shelah, Saharon},
trueauthor = {Kojman, Menachem and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.GN/0112265 },
pages = {1619--1622},
title = {{van der Waerden spaces and Hindman spaces are not the same}},
volume = {131},
year = {2003},
abstract = {A Hausdorff topological space $X$ is {\em 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 {\em Hindman} if for every
sequence $(x_n)_n$ in $X$ there is an {\em 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.},
},
@article{Sh:782,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Advances in Applied Mathematics},
note = { arxiv:math.LO/0112213 },
pages = {217--251},
title = {{On the Arrow property}},
volume = {34},
year = {2005},
abstract = {Let $X$ be a finite set of alternatives. A choice function
$c$ is a mapping which assigns to nonempty subsets $S$ of $X$ an
element $c(S)$ of $S$. A {\it 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.},
},
@article{Sh:783,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/0406440 },
pages = {1-60},
title = {{Dependent first order theories, continued}},
volume = {173},
year = {2009},
abstract = {A dependent theory is a (first order complete theory)
$T$ which does not have the independence property. A major result here
is: if we expand a model of $T$ by the traces on it of sets definable
in a bigger model then we preserve its being dependent. Another
one justifies the cofinality restriction in the theorem (from a
previous work) saying that pairwise perpendicular indiscernible
sequences, can have arbitrary dual-cofinalities in some models
containing them. We introduce ``strongly dependent'' and look at
definable groups; and also at dividing, forking and relatives.},
},
@article{Sh:784,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0112286 },
pages = {285--295},
title = {{Forcing axiom failure for any $\lambda > \aleph_1$}},
volume = {43},
year = {2004},
abstract = {David Aspero asks on the possibility of having Forcing
axiom $FA_{{\aleph_2}}({\frak K})$, where ${\frak K}$ 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 $<\lambda$.},
},
@article{GShS:785,
author = {Goebel, Ruediger and Shelah, Saharon and Struengmann, Lutz},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon and
Str{\"{u}}ngmann, Lutz},
fromwhere = {D,IL,D},
journal = {Canadian Journal of Mathematics},
note = { arxiv:math.LO/0112214 },
pages = {750--765},
title = {{Almost-Free $E$-Rings of Cardinality $\aleph_1$}},
volume = {55},
year = {2003},
abstract = {An $E$-ring is a unital ring $R$ such that every
endomorphism of the underlying abelian group $R^+$ is multiplication by
some ring-element. The existence of almost-free $E$-rings of
cardinality greater than $2^{\aleph_0}$ is undecidable in ZFC. While
they exist in Goedel's universe, they do not exist in other models of
set theory. For a regular cardinal
$\aleph_1\leq\lambda\leq 2^{\aleph_0}$ we construct $E$-rings of
cardinality $\lambda$ in ZFC which have $\aleph_1$-free additive
structure. For $\lambda= \aleph_1$ we therefore obtain the existence of
almost-free $E$-rings of cardinality $\aleph_1$ in ZFC.},
},
@article{ShSr:786,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
fromwhere = {IL, 3},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/0212233 },
pages = {197--235},
title = {{Possible Cardinalities of Maximal Abelian Subgroups of
Quotients of Permutation Groups of the Integers}},
volume = {196},
year = {2007},
abstract = {The maximality of Abelian subgroups play a role in
various parts of group theory. For example, Mycielski has extended
a classical result of Lie groups and shown that a maximal
Abelian subgroup of a compact connected group is connected and,
furthermore, all the maximal Abelian subgroups are conjugate. For
finite symmetric groups the question of the size of maximal
Abelian subgroups has been examined by Burns and Goldsmith in 1989
and Winkler in 1993. We show that there is not much interest
in generalizing this study to infinite symmetric groups;
the cardinality of any maximal Abelian subgroup of the symmetric
group of the integers is $2^{\aleph_0}$. Our purpose is also to
examine the size of maximal Abelian subgroups for a class of groups
closely related to the the symmetric group of the integers; these arise
by taking an ideal on the integers, considering the subgroup of
all permutations which respect the ideal and then taking the quotient
by the normal subgroup of permutations which fix all integers except
a set in the ideal. We prove that the maximal size of Abelian subgroups
in such groups is sensitive to the nature of the ideal as well as
various set theoretic hypotheses.},
},
@article{ShVs:787,
author = {Shelah, Saharon and Vaisanen, Pauli},
trueauthor = {Shelah, Saharon and V{\"{a}}is{\"{a}}nen, Pauli},
fromwhere = {IL,SF},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/0212063 },
pages = {147--173},
title = {{Almost free groups and Ehrenfeucht-Fra{\"\i}ss\'e games
for successors of singular cardinals}},
volume = {118},
year = {2002},
abstract = {We strengthen non-structure theorems for almost free
Abelian groups by studying long Ehrenfeucht-Fra{\"\i}ss\'e 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.},
},
@article{KoSh:788,
author = {Komjath, Peter and Shelah, Saharon},
trueauthor = {Komj\'{a}th, P\'eter and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Journal of Graph Theory.},
note = { arxiv:math.LO/0212064 },
pages = {28--38},
title = {{Finite subgraphs of uncountably chromatic graphs}},
volume = {49},
year = {2005},
abstract = {It is consistent that for every monotonically
increasing function $f:\omega\to\omega$ there is a graph with size
and chromatic number $\aleph_1$ in which every $n$-chromatic
subgraph has at least $f(n)$ elements ($n\geq 3$). This solves a \$
250 problem of Erd\H{o}s. 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 ${\rm
Chr}(Y)\leq \aleph_2$.},
},
@article{ShUs:789,
author = {Shelah, Saharon and Usvyatsov, Alex},
trueauthor = {Shelah, Saharon and Usvyatsov, Alex},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/0303325 },
pages = {245--270},
title = {{Banach spaces and groups - order properties and universal
models}},
volume = {152},
year = {2006},
abstract = {We deal with two natural examples of almost-elementary
classes: the class of all Banach spaces (over ${\mathbb R}$ or
${\mathbb C}$) and the class of all groups. We show both of these
classes do not have the strict order property, and find the exact place
of each one of them in Shelah's $SOP_n$ (strong order property of order
$n$) hierarchy. Remembering the connection between this hierarchy and
the existence of universal models, we conclude, for example, that
there are ``few'' universal Banach spaces (under isometry) of
regular cardinalities. },
},
@article{ShVa:790,
author = {Shelah, Saharon and Vaananen, Jouko},
trueauthor = {Shelah, Saharon and V{\"{a}}{\"{a}}n{\"{a}}nen, Jouko},
fromwhere = {IL,SF},
journal = {Mathematical Logic Quarterly},
note = { arxiv:math.LO/0405016 },
pages = {151--164},
title = {{Recursive logic frames}},
volume = {52},
year = {2006},
abstract = {We define the concept of a {\em logic frame}, which extends
the concept of an abstract logic by adding the concept of a syntax
and an axiom system. In a {\em 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.},
},
@article{ShZa:791,
author = {Shelah, Saharon and Zapletal, Jindrich},
fromwhere = {IL,1},
journal = {Mathematical Research Letters},
note = { arxiv:math.LO/0212041 },
pages = {585--595},
title = {{Duality and the PCF theory}},
volume = {9},
year = {2002},
abstract = {We consider natural cardinal invariants ${\mathfrak hm}_n$
and prove several duality theorems, saying roughly: if $I$ is a
suitably definable ideal and provably ${\rm cov}(I)\geq{\mathfrak
hm}_n$, then ${\rm non}(I)$ is provably small. The proofs integrate
the determinacy theory, forcing and pcf theory.},
},
@article{ShZa:792,
author = {Shelah, Saharon and Zapletal, Jindrich},
fromwhere = {IL,1},
journal = {Commentationes Mathematicae Universitatis Carolinae},
note = { arxiv:math.LO/0212042 },
pages = {9--21},
title = {{Games with creatures}},
volume = {44,1},
year = {2003},
abstract = {Many forcing notions obtained using the creature
technology are naturally connected with certain integer games},
},
@article{KKSh:793,
author = {Kojman, Menahem and Kubis, Wieslaw and Shelah, Saharon},
trueauthor = {Kojman, Menahem and Kubi\'s, Wies{\l}aw and Shelah,
Saharon},
fromwhere = {IL,IL,IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/0406441 },
pages = {3357--3365},
title = {{On two problems of Erd\H os and Hechler: New methods
in singular Madness}},
volume = {132},
year = {2004},
abstract = {For an infinite cardinal $\mu$, ${\rm MAD}(\mu)$ denotes the
set of all cardinalities of nontrivial maximal almost disjoint
families over $\mu$. Erd{\H o}s 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 $\mu$. \endgraf 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.},
},
@article{Sh:794,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/0404323 },
pages = {95--111},
title = {{Reflection implies the SCH}},
volume = {198},
year = {2008},
abstract = {We prove that, e.g., if $\mu>{\rm cf}(\mu)=\aleph_0$ and
$\mu> 2^{\aleph_0}$ and every stationary family of countable subsets
of $\mu^+$ reflect in some subset of $\mu^+$ of cardinality
$\aleph_1$, then the SCH for $\mu^+$ (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.},
},
@article{JuSh:795,
author = {Juhasz, Istvan and Shelah, Saharon},
trueauthor = {Juh\'asz, Istv\'an and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Topology and its Applications},
note = { arxiv:math.LO/0212027 },
pages = {103--108},
title = {{Generic left-separated spaces and calibers}},
volume = {132},
year = {2003},
abstract = {We use a natural forcing to construct a left-separated
topology on an arbitrary cardinal $\kappa$. The resulting
left-separated space $X_\kappa$ is also 0-dimensional $T_2$,
hereditarily Lindel{\"{o}}f, 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.
\endgraf We also prove it consistent that for every countable set $A$
of uncountable regular cardinals there is a hereditarily
Lindel{\"{o}}f $T_3$ space $X$ such that $\varrho=cf(\varrho) >\omega$
is a caliber of $X$ exactly if $\varrho\not\in A$. },
},
@article{KoSh:796,
author = {Komjath, Peter and Shelah, Saharon},
trueauthor = {Komj\'{a}th, P\'eter and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Combinatorics Probability and Computing},
note = {Special issue on Ramsey theory. arxiv:math.LO/0212022 },
pages = {621--626},
title = {{A partition theorem for scattered order types}},
volume = {12},
year = {2003, no.5-6},
abstract = {If $\phi$ is a scattered order type, $\mu$ a cardinal,
then there exists a scattered order type $\psi$ such that $\psi
\to [\phi]^{1}_{\mu,\aleph_0}$ holds. },
},
@article{Sh:797,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of the American Mathematical Society},
note = { arxiv:1005.2806 },
pages = {395--427},
title = {{Nice infinitary logics}},
volume = {25},
year = {2012},
abstract = {We deal with soft model theory of infinitary logics. We
find a logic between ${mathbb L}_{\infty,\aleph_0}$ and ${\mathbb
L}_{\infty, \infty}$ which has some striking properties. First, it
has interpolations (it was known that each of those logics fail
interpolation though the pair has). Second, well ordering is not
characterized in a strong way. Third, it can be characterized as the
maximal such nice logic (in fact, is the maximal logic stronger than
${\mathbb L}_{\infty, \aleph_0}$ and which satisfies ``well ordering is
not characterized in a strong way'').},
},
@article{ShVa:798,
author = {Shelah, Saharon and Vaananen, Jouko},
trueauthor = {Shelah, Saharon and V{\"{a}}{\"{a}}n{\"{a}}nen, Jouko},
fromwhere = {IL,SF},
journal = {Preprint},
title = {{The $\Delta$--closure of $L(Q_1)$ is not finitely
generated, assuming CH}},
abstract = {We prove that, assuming CH, the logic $\Delta(L(Q_1))$ is
not finitely generated. This answers a long-standing open problem},
},
@article{MRSh:799,
author = {Matet, Pierre and Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Matet, Pierre and Ros{\l}anowski, Andrzej and Shelah,
Saharon},
fromwhere = {F,1,IL},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.LO/0210087 },
pages = {4813--4837},
title = {{Cofinality of the nonstationary ideal}},
volume = {357},
year = {2005},
abstract = {We show that the reduced cofinality of the
nonstationary ideal $NS_\kappa$ on a regular uncountable cardinal
$\kappa$ may be less than its cofinality, where the reduced cofinality
of $NS_\kappa$ is the least cardinality of any family $F$
of nonstationary subsets of $\kappa$ such that every
nonstationary subset of $\kappa$ can be covered by less than $\kappa$
many members of $F$.},
},
@article{Sh:800,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Preprint},
title = {{On complicated models and compact quantifiers}},
abstract = {We look here again at building models $M$ with second
order properties, in particular every isomorphism between
two interpretations of a theory $t$ in $M$ is definable in $M$.},
},
@article{DoSh:801,
author = {Doron, Mor and Shelah, Saharon},
trueauthor = {Doron, Mor and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0405091 },
pages = {1297--1324},
title = {{A dichotomy in classifying quantifiers for finite models}},
volume = {70},
year = {2005},
abstract = {We consider a family $\mathfrak{U}$ of finite universes.
The second order quantifier $Q_{\mathfrak{R}}$, means for each
$U\in {\mathfrak{U}}$ quantifying over a set of
$n({\mathfrak{R}})$-place relations isomorphic to a given relation. We
define a natural partial order on such quantifiers called
interpretability. We show that for every $Q_{\mathfrak {R}}$, ever
$Q_{\mathfrak {R}}$ is interpretable by quantifying over subsets of $U$
and one to one functions on $U$ both of bounded order, or the
logic $L(Q_{\mathfrak{R}})$ (first order logic plus the
quantifier $Q_{\mathfrak{R}}$) is undecidable.},
},
@article{KbSh:802,
author = {Kubis, Wieslaw and Shelah, Saharon},
trueauthor = {Kubi\'s, Wies{\l}aw and Shelah, Saharon},
fromwhere = {PL,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/0212026 },
pages = {145--161},
title = {{Analytic Colorings}},
volume = {121},
year = {2003},
abstract = {We investigate the existence of perfect homogeneous sets
for analytic colorings. An {\em 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.},
},
@article{ShSm:803,
author = {Shelah, Saharon and Struengmann, Lutz},
trueauthor = {Shelah, Saharon and Str{\"{u}}ngmann, Lutz},
fromwhere = {IL,D},
journal = {Quarterly Journal of Mathematics},
pages = {353--365},
title = {{Large Indecomposable Minimal Groups}},
volume = {60},
year = {2009},
abstract = {Assuming $V=L$ we prove that there exist
indecomposable almost-free minimal groups of size $\lambda$ for every
regular cardinal $\lambda$ below the first weakly compact cardinal.
This is to say that there are indecomposable almost-free torsion-free
abelian groups of cardinality $\lambda$ which are isomorphic to all of
their finite index subgroups.},
},
@article{MtSh:804,
author = {Matet, Pierre and Shelah, Saharon},
trueauthor = {Matet, Pierre and Shelah, Saharon},
fromwhere = {F,IL},
journal = {Preprint},
note = { arxiv:math.LO/0407440 },
title = {{Positive partition relations for $P_\kappa(\lambda)$}},
abstract = {Let $\kappa$ a regular uncountable cardinal and $\lambda$
a cardinal $>\kappa$, and suppose $\lambda^{<\kappa}$ is less than
the covering number for category ${\rm cov}({\mathcal
M}_{\kappa, \kappa})$. Then \endgraf (a)
$I_{\kappa,\lambda}^+\mathop{\longrightarrow}\limits^\kappa (I_{\kappa,
\lambda}^+,\omega+1)^2,$ \endgraf (b)
$I_{\kappa,\lambda}^+\mathop{\longrightarrow}\limits^\kappa [I_{\kappa,
\lambda}^+]_{\kappa^+}^2$ if $\kappa$ is a limit cardinal, and
\endgraf (c) $I_{\kappa,\lambda}^+
\mathop{\longrightarrow}\limits^\kappa (I_{\kappa,\lambda}^+)^2$ if
$\kappa$ is weakly compact.},
},
@incollection{GSSh:805,
author = {Gitik, Moti and Schindler, Ralf and Shelah, Saharon},
booktitle = {Proceedings of the Logic Colloquium'2002 (ASL)},
fromwhere = {IL,AT,IL},
note = { arxiv:math.LO/0211439 },
pages = {172--206},
title = {{Pcf theory and Woodin cardinals}},
year = {2006},
abstract = {We prove the following two results. Theorem A: Let $\alpha$
be a limit ordinal. Suppose that $2^{|\alpha|}<\aleph_\alpha$ and
$2^{|\alpha|^+}<\aleph_{| \alpha|^+}$, whereas
$\aleph_\alpha^{|\alpha|}>\aleph_{| \alpha|^+}$. Then for all
$n<\omega$ and for all bounded $X\subset \aleph_{|\alpha|^+}$,
$M_n^\#(X)$ exists. \endgraf 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. \endgraf Theorem A answers a question of Gitik and
Mitchell, and Theorem B yields a lower bound for an assertion discussed
in Gitik, M., {\em Introduction to Prikry type forcing notions}, in:
Handbook of set theory, Foreman, Kanamori, Magidor (see Problem 4
there). \endgraf 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.},
},
@article{Sh:806,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Real Analysis Exchange},
note = { arxiv:math.LO/0211438 },
pages = {477--480},
title = {{Martin's Axiom and Maximal Orthogonal Families}},
volume = {28},
year = {2002/03, no.2},
abstract = {It is shown that Martin's Axiom for $\sigma$-centred partial
orders implies that every maximal orthogonal family in ${\mathbb
R}^{\mathbb N}$ is of size $2^{\aleph_0}$.},
},
@article{BrSh:807,
author = {Bartoszynski, Tomek and Shelah, Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0211023 },
pages = {769--779},
title = {{Strongly meager sets of size continuum}},
volume = {42},
year = {2003},
abstract = {We will construct several models where there are no strongly
meager sets of size $2^{\aleph_0}$.},
},
@article{GoSh:808,
author = {Goldstern, Martin and Shelah, Saharon},
trueauthor = {Goldstern, Martin and Shelah, Saharon},
fromwhere = {AT, IL},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math.RA/0212379 },
pages = {3525--3551},
title = {{Clones from Creatures}},
volume = {357},
year = {2005},
abstract = {We show that (consistently) there is a clone $C$ on a
countable set such that the interval of clones above $C$ is linearly
ordered and has no coatoms.},
},
@article{ShSr:809,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
fromwhere = {IL, 3},
journal = {Advances in Mathematics},
note = { arxiv:math.LO/0405092 },
pages = {403--426},
title = {{Comparing the uniformity invariants of null sets for different
measures}},
volume = {192},
year = {2005},
abstract = {It is shown to be consistent with set theory that
the uniformity invariant for Lebesgue measure is strictly greater
than the corresponding invariant for Hausdorff $r$-dimensional
measure where $0 \kappa^+$ (improving the lower bound). Second we show that for many
such $\kappa$ there is a group of height $> 2^\kappa$, so proving
that the upper bound essentially cannot be improved.},
},
@article{GeSh:811,
author = {Geschke, Stefan and Shelah, Saharon},
trueauthor = {Geschke, Stefan and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Topology and its Applications},
note = { arxiv:math.LO/0211399 },
pages = {241--253},
title = {{Some notes concerning the homogeneity of Boolean algebras and
Boolean spaces}},
volume = {133},
year = {2003},
abstract = {We consider homogeneity properties of Boolean algebras that
have nonprincipal ultrafilters which are countably generated. It is
shown that a Boolean algebra $B$ is homogeneous if it is the union of
countably generated nonprincipal ultrafilters and has a dense subset
$D$ such that for every $a\in D$ the relative algebra $B\restriction
a:=\{b\in B:b\leq a\}$ is isomorphic to $B$. In particular, the free
product of countably many copies of an atomic Boolean algebra is
homogeneous. Moreover, a Boolean algebra $B$ is homogeneous if it
satisfies the following conditions: (i) $B$ has a countably generated
ultrafilter, (ii) $B$ is not c.c.c., and (iii) for every $a\in
B\setminus\{0\}$ there are finitely many automorphisms $h_1,\dots,h_n$
of $B$ such that $1=h_1(a)\cup\dots\cup h_n(a)$.},
},
@article{ShVV:812,
author = {Shelah, Saharon and Vaisanen, Pauli and Vaananen, Jouko},
trueauthor = {Shelah, Saharon and V{\"{a}}is{\"{a}}nen, Pauli and
V{\"{a}}{\"{a}}n{\"{a}}nen, Jouko},
fromwhere = {IL,SF,SF},
journal = {Fundamenta Mathematicae},
pages = {193--214},
title = {{On Ordinals Accessible by Infinitary Languages}},
volume = {186},
year = {2005},
abstract = {Let $\lambda$ be an infinite cardinal number. The
ordinal number $\delta(\lambda)$ is the least ordinal $\gamma$ such
that if $\phi$ is any sentence of $L_{\lambda^+\omega}$, with a
unary predicate $D$ and a binary predicate $\prec$, and $\phi$ has a
model $M$ with $\langle D^M,\prec^M\rangle$ a well-ordering of
type $\ge\gamma$, then $\phi$ has a model $M'$ where $\langle
D^{M'}, \prec^{M'}\rangle$ is non-well-ordered. One of the
interesting properties of this number is that the Hanf number
of $L_{\lambda^+\omega}$ is exactly $\beth_{\delta(\lambda)}$. We
show the following theorem. \endgraf {\bf 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 $\delta(\lambda)= \theta$.},
},
@article{MPSh:813,
author = {Matet, Pierre and Pean, Cedric and Shelah, Saharon},
trueauthor = {Matet, Pierre and P\'ean, C\'edric and Shelah, Saharon},
fromwhere = {F,F,IL},
journal = {Israel Journal of Mathematics},
pages = {253--283},
title = {{Cofinality of normal ideals on $P_\kappa(\lambda)$, II}},
volume = {150},
year = {2005},
abstract = {We investigate sufficient conditions for the existence
of members of ${\mathcal E}$, a normal filter on $[\lambda]^{<\kappa}$,
which contains no unbounded, nonstationary subsets
continuing [Sh:698].},
},
@article{EShT:814,
author = {Eklof, Paul C. and Shelah, Saharon and Trlifaj, Jan},
trueauthor = {Eklof, Paul C. and Shelah, Saharon and Trlifaj, Jan},
fromwhere = {1,IL},
journal = {Journal of Algebra},
note = { arxiv:math.LO/0405117 },
pages = {572--578},
title = {{On the cogeneration of cotorsion pairs}},
volume = {227},
year = {2004},
abstract = {Let $R$ be a Dedekind domain. Enochs' solution of the Flat
Cover Conjecture was extended as follows: ($*$) If $\mathfrak C$ is
a cotorsion pair generated by a class of cotorsion modules,
then $\mathfrak C$ is cogenerated by a set. We show that ($*$) is
the best result provable in ZFC in case $R$ has a countable
spectrum: the Uniformization Principle $UP^{+}$ implies that $\mathfrak
C$ is not cogenerated by a set whenever $\mathfrak C$ is a cotorsion
pair generated by a set which contains a non-cotorsion module.},
},
@article{BBSh:815,
author = {Baizhanov, Bektur and Baldwin, John and Shelah, Saharon},
trueauthor = {Baizhanov, Bektur and Baldwin, John and Shelah, Saharon},
fromwhere = {K,1,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0303324 },
pages = {142--150},
title = {{Subsets of superstable structures are weakly benign}},
volume = {70},
year = {2005},
abstract = {Baizhanov and Baldwin introduced the notion of benign
and weakly benign sets to investigate the preservation of stability
by naming arbitrary subsets of a stable structure. They connected
the notion with works of Baldwin, Benedikt, Bouscaren,
Casanovas, Poizat, and Ziegler. Stimulated by those results, we
investigate here the existence of benign or weakly benign sets.},
},
@article{Sh:816,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Discrete Mathematics},
note = { arxiv:math.CO/0405119 },
pages = {2349--2364},
title = {{What majority decisions are possible}},
volume = {309},
year = {2009},
abstract = {Suppose we are given a family of choice functions on pairs
from a given finite set (with at least three elements) closed
under permutations of the given set. The set is considered the set
of alternatives (say candidates for an office). The question is,
what are the choice functions $\bold c$ 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, $\bold c\{x,y\}$ is chosen by the preference of the majority
of voters. We give full characterization.},
},
@article{Sh:817,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Scientiae Mathematicae Japonicae},
note = { arxiv:math.LO/0405158 },
pages = {351--355},
title = {{Spectra of monadic second order sentences}},
volume = {59, No. 2; (special issue: e9, 555--559)},
year = {2004},
abstract = {For a monadic sentence $\psi$ in the finite vocabulary we
show that the spectra, the set of cardinalities of models of $\psi$
is almost periodic under reasonable conditions. The first is
that every model is so called ``weakly $k$-decomposable''. The second
is that we restrict ourselves to a nice class of models constructed
by some recursion.},
},
@article{KShTS:818,
author = {Kramer, Linus and Shelah, Saharon and Tent, Katrin and
Thomas, Simon},
trueauthor = {Kramer, Linus and Shelah, Saharon and Tent, Katrin and
Thomas, Simon},
fromwhere = {D,IL,D,1},
journal = {Advances in Mathematics},
note = { arxiv:math.GT/0306420 },
pages = {142--173},
title = {{Asymptotic cones of finitely presented groups}},
volume = {193},
year = {2005},
abstract = {Let $G$ be a connected semisimple Lie group with at least
one absolutely simple factor $S$ such that ${\mathbb
R}\mbox{-rank}(S) \geq 2$ and let $\Gamma$ be a uniform lattice in
$G$. \endgraf (a) If $CH$ holds, then $\Gamma$ has a unique asymptotic
cone up to homeomorphism. \endgraf (b) If $CH$ fails, then $\Gamma$
has $2^{2^{\omega}}$ asymptotic cones up to homeomorphism.},
},
@article{EiSh:819,
author = {Eisworth, Todd and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
pages = {1287--1309},
title = {{Successors of singular cardinals and coloring theorems. II}},
volume = {74},
year = {2009},
abstract = {In this paper, we investigate the extent to which techniques
used in [10], [2], and [3] -- developed to prove coloring theorems at
successors of singular cardinals of uncountable cofinality -- can be
extended to cover the countable cofinality case.},
},
@article{Sh:820,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Notre Dame Journal of Formal Logic},
note = { arxiv:math.LO/0405159 },
pages = {159--177},
title = {{Universal Structures}},
volume = {58},
year = {2017},
abstract = {We deal with the existence of universal members in a
given cardinality for several classes. First we deal with classes of
Abelian groups, specifically with the existence of universal members
in cardinalities which are strong limit singular of countable
cofinality. Second, we then deal with (variants of) the oak property
(from a work of Dzamonja and the author), a property of complete
first order theories, sufficient for the non-existence of universal
models under suitable cardinal assumptions. Third, we prove that the
oak property holds for the class of groups (naturally interpreted, so
for quantifier free formulas) and deal more with the existence of
universals.},
},
@article{HLSh:821,
author = {Hyttinen, Tapani and Lessmann, Olivier and Shelah, Saharon},
trueauthor = {Hyttinen, Tapani and Lessmann, Olivier and Shelah,
Saharon},
fromwhere = {F,UK,IL},
journal = {Journal of Mathematical Logic},
note = { arxiv:math.LO/0406481 },
pages = {1--47},
title = {{Interpreting groups and fields in some nonelementary
classes}},
volume = {5},
year = {2005},
abstract = {This paper is concerned with extensions of geometric
stability theory to some nonelementary classes. We prove the
following theorem: \endgraf {\bf Theorem}: Let $\mathfrak{C}$ be a
large homogeneous model of a stable diagram $D$. Let $p, q \in S_D(A)$,
where $p$ is quasiminimal and $q$ unbounded. Let $P = p(\mathfrak{C})$
and $Q = q(\mathfrak{C})$. Suppose that there exists an integer $n <
\omega$ such that $dim(a_1\dots a_{n}/A \cup C)=n$, for any
independent $a_1,\dots, a_{n} \in P$ and finite subset $C \subseteq Q$,
but $dim(a_1\dots a_n a_{n+1}/A\cup C)\leq n$, for some
independent $a_1,\dots,a_n,a_{n+1}\in P$ and some finite subset
$C\subseteq Q$. Then $\mathfrak{C}$ interprets a group $G$ which
acts on the geometry $P'$ obtained from $P$. Furthermore, either
$\mathfrak{C}$ interprets a non-classical group, or $n = 1,2,3$ and
\endgraf If $n = 1$ then $G$ is abelian and acts regularly on
$P'$. \endgraf If $n = 2$ the action of $G$ on $P'$ is isomorphic to
the affine action of $K \rtimes K^*$ on the algebraically closed field
$K$. \endgraf 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 $K$.},
},
@article{BGSh:822,
author = {Boerner, Ferdinand and Goldstern, Martin and Shelah,
Saharon},
trueauthor = {B{\"{o}}rner, Ferdinand and Goldstern, Martin and Shelah,
Saharon},
fromwhere = {D,A,IL},
journal = {Algebra Universalis},
note = { arxiv:math.LO/0309165 },
title = {{Automorphisms and strongly invariant relations}},
volume = {accepted},
abstract = {We investigate characterizations of the Galois
connection ${\rm sInv}$--${\rm Aut}$ between sets of finitary relations
on a base set $A$ and their automorphisms. In particular,
for $A=\omega_1$, we construct a countable set $R$ of relations that
is closed under all invariant operations on relations and under
arbitrary intersections, but is not closed under ${\rm sInv}{\rm Aut}$.
\endgraf Our structure $(A,R)$ has an $\omega$-categorical first
order theory. A higher order definable well-order makes it
rigid, but any reduct to a finite language is homogeneous.},
},
@article{BmSh:823,
author = {Bergman, George M. and Shelah, Saharon},
trueauthor = {Bergman, George M. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Algebra Universalis},
note = {a special issue in honor of Walter Taylor.
arxiv:math.GR/0401305 },
pages = {137--173},
title = {{Closed subgroups of the infinite symmetric group}},
volume = {55},
year = {2006},
abstract = {Let $S={\rm Sym}(\omega)$ be the group of all permutations
of the natural numbers, and for subgroups $G_1,G_2\leq S$ let us
write $G_1\approx G_2$ if there exists a finite set $U\subseteq S$
such that $\langle G_1\cup U\rangle=\langle G_2\cup U\rangle$. It
is shown that the subgroups closed in the function topology on $S$
lie in precisely four equivalence classes under this relation. Given
an arbitrary subgroup $G\leq S$, which of these classes the closure
of $G$ belongs to depends on which of the following statements
about pointwise stabilizer subgroups $G_{(\Gamma)}$ of finite
subsets $\Gamma\subseteq\omega$ holds: \endgraf (i) For every finite
set $\Gamma$, the subgroup $G_{(\Gamma)}$ has at least one infinite
orbit in $\omega$. \endgraf (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.
\endgraf (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 $G_{(\Gamma)}=\{1\}.$ \endgraf (iv) There
exist finite sets $\Gamma$ such that $G_{(\Gamma)}=\{1\}$.},
},
@article{Sh:824,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:math.LO/0404149 },
pages = {437--447},
title = {{Two cardinals models with gap one revisited}},
volume = {51},
year = {2005},
abstract = {We succeed to say something on the identities of
$(\mu^+,\mu)$ when $\mu>\theta>{\rm cf}(\mu)$, $\mu$ strong
limit $\theta$--compact. This hopefully will help to prove the
consistency of ``some pair $(\mu^+,\mu)$ is not compact'', however,
this has not been proved.},
},
@article{KnSh:825,
author = {Kanovei, Vladimir and Shelah, Saharon},
trueauthor = {Kanovei, Vladimir and Shelah, Saharon},
fromwhere = {RU,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0311165 },
pages = {159--164},
title = {{A definable nonstandard model of the reals}},
volume = {69},
year = {2004},
abstract = {We prove, in ZFC, the existence of a definable,
countably saturated elementary extension of the reals.},
},
@incollection{BrSh:826,
author = {Bartoszynski, Tomek and Shelah, Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Shelah, Saharon},
booktitle = {Logic Colloquium 2004},
fromwhere = {1,IL},
note = {Proceedings of the AMS. arxiv:math.LO/0311064 },
pages = {18--32},
publisher = {Association of Symbolic Logic, Chicago},
series = {Lecture Notes in Logic, 29},
title = {{On the density of Hausdorff ultrafilters}},
year = {2008},
abstract = {An ultrafilter $U$ is Hausdorff if for any two functions
$f,g \in \omega^\omega$, $f(U)=g(U)$ iff $f \restriction
X=g\restriction X$ for some $X \in U$. We will show that the statement
that Hausdorff ultrafilters are dense in the Rudin-Keisler order
is independent of ZFC},
},
@article{KjSh:827,
author = {Kojman, Menachem and Shelah, Saharon},
trueauthor = {Kojman, Menachem and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/0406530 },
pages = {309--334},
title = {{Almost Isometric Embeddings of Metric Spaces}},
volume = {155},
year = {2006},
abstract = {We investigate a relations of {\em almost isometric
embedding} and {\em almost isometry} between metric spaces and prove
that with respect to these relations: \endgraf (1) There is a
countable universal metric space. \endgraf (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. \endgraf (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
$\aleph_2<2^{\aleph_0}$. \endgraf We also prove that various spaces
$X$ satisfy that if a space $X$ is almost isometric to $X$ than $Y$ is
isometric to $X$.},
},
@article{KrSh:828,
author = {Kellner, Jakob and Shelah, Saharon},
trueauthor = {Kellner, Jakob and Shelah, Saharon},
fromwhere = {A,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0405081 },
pages = {914--945},
title = {{Preserving Preservation}},
volume = {70, 3},
year = {2005},
abstract = {We prove that the property ``$P$ doesn't make the old reals
Lebesgue null'' is preserved under countable support iterations of
proper forcings, under the additional assumption that the forcings are
nep (a generalization of Suslin proper) in an absolute way. We also
give some results for general Suslin ccc ideals.},
},
@article{Sh:829,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/0406482 },
pages = {133--160},
title = {{More on the Revised GCH and the Black Box}},
volume = {140},
year = {2006},
abstract = {We strengthen the revised GCH theorem by showing, e.g.,
that for $\lambda={\rm cf}(\lambda)>\beth_\omega$, for all but
finitely many regular $\kappa<\beth_\omega$, $\lambda$ is accessible
on cofinality $\kappa$ in a weak version of it holds. In
particular, $\lambda=2^\mu=\mu^+>\beth_\omega$ implies the diamond on
$\lambda$ is restricted to cofinality $\kappa$ for all but finitely
many $\kappa\in{\rm Reg}\cap \beth_\omega$ and we strengthen the
results on the middle diamond. Moreover, we get stronger results on
the middle diamond.},
},
@article{Sh:830,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/0407498 },
pages = {1--23},
title = {{The combinatorics of reasonable ultrafilters}},
volume = {192},
year = {2006},
abstract = {We are interested in generalizing part of the theory
of ultrafilters on $\omega$ to larger cardinals. Here we set the scene
for further investigations introducing properties of ultrafilters in
strong sense dual to being normal.},
},
@article{GbSh:831,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Journal of Pure and Applied Algebra},
pages = {230--258},
title = {{How rigid are reduced products?}},
volume = {202},
year = {2005},
abstract = {For any cardinal $\mu$ let ${\mathbb Z}^\mu$ be the
additive group of all integer-valued functions $f:\mu\to {\mathbb Z}$.
The support of $f$ is $[f]=\{i\in\mu: f(i)=f_i\ne 0\}$. Also let
${\mathbb Z}_\mu= {\mathbb Z}^\mu/{\mathbb Z}^{<\mu}$ with ${\mathbb
Z}^{<\mu}= \{f\in {\mathbb Z}^\mu: \left|[f]\right|<\mu\}$. If $\mu\le
\chi$ are regular cardinals we analyze the question when Hom$({\mathbb
Z}_\mu,{\mathbb Z}_\chi) = 0$ and obtain a complete answer under
GCH and independence results in Section 8. These results and some
extensions are applied to a problem on groups: Let the norm $\|G\|$ of
a group $G$ be the smallest cardinal $\mu$ with Hom$({\mathbb Z}_\mu,G)
\ne 0$ - this is an infinite, regular cardinal (or $\infty$). As a
consequence we characterize those cardinals which appear as norms of
groups. This allows us to analyze another problem on radicals: The norm
$\|R\|$ of a radical $R$ is the smallest cardinal $\mu$ for which there
is a family $\{ G_i: i\in \mu\}$ of groups such that $R$ does not
commute with the product $\prod_{i\in\mu}G_i$. Again these norms are
infinite, regular cardinals and we show which cardinals appear as norms
of radicals. The results extend earlier work (Arch. Math. 71 (1998)
341--348; Pacific J. Math. 118(1985) 79--104; Colloq. Math. Soc.
J{\'a}nos Bolyai 61 (1992) 77--107) and a seminal result by {\L}o{\'s}
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.},
},
@article{Sh:832,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Preprint},
note = { arxiv:math.LO/1306.5399 },
title = {{Many forcing axioms for all regular uncountable cardinals}},
abstract = {Our original aim was, in Abelian group theory to prove the
consistency of: $\lambda$ is strong limit singular and for some
properties of abelian groups which are relatives of being free, the
compactness in singular fails. In fact this should work for
$R$-modules, etc. As in earlier cases part of the work is analyzing
how to move between the set theory and the algebra. \endgraf Set
theoretically we try to force a universe which satisfies G.C.H. and
diamond holds for many stationary sets but, for every regular
uncountable $\lambda$, in some sense anything which ``may'' hold for
some stationary set, does hold for some stationary set. More
specifically we try to get a universe satisfying GCH such that e.g. for
regular $\kappa < \lambda$ there are pairs $(S,B),S \subseteq
S^\lambda_\kappa$ stationary, $B \subseteq {\mathcal{H}}(\lambda)$,
which satisfies some pregiven forcing axiom related to $(S,B)$, (so
$(\lambda \backslash S)$-complete, i.e. ``trivial outside $S$'')
\underline{but} no more, i.e. slightly stronger versions fail (for
this $S$). So set theoretically we try to get a universe satisfying
G.C.H. but still satisfies ``many'', even for a maximal family in some
sense, of forcing axioms of the form ``for some stationary'' while
preserving GCH. As completion of the work lag for long, here we deal
only with the set theory.},
},
@article{GbSh:833,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Communications in Algebra},
note = { arxiv:math.LO/0504198 },
pages = {4211--4218},
title = {{On Crawley Modules}},
volume = {53},
year = {2005},
abstract = {This continues recent work in a paper by Corner, G{\"{o}}bel
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 $CH$.},
},
@incollection{GbSh:834,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
booktitle = {Abelian groups, rings, modules, and homological algebra},
fromwhere = {D,IL},
pages = {153--158},
series = {Lect. Notes Pure Appl. Math},
title = {{Torsionless linearly compact modules}},
volume = {249},
year = {2006},
},
@article{Sh:835,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0510229 },
title = {{PCF without choice}},
volume = {submitted},
abstract = {We mainly investigate model of set theory, e.g., ZF + DC +
``the family of countable subsets of $\lambda$ is well ordered for
every $\lambda$'' (really local version for a given $\lambda$). In
this frame much of pcf theory can be generalized. E.g., there is a
class of regular cardinals, and we can prove cardinal inequality.},
},
@incollection{Sh:836,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
booktitle = {Proceedings of Logic Colloquium, Helsinki, August 2003},
fromwhere = {IL},
note = { arxiv:math.LO/0404222 },
pages = {315--325},
publisher = {ASL},
title = {{On long EF-equivalence in non-isomorphic models}},
volume = {Lecture Notes in Logic 24},
year = {2006},
abstract = {There has been much interest on constructing models which
are not isomorphic of cardinality $\lambda$ but are equivalent under
the Ehrenfeucht--Fraiss\'e 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.},
},
@article{ShUs:837,
author = {Shelah, Saharon and Usvyatsov, Alex},
trueauthor = {Shelah, Saharon and Usvyatsov, Alex},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/0612350 },
pages = {157--198},
title = {{Model theoretic stability and categoricity for complete metric
spaces}},
volume = {182},
year = {2011},
abstract = {We deal with the systematic development of stability for the
context of approximate elementary submodels of a monster metric space,
which is not far, but still very distinct from the first order case.
In particular we prove the analogue of Morley's theorem for the
classes of complete metric spaces},
},
@inbook{Sh:838,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
booktitle = {Classification theory for abstract elementary classes II},
fromwhere = {IL},
note = {Chapter VII, in series Studies in Logic, volume 20, College
Publications. arxiv:0808.3020 },
title = {{Non-structure in $\lambda^{++}$ using instances of WGCH}},
abstract = {Here we try to redo, improve and continue the non-structure
parts in some works on a.e.c., which uses weak diamond, in $\lambda^+$
and $\lambda^{++}$ getting better and more results and do what is
necessary for the book on a.e.c. So we rework and improve
non-structure proofs from \cite[\S6]{Sh:87b}, \cite{Sh:88r} (or
\cite{Sh:88}), \cite{Sh:E46}, (or \cite{Sh:576}, \cite{Sh:603}) and
fulfill promises from \cite{Sh:88r},
\cite{Sh:600}, \cite{Sh:705}. Comparing with \cite{Sh:576} we make the
context closer to the examples, hence hopefully improve transparency,
though losing some generality. Toward this we work also on the
positive theory, i.e. structure side of ``low frameworks'' like almost
good $\lambda$-frames. },
},
@article{Sh:839,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Preprint},
title = {{Stable frames and weights}},
abstract = {Part I: We would like to generalize imaginary elements,
weight of ${\rm ortp}(a,M,N),{\bold P}$-weight, ${\bold P}$-simple
types, etc. from \cite[Ch.III,V,\S4]{Sh:c} to the context of good
frames. This requires allowing the vocabulary to have predicates and
function symbols of infinite arity, but it seemed that we do not
suffer any real loss. \endgraf Part II: Good frames were suggested in
\cite{Sh:h} as the (bare bones) right parallel among a.e.c. to
superstable (among elementary classes). Here we consider
$(\mu,\lambda,\kappa)$-frames as candidates for being the right
parallel to the class of $|T|^+$-saturated models of a stable theory
(among elementary classes). A loss as compared to the superstable
case is that going up by induction on cardinals is problematic (for
cardinals of small cofinality). But this arises only when we try to
lift. But this context we investigate the dimension. \endgraf Part
III: In the context of Part II, we consider the main gap problem for
the parallel of somewhat saturated model; showing we are not worse than
in the first order case.},
},
@article{Sh:840,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0504196 },
pages = {361--401},
title = {{Model theory without choice: Categoricity}},
volume = {74},
year = {2009},
abstract = {The main result is \L os 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.},
},
@article{SaSh:841,
author = {Sagi, Gabor and Shelah, Saharon},
trueauthor = {S\'agi, G\'abor and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:math.LO/0404148 },
pages = {254--257},
title = {{On topological properties of ultraproducts of finite sets}},
volume = {51},
year = {2005},
abstract = {Motivated by the model theory of higher order logics, a
certain kind of topological spaces had been introduced
on ultraproducts. These spaces are called ultratopologies.
Ultratopologies provide a natural extra topological structure for
ultraproducts and using this extra structure some preservation and
characterization theorems had been obtained for higher order logics.
\endgraf The purely topological properties of ultratopologies
seem interesting on their own right. Here we present the solutions of
two problems of Gerlits and S\'agi. More concretely we show
that \endgraf (1) there are sequences of finite sets of pairwise
different cardinality such that in their certain ultraproducts there
are homeomorphic ultratopologies and \endgraf (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.},
},
@article{Sh:842,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Preprint},
title = {{Eventual categoricity spectrum and Frames}},
abstract = {This work aims at doing parallel of \cite{Sh:705} in a wider
and axiomatic way. First, we deal with frames (as in \cite{Sh:600},
\cite{Sh:705}), starting for convenience with a fractured a.e.c.
${\frak K}$. Our basic frames ${\frak s}$ ``live'' on such ${\frak
K}$, so on some cardinals, not just one and possibly not even on an
interval of cardinals and types are only formal (and amalgamation is
not required). We then add axioms as proved to hold in
\cite{Sh:705,\S1-\S11} and proved some things. Second, we deal with
[stable] $(I,{\frak s})$-systems and the related property up to beauty
parallely to \cite[\S12]{Sh:705}, including a pure existential version.
Third, we prove the main-gap for beautiful frames; this exemplifies
that for them we have a full-fledge parallel of the theory of
superstable elementary classes. Fourth, we hope to start with an
a.e.c. definable in $\Bbb L_{\kappa,\omega},\kappa$ measurable and get
beauty from solvability in not too small a cardinal. Fifth, we hope to
prove that the categoricity and/or solvability spectrum contains or is
disjoint to an end segment of the class of cardinals.},
},
@article{MdSh:843,
author = {Mildenberger, Heike and Shelah, Saharon},
trueauthor = {Mildenberger, Heike and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/0404147 },
pages = {7--13},
title = {{Increasing the groupwise density number by c.c.c. forcing}},
volume = {149},
year = {2007},
abstract = {We try to control many cardinal characteristics by working
with a notion of orthogonality between two families of forcings. We
show that ${\mathfrak b}^+<{\mathfrak g}$ is consistent},
},
@article{ShUs:844,
author = {Shelah, Saharon and Usvyatsov, Alex},
trueauthor = {Shelah, Saharon and Usvyatsov, Alex},
fromwhere = {IL,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/0404178 },
pages = {16--31},
title = {{More on ${\rm SOP}_1$ and ${\rm SOP}_2$}},
volume = {155},
year = {2008},
abstract = {This paper continues \cite{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 \cite{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.},
},
@article{RoSh:845,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0404146 },
pages = {179--196},
title = {{Universal forcing notions and ideals}},
volume = {46},
year = {2007},
abstract = {The main result of this paper is a partial answer to
[RoSh:672, Problem 5.5]: a finite iteration of Universal Meager forcing
notions adds generic filters for many forcing notions determined
by universality parameters. We also give some results
concerning cardinal characteristics of the $\sigma$--ideals determined
by those universality parameters.},
},
@article{Sh:846,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Colloquium Mathematicum},
note = { arxiv:math.LO/0612240 },
pages = {213--220},
title = {{The spectrum of characters of ultrafilters on $\omega$}},
volume = {111, No.2},
year = {2008},
abstract = {We show the consistency of the statement: ``the set of
regular cardinals which are the characters of ultrafilters on
$\omega$ is not convex''. We also deal with the set of
$\pi$-characters of ultrafilters on $\omega$.},
},
@article{MShT:847,
author = {Mildenberger, Heike and Shelah, Saharon and Tsaban, Boaz},
trueauthor = {Mildenberger, Heike and Shelah, Saharon and Tsaban, Boaz},
fromwhere = {A,IL,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/0407487 },
pages = {60--71},
title = {{Covering the Baire space by families which are not
finitely dominating}},
volume = {140},
year = {2006},
abstract = {We show that the mentioned constellation of three
cardinal characteristices is relatively consistent to ZFC},
},
@article{MdSh:848,
author = {Mildenberger, Heike and Shelah, Saharon},
trueauthor = {Mildenberger, Heike and Shelah, Saharon},
fromwhere = {A,IL},
journal = {Journal of Applied Analysis},
pages = {47--78},
title = {{Specializing Aronszajn trees and preserving certain weak
diamonds}},
volume = {15},
year = {2009},
abstract = {We show that $\diamondsuit({\mathbb R}, {\mathcal L},
\not\in)$ together with ``all Aronszajn trees are special'' is
consistent relative to ZFC. The weak diamond for the uniformity of
Lebegue null sets was the only weak diamond in the Cicho\'n 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]},
},
@article{Sh:849,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
pages = {507--537},
title = {{Beginning of stability theory for Polish Spaces}},
volume = {214},
year = {2016},
abstract = {We consider stability theory for Polish spaces, and more
generally, for definable structures (say with elements of a set of
reals). We clarify by proving some equivalent conditions for
$\aleph_0$-stability. We succeed to prove existence of
indiscernibles under reasonable conditions; this provides strong
evidence that such a theory exists.},
},
@article{ChSh:850,
author = {Cherlin, Gregory and Shelah, Saharon},
trueauthor = {Cherlin, Gregory and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Combinatorial Theory. Ser. B},
note = { arxiv:math.LO/0512218 },
pages = {293--333},
title = {{Universal graphs with a forbidden subtree}},
volume = {97},
year = {2007},
abstract = {We show that the problem of the existence of universal
graphs with specified forbidden subgraphs can be systematically
reduced to certain critical cases by a simple pruning technique which
simplifies the underlying structure of the forbidden graphs, viewed as
trees of blocks. As an application, we characterize the trees T for
which a universal countable T-free graph exists.},
},
@article{LwSh:851,
author = {Laskowski, Michael C. and Shelah, Saharon},
trueauthor = {Laskowski, Michael C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Fundamenta Mathematicae},
pages = {95--124},
title = {{Decompositions of saturated models of stable theories}},
volume = {191},
year = {2006},
abstract = {We characterize the stable theories $T$ for which the
saturated models of $T$ admit decompositions. Additionally, we show
that when $T$ is countable the criterion can be weakened.},
},
@article{KeSh:852,
author = {Kennedy, Juliette and Shelah, Saharon},
trueauthor = {Kennedy, Juliette and Shelah, Saharon},
fromwhere = {F,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0504200 },
pages = {1261--1266},
title = {{More on regular reduced products}},
volume = {69},
year = {2004},
abstract = {The authors show, by means of a finitary
version $\square^{fin}_{\lambda,D}$ of the combinatorial
principle $\square^{b^*}_{\lambda}$, the consistency of the failure,
relative to the consistency of supercompact cardinals, of the
following: for all regular filters $D$ on a cardinal $\lambda$, if
$M_i$ and $N_i$ are elementarily equivalent models of a language of
size $\le\lambda$, then the second player has a winning strategy in
the Ehrenfeucht-Fra{\"\i}ss\'e 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.},
},
@article{Sh:853,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Algebra Universalis},
note = { arxiv:math.LO/0406531 },
pages = {91--96},
title = {{The depth of ultraproducts of Boolean Algebras}},
volume = {54},
year = {2005},
abstract = {We show that if $\mu$ is a compact cardinal then the depth
of ultraproducts of less than $\mu$ many Boolean algebras is at most
$\mu$ plus the ultraproduct of the depths of those Boolean algebras},
},
@article{BsSh:854,
author = {Blass, Andreas and Shelah, Saharon},
trueauthor = {Blass, Andreas and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Communications in Algebra},
note = { arxiv:math.LO/0504199 },
pages = {1997--2007},
title = {{Ultrafilters and partial products of infinite cyclic groups}},
volume = {33},
year = {2005},
abstract = {We consider, for infinite cardinals $\kappa$
and $\alpha\leq\kappa^+$, the group $\Pi(\kappa,<\alpha)$ of
sequences of integers, of length $\kappa$, with non-zero entries in
fewer than $\alpha$ positions. Our main result tells when
$\Pi(\kappa,<\alpha)$ can be embedded in $\Pi(\lambda,<\beta)$. The
proof involves some set-theoretic results, one about familes of finite
sets and one about families of ultrafilters. },
},
@article{ShSm:855,
author = {Shelah, Saharon and Struengmann, Lutz},
trueauthor = {Shelah, Saharon and Str{\"{u}}ngmann, Lutz},
fromwhere = {IL,D},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/0612241 },
pages = {935--943},
title = {{Filtration-equivalent $aleph_1$ separable abelian groups of
cardinality $\aleph_1$}},
volume = {161},
year = {2010},
abstract = {We show that it is consistent with ordinary set theory
$ZFC$ and the generalized continuum hypothesis that there exist
two $aleph_1$ separable abelian groups of cardinality $\aleph_1$ which
are filtration-equivalent and one is a Whitehead group but the other
is not. This solves one of the open problems of Eklof and Mekler.},
},
@article{RoSh:856,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:math.LO/0406612 },
pages = {71--86},
title = {{How much sweetness is there in the universe?}},
volume = {52},
year = {2006},
abstract = {We continue investigations of forcing notions with strong
ccc properties introducing new methods of building sweet
forcing notions. We also show that quotients of topologically sweet
forcing notions over Cohen reals are topologically sweet.},
},
@article{KuSh:857,
author = {Kuhlmann, Salma and Shelah, Saharon},
trueauthor = {Kuhlmann, Salma and Shelah, Saharon},
fromwhere = {2,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/0512220 },
pages = {284--296},
title = {{$\kappa$-bounded Exponential-Logarithmic Power Series
Fields}},
volume = {136},
year = {2005},
abstract = {In [KKSh:601] it was shown that fields of generalized
power series cannot admit an exponential function. In this paper,
we construct fields of generalized power series with {\em
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 {\em growth rates}.
This method relies on constructing lexicographic chains with many
automorphisms.},
},
@article{MShT:858,
author = {Mildenberger, Heike and Shelah, Saharon and Tsaban, Boaz},
trueauthor = {Mildenberger, Heike and Shelah, Saharon and Tsaban, Boaz},
fromwhere = {A,IL,IL},
journal = {Topology and its Applications},
note = { arxiv:math.GN/0409068 },
pages = {263--276},
title = {{The combinatorics of $\tau$-covers}},
volume = {154},
year = {2007},
abstract = {We compute the critical cardinalities for some
topological properties involving $\tau$-covers. A surprising
structure counterpart of one of the computations is that, using
Scheepers' notation: \endgraf If $X^2$ satisfies $S_{\rm
fin}(\Gamma,T)$, then $X$ satisfies Hurewicz' property $U_{\rm
fin}(\Gamma,\Gamma)$. \endgraf Some results on the additivity of these
properties and special elements are also provided. },
},
@article{KrSh:859,
author = {Kellner, Jakob and Shelah, Saharon},
trueauthor = {Kellner, Jakob and Shelah, Saharon},
fromwhere = {A,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0511330 },
pages = {1153--1183},
title = {{Saccharinity}},
volume = {74},
year = {2011},
abstract = {We present a method to iterate finitely splitting
lim-sup tree forcings along non-wellfounded linear orders. We
apply this method to construct a forcing (without using an inaccessible
or amalgamation) that makes all definable sets of reals measurable with
respect to a certain (non-ccc) ideal.},
},
@article{RoSh:860,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Quaderni di Matematica},
note = { arxiv:math.LO/0508272 },
pages = {195--239},
series = {Set Theory: Recent Trends and Applications (A. Andretta,
ed.)},
title = {{Reasonably complete forcing notions}},
volume = {17},
year = {2006},
abstract = {We introduce more properties of forcing notions which
imply that their $\lambda$--support iterations are
$\lambda$--proper, where $\lambda$ is an inaccessible cardinal. This
paper is a direct continuation of Ros{\l}anowski and Shelah [RoSh:777,
\S 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.},
},
@article{Sh:861,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0612243 },
pages = {226--242},
title = {{Power set modulo small, the singular of uncountable
cofinality}},
volume = {72},
year = {2007},
abstract = {Let $\mu$ be singular of uncountable cofinality. If
$\mu> 2^{{\rm cf}(\mu)}$, we prove that in ${\mathbb
P}=([\mu]^\mu, \supseteq)$ as a forcing notion we have a natural
complete embedding of ${\rm Levy}(\aleph_0,\mu^+)$ (so ${\mathbb P}$
collapses $\mu^+$ to $\aleph_0$) and even ${\rm Levy}(\aleph_0, {\bf
U}_{J^{{\rm bd}}_\kappa}(\mu))$. The ``natural'' means that the
forcing $(\{p \in [\mu]^\mu:p$ closed$\},\supseteq)$ is naturally
embedded and is equivalent to the Levy algebra. If $\mu<2^{{\rm
cf}(\mu)}$ we have weaker results.},
},
@article{BlSh:862,
author = {Baldwin, John and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
pages = {765--782},
title = {{Examples of non-locality}},
volume = {73},
year = {2008},
abstract = {We use $\kappa$-free but not Whitehead Abelian groups
to construct abstract elementary classes which satisfy the
amalgamation property but fail various conditions on the locality of
types. We show in fact that in a large class of cases amalgamation can
have no positive effect on locality by exhibiting a transformation of
aec's which preserves non-locality but takes any aec satisfying a
new property called admitting closures to one with amalgamation.},
},
@article{Sh:863,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/0504197 },
pages = {1--83},
title = {{Strongly dependent theories}},
volume = {204},
year = {2014},
abstract = {We further investigate the class of models of a
strongly dependent (first order complete) theory $T$, continuing
[Sh:783]. If $|A|+|T|\le\mu$, $I\subseteq {\frak C}$,
$|I|\ge \beth_{|T|^+}(\mu)$ then some $J\subseteq I$ of cardinality
$\mu^+$ is an indiscernible sequence over $A$.},
},
@article{SaSh:864,
author = {Sagi, Gabor and Shelah, Saharon},
trueauthor = {S\'agi, G\'abor and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0612244 },
pages = {104--118},
title = {{On Weak and Strong Interpolation in Algebraic Logics}},
volume = {71},
year = {2006},
abstract = {We show that there is a restriction, or modification of
the finite-variable fragments of First Order Logic in which a weak
form of Craig's Interpolation Theorem holds, but a strong form of
this theorem does not hold. Translating these results into
Algebraic Logic we obtain a finitely axiomatizable subvariety of
finite dimensional Representable Cylindric Algebras that has the
Strong Amalgamation Property, but does not have the
Superamalgamation Property. This settles a conjecture of Pigozzi},
},
@article{DoSh:865,
author = {Doron, Mor and Shelah, Saharon},
trueauthor = {Doron, Mor and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0607375 },
pages = {1283--1298},
title = {{Relational structures constructible by quantifier
free definable operations}},
volume = {72},
year = {2007},
abstract = {We consider three classes of models: models of bounded
patch width defined Fisher and Makowsky, models constructible by
addition operations that preserve monadic theories defined in [Sh:817],
and models monadicaly interpretable in trees defined below. For
all three classes we have eventual periodicity of restricted spectrum
of MSO sentences. We show that the second and third classes are in
some sence equivalent, while the first is essentially smaller.},
},
@article{HvSh:866,
author = {Havlin, Chanoch and Shelah, Saharon},
trueauthor = {Havlin, Chanoch and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:math.LO/0612245 },
pages = {111--127},
title = {{Existence of EF-equivalent Non-Isomorphic Models}},
volume = {53},
year = {2007},
abstract = {We prove the existence of pairs of models of the
same cardinality $\lambda$ which are very equivalent according to
EF games, but not isomorphic. We continue the paper [Sh:836], but we
don't rely on it.},
},
@article{GbSh:867,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:0711.3045 },
pages = {155--181},
title = {{Generalized $E$-Algebras via $\lambda$-Calculus I}},
volume = {192},
year = {2006},
abstract = {An $R$-algebra $A$ is called $E(R)$--algebra if the
canonical homomorphism from $A$ to the endomorphism algebra $End_R A$
of the $R$-module ${}_R A$, taking any $a\in A$ to the right
multiplication $a_r\in End_R A$ by $a$ is an isomorphism of algebras.
In this case ${}_R A$ is called an $E(R)$--module. $E(R)$-algebras come
up naturally in various topics of algebra, so it's not surprising
that they were investigated thoroughly in the last decade. Despite
some efforts it remained an open question whether proper
generalized $E(R)$-algebras exist. These are $R$--algebras $A$
isomorphic to $End_R A$ but not under the above canonical isomorphism,
so not $E(R)$--algebras. This question was raised about 30 years ago
(for $R={\mathbb Z}$) by Phil Schultz and we will answer it. For PIDs
$R$ of characteristic $0$ that are neither quotient fields nor
complete discrete valuation rings - we will establish the existence
of generalized $E(R)$-algebras. It can be shown that
$E(R)$-algebras over rings $R$ that are complete discrete valuation
rings or fields must trivial (copies of $R$). The main tool is an
interesting connection between $\lambda$-calculus (used in theoretical
computer sciences) and algebra. It seems reasonable to divide the work
into two parts, in this paper we will work in V=L (Godels
universe) hence stronger combinatorial methods make the final arguments
more transparent. The proof based entirely on ordinary set theory
(the axioms of ZFC) will appear in a subsequent paper.},
},
@article{Sh:868,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Colloquium Mathematicum},
note = { arxiv:math.LO/0603651 },
pages = {187--204},
title = {{When first order $T$ has limit models}},
volume = {126},
year = {2012},
abstract = {We to a large extent sort out when does a (first order
comp- lete theory) $T$ have a superlimit model in a cardinal $\lambda$.
Also we deal with relation notions of being limit.},
},
@article{MtSh:869,
author = {Matet, Pierre and Shelah, Saharon},
trueauthor = {Matet, Pierre and Shelah, Saharon},
fromwhere = {F,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0612246 },
pages = {24 pps},
title = {{The nonstationary ideal on $P_\kappa(\lambda)$ for
$\lambda$ singular}},
volume = {accepted, Baumgartner memorial issue},
year = {2017},
abstract = {Let $\kappa$ be a regular uncountable cardinal
and $\lambda>\kappa$ a singular strong limit cardinal. We give a
new characterization of the nonstationary subsets of
$P_\kappa(\lambda)$ and use this to prove that the nonstationary ideal
on $P_\kappa(\lambda)$ is nowhere precipitous.},
},
@article{BsSh:870,
author = {Blass, Andreas and Shelah, Saharon},
trueauthor = {Blass, Andreas and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Quaderni di Matematica},
note = { arxiv:math/0509406 },
pages = {1--24},
series = {Set Theory: Recent Trends and Applications (A. Andretta,
ed.)},
title = {{Disjoint Non-Free Subgoups of Abelian Groups}},
volume = {17},
year = {2006},
abstract = {Let $G$ be an abelian group and let $\lambda$ be the
smallest rank of any group whose direct sum with a free group is
isomorphic to $G$. If $\lambda$ is uncountable, then $G$ has
$\lambda$ pairwise disjoint, non-free subgroups. There is an example
where $\lambda$ is countably infinite and $G$ does not have even
two disjoint, non-free subgroups.},
},
@article{LwSh:871,
author = {Laskowski, Michael C. and Shelah, Saharon},
trueauthor = {Laskowski, Michael C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:0711.3043 },
pages = {1619--1629},
title = {{A trichotomy of countable, stable, unsuperstable theories}},
volume = {363},
year = {2011},
abstract = {A trichotomy theorem for countable, stable,
unsuperstable theories is offered. We develop the notion of a `regular
ideal' of formulas and study types that are minimal with respect to
such an ideal.},
},
@article{KrSh:872,
author = {Kellner, Jakob and Shelah, Saharon},
trueauthor = {Kellner, Jakob and Shelah, Saharon},
fromwhere = {A,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0601083 },
pages = {73--104},
title = {{Decisive creatures and large continuum}},
volume = {74},
year = {2009},
abstract = {For $f>g\in\omega^\omega$ let $c^{\forall}_{f,g}$ be
the minimal number of uniform trees with $g$-splitting needed
to $\forall^\infty$-cover the uniform tree with $f$-splitting.
$c^{\exists}_{f,g}$ is the dual notion for the $\exists^\infty$-cover.
\endgraf 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
$c^{\exists}_{f_\epsilon,g_\epsilon}= c^{\forall}_{f_\epsilon,g_\epsilo
n}=\kappa_\epsilon$. \endgraf 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.},
},
@article{ShSm:873,
author = {Shelah, Saharon and Struengmann, Lutz},
trueauthor = {Shelah, Saharon and Str{\"{u}}ngmann, Lutz},
fromwhere = {IL,D},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/0609638 },
pages = {141--151},
title = {{A characterization of ${\rm Ext}(G,{\mathbb Z})$ assuming
$(V=L)$}},
volume = {193},
year = {2007},
abstract = {In this paper we complete the characterization of
${\rm Ext}(G,{\mathbb Z})$ under G{\"{o}}del'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 ${\rm Ext}(G,{\mathbb Z})$.},
},
@article{ShSm:874,
author = {Shelah, Saharon and Struengmann, Lutz},
trueauthor = {Shelah, Saharon and Str{\"{u}}ngmann, Lutz},
fromwhere = {IL,D},
journal = {Algebra and Logic},
note = { arxiv:math.LO/0609637 },
pages = {200--215},
title = {{On the $p$-rank of ${\rm Ext}_{\mathbb Z}(G,{\mathbb Z})$
in certain models of ZFC}},
volume = {46},
year = {2007},
abstract = {We show that if the existence of a supercompact cardinal
is consistent with $ZFC$, then it is consistent with $ZFC$ that
the $p$-rank of ${\rm Ext}_{\mathbb Z}(G,\mathbb Z)$ is as large
as possible for every prime $p$ and any torsion-free abelian group $G$.
Moreover, given an uncountable strong limit cardinal $\mu$ of countable
cofinality and a partition of $\Pi$ (the set of primes) into two
disjoint subsets $\Pi_0$ and $\Pi_1$, we show that in some model which
is very close to $ZFC$ there is an almost-free abelian group $G$ of
size $2^{\mu}=\mu^+$ such that the $p$-rank of ${\rm Ext}_{\mathbb
Z}(G,{\mathbb Z})$ equals $2^{\mu}=\mu^+$ for every $p\in\Pi_0$ and $0$
otherwise, i.e. for $p\in\Pi_1$.},
},
@article{JrSh:875,
author = {Jarden, Adi and Shelah, Saharon},
trueauthor = {Jarden, Adi and Shelah, Saharon},
fromwhere = {IL, IL},
journal = {Annals of Pure and Applied Logic},
pages = {135--191},
title = {{Non-forking frames in abstract elementary classes}},
volume = {164},
year = {2013},
abstract = {The stability theory of first order theories was initiated
by Saharon Shelah in 1969. The classification of abstract elementary
classes was initiated by Shelah, too. In several papers, he
introduced non-forming relations. Later, Shelah (2009) [17,
11] introduced the good non-forking frame, an axiomatization of the
non-forking notion. We improve results of Shelah on good non-forming
grames, mainly by weakening the stability hypothesis in several
important theorems, replacing it by the almost $\lambda$-stability
hypothesis: The number of types over a model of cardinality $\lambda$
is at most $\lambda^+$. We present conditions on $K_\lambda$, that
imply the existence of a model in $K_{\lambda^{+n}}$ for all $n$. We
do this by providing sufficiently strong conditions on $K_\lambda$,
that they are inherited by a properly chosen subclass of
$K_{\lambda^+}$. What are these conditions? We assume that there is
a `non-forking' relation which satisfies the properties of the
non-forking relation on superstable first order theories. Note that
here we deal with models of fixed cardinality $\lambda$. While in
Shelah (2009) [17,II] we assume stability in $\lambda$, so we can use
brimmed (=limit) models, here we assume almost stability only, but we
add an assumption: The conjugation property. In the context of
elementary classes, the superstability assumption gives the existence
of types with well-defined dimension and the $\omega$-stability
assumption gives the existence and uniqueness of models prime over
sets. In our context, the local character assumption is an
analog to superstability and the density of the class of uniqueness
triples with respect to the relation $ \preccurlyeq $ is the analog to
$omega$-stability.},
},
@article{Sh:876,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO/0603652 },
pages = {1087-1091},
title = {{Minimal bounded index subgroup for dependent theories}},
volume = {136},
year = {2008},
abstract = {For a dependent theory $T$, in ${\mathfrak C}_T$ for every
type definable group $G$, the intersection of type definable
subgroups with bounded index is a type definable subgroup with
bounded index.},
},
@article{Sh:877,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Tbilisi Mathematical Journal},
note = { arxiv:math.LO/0609636 },
pages = {99--128},
title = {{Dependent $T$ and Existence of limit models}},
volume = {7},
year = {2014},
abstract = {We continue [Sh:868] and [Sh:783]. The problem there is
when does (first order) $T$ have a model $M$ of cardinality
$\lambda$ which is (one of the variants of) a limit model for
cofinality $\kappa$, and the most natural case to try is
$\lambda=\lambda^{< \lambda}>\kappa={\rm cf}(\kappa)>|T|$. The stable
theories has one; are there unstable $T$ whnce of limit models AUTHORS:
Saharon Shelah ich has such limit models? We find one: the theory
$T_{\rm ord}$ of dense linear orders. So does this hold for all
unstable $T$? As $T_{\rm ord}$ is prototypical of dependent theories,
it is natural to look for independent theories. A strong, explicit
version of $T$ being independent is having the strong independence
property. We prove that for such $T$ there are no limit models. We
work harder to prove this for every dependent $T$, i.e., with the
independence property though a weaker version. This makes us conjecture
that any dependent $T$ has such models. Toward this end we continue the
investigation of types for dependent $T$.},
},
@article{GaSh:878,
author = {Garti, Shimon and Shelah, Saharon},
trueauthor = {Garti, Shimon and Shelah, Saharon},
fromwhere = {IL, IL},
journal = {Algebra Universalis},
note = { arxiv:math.LO/0512217 },
pages = {243--248},
title = {{On ${\rm DEPTH}$ and ${\rm DEPTH}^+$ of Boolean Algebras}},
volume = {58},
year = {2008},
abstract = {We show that the ${\rm Depth}^+$ of an ultraproduct of
Boolean Algebras, can not jump over the ${\rm Depth}^+$ of every
component by more than one cardinality. We can have, consequently,
similar results for the Depth invariant},
},
@article{EFSh:879,
author = {Eklof, Paul C. and Fuchs, Laszlo and Shelah, Saharon},
trueauthor = {Eklof, Paul C. and Fuchs, Laszlo and Shelah, Saharon},
fromwhere = {1,1,IL},
journal = {Rocky Mountain Journal of Mathematics},
note = { arxiv:math.LO/0702293 },
pages = {1863--1873},
title = {{Test Groups for Whitehead Groups}},
volume = {42},
year = {2012},
abstract = {We consider the question of when the dual of a
Whitehead group is a test group for Whitehead groups. This turns
out to be equivalent to the question of when the tensor product of two
Whitehead groups is Whitehead. We investigate what happens in
different models of set theory.},
},
@article{GbSh:880,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D, IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:0711.3011 },
pages = {1641--1649},
title = {{Absolutely Indecomposable Modules}},
volume = {135},
year = {2007},
abstract = {A module is called {\em 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 {\em 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.},
},
@article{Sh:881,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Canadian Mathematical Bulletin},
note = { arxiv:math.LO/0605385 },
pages = {127--131},
title = {{The Erdos-Rado Arrow for singular}},
volume = {52},
year = {2009},
abstract = {We try to prove that if $\mathrm{cf}(\lambda) >
\aleph_0$ and $2^{\mathrm{cf}(\lambda)} < \lambda$ then
$\lambda \to(\lambda, \omega +1)^2$},
},
@article{KpSh:882,
author = {Kaplan, Itay and Shelah, Saharon},
trueauthor = {Kaplan, Itay and Shelah, Saharon},
fromwhere = {IL, IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/0606216 },
pages = {799--815},
title = {{The automorphism tower of a centerless group without
choice}},
volume = {48},
year = {2009},
abstract = {For a centerless group $G$, we can define its
automorphism tower. We define $G^{\alpha}$: $G^0=G$,
$G^{\alpha+1}= Aut(G^\alpha)$ and for limit ordinals
$G^\delta= \bigcup_{\alpha<\delta}G^\alpha$. Let $\tau_G$ be the
ordinal when the sequence stabilizes. Thomas' celebrated theorem says
$\tau_G< 2^{|G|})^{+}$ and more. If we consider Thomas' proof too
set theoretical, we have here a shorter proof with little set theory.
However, set theoretically we get a parallel theorem without the axiom
of choice. We attach to every element in $G^\alpha$, the $\alpha$-th
member of the automorphism tower of $G$, a unique quantifier free
type over $G$ (whish is a set of words from $G* \langle x\rangle$).
This situation is generalized by defining ``$(G,A)$ is a special
pair''.},
},
@article{Sh:883,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {CUBO, A Mathematical Journal},
note = { arxiv:math.LO/0609634 },
pages = {59--79},
title = {{$\aleph_n$-free abelain group with no non-zero homomorphism to
$\Bbb Z$}},
volume = {9},
year = {2007},
abstract = {For any natural $n$, we construct an $\aleph_n$-free
abelian groups which have few homomorphisms to $\Bbb Z$. 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.},
},
@article{GoSh:884,
author = {Goldstern, Martin and Shelah, Saharon},
trueauthor = {Goldstern, Martin and Shelah, Saharon},
fromwhere = {AT, IL},
journal = {TAMS},
pages = {7551--7577},
title = {{All creatures great and small}},
volume = {368},
year = {2016},
abstract = {Let $\lambda$ be an uncountable regular cardinal. Assuming
$2^\lambda=\lambda^+$, we show that the clone lattice on a set of size
$\lambda$ is not dually atomic},
},
@article{Sh:885,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Canadian Mathematical Bulletin},
note = { arxiv:math.LO/0404220 },
pages = {303--314},
title = {{A comment on ``${\mathfrak p}<{\mathfrak t}$''}},
volume = {52},
year = {2009},
abstract = {Dealing with the cardinal invariants ${\mathfrak p}$
and ${\mathfrak t}$ of the continuum we prove that ${\mathfrak m}\geq
{\mathfrak p} = \aleph_2 \Rightarrow {\mathfrak t} = \aleph_1$. In
other words if ${\bf MA}_{\aleph_1}$ (or a weak version of this) then
(of course $\aleph_2 \leq {\mathfrak p}\leq {\mathfrak t}$ and)
${\mathfrak p} = \aleph_2 \Rightarrow {\mathfrak p} = {\mathfrak t}$.
This is based on giving a consequence.},
},
@article{Sh:886,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Sarajevo Journal of Mathematics},
note = { arxiv:math.LO/0703045 },
pages = {3-25},
title = {{Definable groups for dependent and 2-dependent theories}},
volume = {13(25)},
year = {2017},
abstract = {Let $T$ be a (first order complete) dependent
theory, ${\mathfrak{C}}$ a $\bar{\kappa}$-saturated model of $T$ and
$G$ a definable subgroup which is abelian. Among subgroups of
bounded index which are the union of $<\kappa$ type definable subsets
there is a minimal one, i.e. their intersection has bounded index.
In fact, the bound is $\leq 2^{|T|}$. We then deal with
2-dependent theories, a wider class of first order theories.},
},
@article{Sh:887,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:math.LO/0612353 },
pages = {340--344},
title = {{Groupwise density cannot be much bigger than the
unbounded number}},
volume = {54},
year = {2008},
abstract = {We prove that ${\mathfrak g}$, (the groupwise density)
is smaller or equal to ${\mathfrak b}^+$, successor of the minimal
cardinality of a non-dominated subset of ${}^\omega \omega$.},
},
@incollection{RoSh:888,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
booktitle = {Set Theory and Its Applications},
fromwhere = {1,IL},
note = { arxiv:math.LO/0611131 },
pages = {287--330},
publisher = {Amer. Math. Soc.},
series = {Contemporary Mathematics (CONM)},
title = {{Lords of the iteration}},
volume = {533},
year = {2011},
abstract = {We introduce several properties of forcing notions which
imply that their $\lambda$--support iterations are $\lambda$--proper.
Our methods and techniques refine those studied in
[RoSh:655], [RoSh:777], [RoSh:860] and [RoSh:890], covering some new
forcing notions (though the exact relation of the new properties to the
old ones remains undecided).},
},
@article{RoSh:889,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:math.LO/0607218 },
pages = {202--220},
title = {{Generating ultrafilters in a reasonable way}},
volume = {54},
year = {2008},
abstract = {We continue investigations of reasonable ultrafilters
on uncountable cardinals defined in Shelah [Sh:830]. We introduce
a general scheme of generating a filter on $\lambda$ from filters
on smaller sets and we investigate the combinatorics of
objects obtained this way.},
},
@article{RoSh:890,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Notre Dame Journal of Formal Logic},
note = { arxiv:math.LO/0605067 },
pages = {113--147},
title = {{Reasonable ultrafilters, again}},
volume = {52},
year = {2011},
abstract = {We continue investigations of {\em 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). },
},
@article{GaSh:891,
author = {Garti, Shimon and Shelah, Saharon},
trueauthor = {Garti, Shimon and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:math.LO/0612247 },
pages = {636--641},
title = {{Two cardinal models for singular $\mu$}},
volume = {53},
year = {2007},
abstract = {We deal here with colorings of the pair $(\mu^+,\mu)$,
when $\mu$ is a strong limit and singular cardinal. We show that
there exists a coloring $c$, with no refinement. It follows, that
the properties of identities of $(\mu^+,\mu)$ when $\mu$ is
singular, differ in an essential way from the case of regular $\mu$.},
},
@article{FGSSh:892,
author = {Dror Farjoun, Emmanuel and Goebel, Ruediger and Segev, Yoav
and Shelah, Saharon},
trueauthor = {Dror Farjoun, Emmanuel and G{\"{o}}bel, R{\"{u}}diger
and Segev, Yoav and Shelah, Saharon},
fromwhere = {IL,D,IL,IL},
journal = {Groups, Geometry, and Dynamics},
note = { arxiv:math.GR/0702294 },
pages = {409--419},
title = {{On kernels of cellular covers}},
volume = {1},
year = {2007},
abstract = {In the present paper we continue to examine cellular
covers of groups, focusing on the cardinality and the structure of the
kernel $K$ of the cellular map $G\to M$. We show that in general a
torsion free reduced abelian group $M$ may have a proper class of
non-isomorphic cellular covers. In other words, the cardinality of the
kernels is unbounded. In the opposite direction we show that if the
kernel of a cellular cover of any group $M$ has certain ``freeness''
properties, then its cardinality must be bounded.},
},
@incollection{Sh:893,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
booktitle = {Logic Without Borders},
fromwhere = {IL},
note = { arxiv:math.LO/1302.4841 },
pages = {367--402},
publisher = {Berlin, Boston: DeGruyter [2015]},
title = {{A.E.C. with not too many models}},
volume = {Ontos Mathematical Logic, vol. 5},
year = {2015},
abstract = {Consider an a.e.c. ${\mathfrak K}$ and the class of ${\bold
C}_{\aleph_0}$ cardinals of cofinality $\aleph_0$. A nicely stated
consequence of this work is for some closed unbounded class $C$ of
cardinals we have \endgraf (a)\quad $\dot I(\lambda,{\mathfrak K})\geq
\lambda$ for $\lambda \in C \cap {\bold
C_{\aleph_0}}$ \endgraf or \endgraf (b)\quad if $M \in K_\lambda$ and
$\lambda \in {\bold C}_{\aleph_0}$, then $M$ has $\le_{\mathfrak
K}$-extension (so in ${\mathfrak K}$) of arbitrarily large cardinals.},
},
@article{MdSh:894,
author = {Mildenberger, Heike and Shelah, Saharon},
trueauthor = {Mildenberger, Heike and Shelah, Saharon},
fromwhere = {A, IL},
journal = {Transactions of the American Mathematical Society},
pages = {2305--2317},
title = {{The Near Coherence of Filters Principle does not imply
the Filter Dichotomy Principle}},
volume = {361},
year = {2009},
abstract = {We show that there is a forcing extension in which any two
ultrafilters on $\omega$ are nearly coherent and there is a non-meagre
filter that is not nearly ultra. This answers Blass' longstanding
question (cite[3]) whether the principle of near coherence of filters
is strictly weaker than the filter dichotomy principle.},
},
@article{Sh:895,
author = {Shelah, Saharon},
fromwhere = {IL},
journal = {Central European Journal of Mathematics},
note = { arxiv:0707.1818 },
pages = {213--234},
title = {{Large continuum, oracles}},
volume = {8},
year = {2010},
abstract = {Our main theorem is about iterated forcing for making
the continuum larger than $\aleph_2$. We present a generalization of
[Sh:669] which is dealing with oracles for random, etc.,
replacing $\aleph_1,\aleph_2$ by $\lambda,\lambda^+$ (starting with
$\lambda = \lambda^{< \lambda} > \aleph_1$). Instead of properness we
demand absolute c.c.c. So we get, e.g. the continuum is $\lambda^+$
but we can get cov(meagre) $=\lambda$. We also give some
applications related to peculiar cuts of [Sh:885].},
},
@article{KPSh:896,
author = {Kellner, Jakob and Pauna, Matti and Shelah, Saharon},
trueauthor = {Kellner, Jakob and Pauna, Matti and Shelah, Saharon},
fromwhere = {A, F, IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0609655 },
pages = {1323--1335},
title = {{Winning the pressing down game but not Banach Mazur}},
volume = {72},
year = {2007},
abstract = { Let $S$ be the set of those $\alpha\in\omega_2$ that
have cofinality $\omega_1$. It is consistent relative to a measurable
that player II (the nonempty player) wins the pressing down game of
length $\omega_1$, but not the Banach Mazur game of length
$\omega+1$ (both starting with $S$).},
},
@article{Sh:897,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Tbilisi Mathematical Journal},
note = { arxiv:math.LO/0703477 },
pages = {133--164},
title = {{Theories with EF-Equivalent Non-Isomorphic Models}},
volume = {1},
year = {2008},
abstract = {Our ``large scale'' aim is to characterize the first order
$T$ (at least the countable ones) such that: for every ordinal
$\alpha$ there $\lambda,M_1,M_2$ such that $M_1,M_2$ are
non-isomorphic models of $T$ of cardinality $\lambda$ which
are EF$_{\alpha,\lambda}$-equivalent. We expect that as in the main
gap ([Sh:c,XII]) we get a strong dichotomy, so in the non-structure
side we have more, better example, and in the structure side we have
a parallel of [Sh:c,XIII]. We presently prove the consistency of
the non-structure side for $T$ which is $\aleph_0$-independent (=
not strongly dependent) or just not strongly stable, even
for PC$(T_1,T)$ and more for unstable $T$ (see [Sh:c,VII] or [Sh:h])
and infinite linear order $I$.},
},
@article{Sh:898,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Forum Mathematicum},
note = { arxiv:arxiv:0710.0157 },
pages = {967--1038},
title = {{PCF and abelian groups}},
volume = {25},
year = {2013},
abstract = {We deal with some pcf investigations mostly motivated by
abelian group theory problems and deal their applications to test
problems (we expect reasonably wide applications). We prove almost
always the existence of $\aleph_\omega$-free abelian groups with
trivial dual, i.e. no non-trivial homomorphisms to the integers. This
relies on investigation of pcf; more specifically, for this we prove
that ``almost always'' there are ${\Cal F} \subseteq {}^\kappa
\lambda$ 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.},
},
@article{JuSh:899,
author = {Juhasz, Istvan and Shelah, Saharon},
trueauthor = {Juh\'asz, Istv\'an and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Studia Scientiarum Mathematicarum Hungarica},
note = { arxiv:math.LO/0702295 },
pages = {557--562},
title = {{Hereditarily Lindel{\"{o}}f spaces of singular density}},
volume = {45},
year = {2008},
abstract = {A cardinal $\lambda$ is called $\omega$-inaccessible if for
all $\mu<\lambda$ we have $\mu^\omega < \lambda.$ We show that for
every $\omega$-inaccessible cardinal $\lambda$ there is a CCC
(hence cardinality and cofinality preserving) forcing that adds
a hereditarily Lindel\'' of regular space of density $\lambda$.
This extends an analogous earlier result of ours that only worked
for regular $\lambda$.},
},
@article{Sh:900,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Communications in Contemporary Mathematics},
note = { arxiv:math.LO/0702292 },
pages = {1550004 (64 pps.)},
title = {{Dependent theories and the generic pair conjecture}},
volume = {17},
year = {2015},
abstract = {On the one hand we try to understand complete types
over somewhat saturated model of a complete first order theory which
is dependent, by ``decomposition theorems for such types''. Our
thesis is that the picture of dependent theory is the combination of
the one for stable theories and the one for the theory of dense
linear order or trees (and first we should try to understand the
quite saturated case). On the other hand as a measure of our progress,
we give several applications considering some test questions;
in particular we try to prove the generic pair conjecture and do it
for measurable cardinals. The order of the sections is by
their conceptions, so there are some repetitions.},
},
@article{JShS:901,
author = {Juhasz, Istvan and Shelah, Saharon and Soukup, Lajos},
trueauthor = {Juh\'asz, Istv\'an and Shelah, Saharon and Soukup, Lajos},
fromwhere = {H,IL,IL},
journal = {Topology and its Applications},
note = { arxiv:math.GN/0702296 },
pages = {1966--1969},
title = {{Resolvability vs. almost resolvability}},
volume = {156},
year = {2009},
abstract = {A space $X$ is {\em $\kappa$-resolvable} (resp. {\em
almost $\kappa$-resolvable}) if it contains $\kappa$ dense sets that
are pairwise disjoint (resp. almost disjoint over the ideal of
nowhere dense subsets of $X$). \endgraf Answering a problem raised by
Juh\'asz, Soukup, and Szentmikl\'ossy, 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 ${\kappa}$.},
},
@article{LrSh:902,
author = {Larson, Paul and Shelah, Saharon},
trueauthor = {Larson, Paul and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Mathematical Logic Quarterly},
pages = {187--193},
title = {{The stationary set splitting game}},
volume = {54},
year = {2008},
abstract = {The \emph{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.},
},
@article{MShT:903,
author = {Machura, Michal and Shelah, Saharon and Tsaban, Boaz},
trueauthor = {Machura, Michal and Shelah, Saharon and Tsaban, Boaz},
fromwhere = {P,IL,IL},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:math/0611353 },
pages = {1751--1764},
title = {{Squares of Menger-Bounded Groups}},
volume = {362},
year = {2010},
abstract = {Using a portion of the Continuum Hypothesis, we prove that
there is a Menger-bounded (also called $ o$-bounded) subgroup of the
Baer-Specker group $ \mathbb{Z}^{\mathbb{N}}$, whose square is
not Menger-bounded. This settles a major open problem concerning
boundedness notions for groups and implies that Menger-bounded groups
need not be Scheepers-bounded. This also answers some questions of
Banakh, Nickolas, and Sanchis.},
},
@article{Sh:904,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Algebra Universalis},
note = { arxiv:math.LO/0703493 },
pages = {351--366},
title = {{Reflexive abelian groups and measurable cardinals and full MAD
families}},
volume = {63},
year = {2010},
abstract = {Answering problem (DG) of \cite{EM90}, \cite{EM02}, we
show that there is a reflexive group of cardinality $\ge$
first measurable.},
},
@article{KrSh:905,
author = {Kellner, Jakob and Shelah, Saharon},
trueauthor = {Kellner, Jakob and Shelah, Saharon},
fromwhere = {A,IL},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/0703302 },
pages = {51--76},
title = {{A Sacks real out of Nowhere}},
volume = {75},
year = {2010},
abstract = {There is a proper countable support iteration of length
$\omega$ adding no new reals at finite stages and adding a Sacks real
in the limit.},
},
@article{Sh:906,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {CRM Proceedings and Lecture Notes},
note = { arxiv:0705.4131 },
pages = {277--290},
title = {{No limit model in inaccessibles}},
volume = {53},
year = {2011},
abstract = {Our aim is to improve the negative results i.e.
non-existence of limit models, and the failure of the generic pair
property from [Sh:877] to inaccessible $\lambda$ as promised there. The
motivation is that in [Sh:F756] the positive results are for
$\lambda$ measurable hence inaccessible, whereas in [Sh:877] in the
negative results obtained only on non-strong limit cardinals.},
},
@article{Sh:907,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:0705.4126 },
pages = {4405--4412},
title = {{EF equivalent not isomorphic pair of models}},
volume = {136},
year = {2008},
abstract = {We construct non-isomorphic models $M, N$, e.g.
of cardinality $\aleph_1$ such that in the Ehrenfeucht-Fraiss\'e game
of length $\zeta < \omega_1$ the isomorphism player wins},
},
@article{Sh:908,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:0705.4130 },
pages = {397--399},
title = {{On long increasing chains modulo flat ideals}},
volume = {56},
year = {2010},
abstract = {We prove that e.g. there is no $\omega_4$---sequence in
$(\omega_3)^{\omega_3}$ increasing mod countable.},
},
@incollection{GhSh:909,
author = {Gruenhut, Esther and Shelah, Saharon},
trueauthor = {Gruenhut, Esther and Shelah, Saharon},
booktitle = {Set Theory and Its Applications},
fromwhere = {IL,IL},
note = { arxiv:0906.3055 },
pages = {267--280},
publisher = {Amer. Math. Soc.},
series = {Contemporary Mathematics (CONM)},
title = {{Uniforming $n$-place functions on well founded trees}},
volume = {533},
year = {2011},
abstract = {In this paper the Erd\H os-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.
},
},
@incollection{BsSh:910,
author = {Blass, Andreas and Shelah, Saharon},
trueauthor = {Blass, Andreas and Shelah, Saharon},
booktitle = {Models, Modules and Abelian Groups, in Memory of A.L.S.
Corner},
fromwhere = {1,IL},
note = { arxiv:0711.3031 },
nt = {Ruediger Goebel and Bendan Goldsmith, editors},
pages = {63--73},
publisher = {Walter de Gruyter, Berlin, NY},
title = {{Basic Subgroups and Freeness, a counterexample}},
year = {2008},
abstract = {We construct a non-free but $\aleph_1$-separable,
torsion-free abelian group $G$ with a pure free subgroup $B$ such that
all subgroups of $G$ disjoint from $B$ are free and such that $G/B$
is divisible. This answers a question of Irwin and shows that
a theorem of Blass and Irwin cannot be strengthened so as to give
an exact analog for torsion-free groups of a result proved
for $p$-groups by Benabdallah and Irwin.},
},
@article{GaSh:911,
author = {Garti, Shimon and Shelah, Saharon},
trueauthor = {Garti, Shimon and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Notre Dame Journal of Formal Logic},
pages = {307--314},
title = {{Depth of Boolean Algebras}},
volume = {52},
year = {2011},
abstract = {We show under the assumption $\bf {V} = \bf {L}$ that the
${\rm Depth}$ of an ultraproduct of Boolean Algebras cannot jump over
the ${\rm Depth}$ of every component by more than one cardinal. This is
done for every cardinal. We also get better results for
singular cardinals with countable cofinality.},
},
@article{KShV:912,
author = {Kennedy, Juliette and Shelah, Saharon and Vaananen, Jouko},
trueauthor = {Kennedy, Juliette and Shelah, Saharon
and V{\"{a}}{\"{a}}n{\"{a}}nen, Jouko},
fromwhere = {F,IL,SF},
journal = {Journal of Symbolic Logic},
pages = {817--823},
title = {{Regular Ultrafilters and Finite Square Principles}},
volume = {73},
year = {2008},
abstract = {We show that many singular cardinals $\lambda$ above a
strongly compact cardinal have regular ultrafilters $D$ that violate
the finite square principle $\square^{fin}_{\lambda, D}$ introduced in
[3]. For such ultrafilters $D$ and cardinals $\lambda$ there are
models of size $\lambda$ for which $M^{\lambda}/D$ is not
$\lambda^{++}$-universal and elementarily equivalent models $M$ and $N$
of size $\lambda$ for which $M^\lambda/D$ and $N^\lambda/D$ are
non-isomorphic. The question of the existence of such ultrafilters and
models was raised in [1].},
},
@article{KpSh:913,
author = {Kaplan, Itay and Shelah, Saharon},
trueauthor = {Kaplan, Itay and Shelah, Saharon},
fromwhere = {IL, IL},
journal = {Contemporary Mathematics},
pages = {187--203},
title = {{Automorphism towers and automorphism groups of fields without
choice}},
volume = {576},
year = {2012},
abstract = {The main result (already achieved in \cite{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.},
},
@article{Sh:914,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Tbilisi Mathematical Journal},
note = { arxiv:0708.1980 },
pages = {17--30},
title = {{The First almost free Whitehead group}},
volume = {4},
year = {2011},
abstract = {Assume G.C.H. and $\kappa$ is the first uncountable cardinal
such that there is a $\kappa$-free abelian group which is not a
Whitehead (abelian) group. We prove that $\kappa$ is necessarily
an inaccessible cardinal.},
},
@article{Sh:915,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Topology and its Applications},
note = { arxiv:1004.2083 },
pages = {2535--2555},
title = {{The character spectrum of $\beta(N)$}},
volume = {158},
year = {2011},
abstract = {We show the consistency of: the set of regular
cardinals which are the character of some ultrafilter on $\omega$ can
be quite chaotic, in particular not only can be not convex but can have
many gaps. We also deal with the set of $\pi$-characters of
ultrafilters on $\omega$},
},
@article{DwSh:916,
author = {Dow, Alan and Shelah, Saharon},
trueauthor = {Dow, Alan and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Topology and its Applications},
note = { arxiv:0711.3037 },
pages = {1661--1671},
title = {{Tie-points and fixed-points in $\mathbb N^*$}},
volume = {155},
year = {2008},
abstract = {A point $x$ is a (bow) tie-point of a space $X$
if $X\setminus \{x\}$ can be partitioned into (relatively) clopen sets
each with $x$ in its closure. Tie-points have appeared in the
construction of non-trivial autohomeomorphisms of $\beta{\mathbb N}
\setminus {\mathbb N}$ (e.g. \cite{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. },
},
@article{DwSh:917,
author = {Dow, Alan and Shelah, Saharon},
trueauthor = {Dow, Alan and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Fundamenta Mathematica},
note = { arxiv:0711.3038 },
pages = {191--210},
title = {{More on Tie-points and homeomorphism in $\mathbb N^*$}},
volume = {203},
year = {2009},
abstract = {A point $x$ is a (bow) tie-point of a space $X$ if
$X\setminus \{x\}$ can be partitioned into (relatively) clopen sets
each with $x$ in its closure. Tie-points have appeared in the
construction of non-trivial autohomeomorphisms of $\beta{\mathbb N}
\setminus {\mathbb N}=\mathbb N^*$ and in the recent study of
(precisely) 2-to-1 maps on $\mathbb N^*$. In these cases the
tie-points have been the unique fixed point of an involution on
$\mathbb N^*$. One application of the results in this paper is the
consistency of there being a 2-to-1 continuous image of $\mathbb N^*$
which is not a homeomorph of $\mathbb N^*$.},
},
@article{Sh:918,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:0902.0440 },
pages = {507--543},
title = {{Many partition relations below density}},
volume = {192},
year = {2012},
abstract = {We force $2^\lambda$ to be large and for many pairs in
the interval $(\lambda,2^\lambda)$ a stronger version of the
polarized partition relations hold. We apply this toproblem in
general topology},
},
@article{CoSh:919,
author = {Cohen, Moran and Shelah, Saharon},
trueauthor = {Cohen, Moran and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:0906.3050 },
pages = {140--154},
title = {{Stable theories and representation over sets}},
volume = {62},
year = {2016},
abstract = {In this paper we give a characterization of the class of
first order stable theories using.},
},
@article{GbSh:920,
author = {Goebel, Ruediger and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Results in Mathematics},
pages = {53--64},
title = {{$\aleph_n$-free modules with trivial dual}},
volume = {54},
year = {2009},
abstract = {In the first part of this paper we introduce a
simplified version of a new Black Box from Shelah [Sh:883] which can be
used to construct complicated $\aleph_n$-free abelian groups for any
natural number $n\in N$. In the second part we apply this
prediction principle to derive for many commutative rings $R$ the
existence of $\aleph_n$-free $R$-modules $M$ with trivial dual $M^*=0$,
where $M^*={\rm Hom}(M,R)$. The minimal size of the
$\aleph_n$-free abelian groups constructed below is $\beth_n$, and this
lower bound is also necessary as can be seen immediately if we apply
GCH.},
},
@article{ShTb:921,
author = {Shelah, Saharon and Tsaban, Boaz},
trueauthor = {Shelah, Saharon and Tsaban, Boaz},
fromwhere = {IL, IL},
journal = {Topology Proceedings},
pages = {385--392},
title = {{On a problem of Juh\'asz and Van Mill}},
volume = {36},
year = {2010},
abstract = {A $27$ years old and still open problem of Juh\'asz 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 \emph{regular} is weakened to
\emph{Hausdorff}, and \emph{coutnably compact} is strengthened
to \emph{sequentially compact}.},
},
@article{Sh:922,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:0711.3030 },
pages = {2151--2161},
title = {{Diamonds}},
volume = {138},
year = {2010},
abstract = {We prove, e.g. that if $\lambda=\chi^+=2^\chi$ and $S
\subseteq \{\delta<\lambda:cf(\delta)\ne cf(\chi)\}$ is stationary
then $\diamondsuit_\lambda$.},
},
@article{AMRShS:923,
author = {Ardal, Hayri and Manuch, Jan and Rosenfeld, Moshe and Shelah,
Saharon and Stacho, Ladislav},
trueauthor = {Ardal, Hayri and Manuch, Jan and Rosenfeld, Moshe and
Shelah, Saharon and Stacho, Ladislav},
fromwhere = {3,3,1,IL,3},
journal = {Discrete and Computational Geometry},
pages = {132--141},
title = {{The odd-distance plane graph}},
volume = {42},
year = {2009},
abstract = {The vertices of the odd-distance graph are the points of the
plane $\mathbb{R}^2$. Two points are connected by an edge if their
Euclidean distance is an odd integer. We prove that the chormatic
number of this graph is at least five. We also prove that the
odd-distance graph in $\mathbb{R}^2$ is countably choosable, which such
a graph in $\mathbb{R}^3$ is not.},
},
@article{Sh:924,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:1004.3342 },
pages = {399--417},
title = {{Models of PA: when two elements are necessarily order
automorphic}},
volume = {61},
year = {2015},
abstract = {We are interested in the question of how much the order of
a non-standard model of PA can determine the model. In particular, for
a model $M$, we want to characterize the complete types $p(x,y)$ of
non-standard elements $(a,b)$ such that the linear orders $\{x:x< a\}$
and $\{x:x < b\}$ are necessarily isomorphic. It is proved that this
set includes the complete types $p(x,y)$ such that if the pair $(a,b)$
realizes it (in $M$) then there is an element $c$ such that for all
standard $n,c^n < a,c^n < b,a < bc$ and $b < ac$. We prove that this
is optimal, because if $\diamondsuit_{\aleph_1}$ holds, then there is
$M$ of cardinality $\aleph_1$ for which we get equality. We also deal
with how much the order in a model of PA may determine the addition.},
},
@article{LrSh:925,
author = {Larson, Paul and Shelah, Saharon},
trueauthor = {Larson, Paul and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Mathematical Logic Quarterly},
pages = {299--306},
title = {{Splitting stationary sets from weak forms of Choice}},
volume = {55},
year = {2009},
abstract = {We consider splitting stationary sets under the assumption
of ZF + DC plus the existence of a ladder system for ordinals of
countable cofinality. This is a continuation of \cite{Sh835}.},
},
@article{BrSh:926,
author = {Bartoszynski, Tomek and Shelah, Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
pages = {1293--1310},
title = {{Dual Borel Conjecture and Cohen reals}},
volume = {75},
year = {2010},
abstract = {We construct a model of ZFC satisfying the Dual
Borel Conjecture in which there is a set of size $\aleph_1$ that does
not have measure zero.},
},
@article{BKSh:927,
author = {Baldwin, John and Kolesnikov, Alexei and Shelah, Saharon},
trueauthor = {Baldwin, John and Kolesnikov, Alexei and Shelah, Saharon},
fromwhere = {1,1,IL},
journal = {Journal of Symbolic Logic},
pages = {914--928},
title = {{The amalgamation spectrum}},
volume = {74},
year = {2009},
abstract = {For every natural number $k^*$, there is a class
$\mbox{\boldmath{K}}_*$ defined by a sentence in
$L_{\omega_1,\omega}$ that has no models of cardinality $>
\beth_{k^*+1}$, but $\mbox{\boldmath{K}}_*$ 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
$\mbox{\boldmath{K}}_*$ defined by a sentence in $L_{\omega_1,\omega}$
that has no models of cardinality $> \beth_{\alpha}$, but
$\mbox{\boldmath{K}}_*$ has the disjoint amalgamation property on
models of cardinality $\leq \aleph_{\alpha}$. Similar results hold
for arbitrary $\kappa$ and $L_{\kappa^+,\omega}$.},
},
@article{ShUs:928,
author = {Shelah, Saharon and Usvyatsov, Alex },
trueauthor = {Shelah, Saharon and Usvyatsov, Alex},
fromwhere = {IL,IL},
journal = {preprint},
title = {{Unstable Classes of Metric Structures}},
abstract = { We prove a strong nonstructure theorem for a class
of metric structures with an unstable formula. As a consequence, we
show that weak categoricity (that is, categoricity up to isomorphisms
and not isometries) implies several weak versions of stability. This is
the first step in the direction of the investigation of weak
categoricity of metric classes.},
},
@article{HeSh:929,
author = {Herden, Daniel and Shelah, Saharon},
trueauthor = {Herden, Daniel and Shelah, Saharon},
fromwhere = {D, IL},
journal = {Forum Mathematicum},
pages = {627--640},
title = {{$\kappa$-fold transitive groups}},
volume = {22},
year = {2010},
abstract = {A group $G$ of type $0$ is called $\kappa$-transitive for
some cardinal $\kappa >0$ if for any ordered pair of pure elements
$x,y \in G$ there exist exactly $\kappa$-many $\varphi \in {\rm Aut}
G$ such that $x\varphi =y$. We shows the existence of large
$\kappa$-transitive groups for every $\kappa\ge \aleph_0$ assuming V=L
and ZFC respectively.},
},
@article{HeSh:930,
author = {Herden, Daniel and Shelah, Saharon},
trueauthor = {Herden, Daniel and Shelah, Saharon},
fromwhere = {D, IL},
journal = {Proceedings of the American Mathematical Society},
pages = {2843--2847},
title = {{An upper cardinal bound on absolute E-rings}},
volume = {137},
year = {2009},
abstract = {We show that for every abelian group $A$ of cardinality
$\ge\kappa(\omega)$ there exists a generic extension of the universe,
where $A$ is countable with $2^{\aleph_O}$ injective endomorphisms.
As an immediate consequence of this result there are no absolute
E-rings of cardinality $\ge \kappa (\omega)$. This paper does not
require any specific prior knowledge of forcing or model theory and can
be considered accessible also for graduate students.},
},
@article{ShSr:931,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
fromwhere = {IL, 3},
journal = {Journal of Applied Analysis},
pages = {69--89},
title = {{MASAS in the Calkin algebra without the continuum
hypothesis}},
volume = {17},
year = {2011},
abstract = {Methods for constructing masas in the Calkin algebra
without assuming the Continuum Hypothesis are developed},
},
@article{Sh:932,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {TBA: a volume in honor of Andrzej Mostowski},
note = { arxiv:0903.3614 },
title = {{Maximal failures of sequence locality in a.e.c.}},
volume = {submitted},
abstract = {We are interested in examples of a.e.c. with
amalgamation having some (extreme) behaviour concerning types. Note we
deal with ${\frak k}$ being sequence-local, i.e. local for
increasing chains of length a regular cardinal. For any cardinal
$\theta \ge \aleph_0$ we construct an a.e.c. with amalgamation ${\frak
k}$ with L.S.T.$({\frak k}) = \theta,|\tau_{\frak K}| = \theta$ such
that $\{\kappa:\kappa$ is a regular cardinal and ${\frak K}$ is
not $(2^\kappa,\kappa)$-sequence-local$\}$ is maximal. In fact we have
a direct characterization of this class of cardinals: the regular
$\kappa$ such that there is no uniform $\kappa^+$-complete ultrafilter.
We also prove a similar result to ``$(2^\kappa,\kappa)$-compact for
types''.},
},
@article{LwSh:933,
author = {Laskowski, Michael C. and Shelah, Saharon},
trueauthor = {Laskowski, Michael C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/1206.6028 },
pages = {47--81},
title = {{$\bold P$-NDOP and $\bold P$-decompositions of
$\aleph_\epsilon$-saturated models of superstable theories}},
volume = {229},
year = {2015},
abstract = {Assume a complete first order theory $T$ is superstable.
We generalize revise \cite{Sh:401} in two respects, so do not depend
on it. First issue we deal with a more general case. Let $\bold P$
be a class of regular types in ${\frak C}$, closed under automorphisms
and under $\pm$. We generalize \cite{Sh:401} to this context to
$\bold P^\pm$-saturated $M$'s, assuming $\bold P$-NDOP which is
weaker than NDOP. Second issue, in this content it is more delicate
to find sufficient condition on two $\bold P$-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
${\frak d}_\ell = \langle M^\ell_\eta,a_\eta:\eta \in I_\ell\rangle$
is a $\bold P$-decomposition of $M$ for $\ell=1,2$ and $\{p^{{\frak
d}_\ell}_\eta:\eta \in I_\ell\}/\pm = ({\Cal P} \cap \bold S(M))/\pm$
and show the two trees are quite similar (or isomorphic).},
},
@article{HalSh:934,
author = {Hall, Eric and Shelah, Saharon},
trueauthor = {Hall, Eric and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:0808.0535 },
pages = {207--216},
title = {{Partial choice functions for families of finite sets}},
volume = {220},
year = {2013},
abstract = {Let $p$ be a prime. We show that ZF $+$ ``Every
countable set of $p$-element sets has an infinite partial choice
function'' is not strong enough to prove that every countable set of
$p$-element sets has a choice function, answering an open question
from [1]. The independence result is obtained by way of a
permutation (Fraenkel-Mostowski) model in which the set of atoms has
the structure of a vector space over the field of $p$ elements. By way
of comparison, some simpler permutation models are considered in
which some countable families of $p$-element sets fail to have infinite
partial choice functions.},
},
@article{Sh:935,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Canadian Journal of Mathematics},
note = { arxiv:0904.0816 },
pages = {1416--1435},
title = {{MAD Saturated Families and SANE Player}},
volume = {63},
year = {2011},
abstract = {We throw some light on the question: is there a MAD
family (= a family of infinite subsets of $\Bbb N$, the intersection of
any two is finite) which is completely separable (i.e. any $X \subseteq
\Bbb N$ is included in a finite union of members of the family
\underbar{or} 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.},
},
@article{EnSh:936,
author = {Enayat, Ali and Shelah, Saharon},
trueauthor = {Enayat, Ali and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Topology and its Applications},
pages = {2495--2502},
title = {{An improper arithmetically closed Borel subalgebra
of $P(\omega)$ mod FIN}},
volume = {158},
year = {2011},
abstract = { We show the existence of a subalgebra ${\mathcal
A}\subset {\mathcal P}(\omega)$ that satisfies the following
three conditions. \endgraf ${\mathcal A}$ is Borel (when ${\mathcal
P}(\omega)$ is identified with $2^\omega$). \endgraf ${\mathcal A}$ is
arithmetically closed (i.e., ${\mathcal A}$ is closed under the Turing
jump, and Turing reducibility). \endgraf The forcing notion
$({\mathcal A}, \subset)$ modulo the ideal FIN of finite sets collapses
the continuum to $\aleph_0$. },
},
@article{Sh:937,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:0808.2960 },
pages = {341--365},
title = {{Models of expansions of $\Bbb N$ with no end extensions}},
volume = {57},
year = {2011},
abstract = { We deal with models of Peano arithmetic (specifically
with a question of Ali Enayat). The methods are from creature
forcing. We find an expansion of $\Bbb N$ such that its theory
has models with no (elementary) end extensions. In fact there is a
Borel uncountable set of subsets of $\Bbb N$ such that expanding $\Bbb
N$ by any uncountably many of them suffice. Also we find
arithmetically closed ${\Cal A}$ with no definably closed ultrafilter
on it},
},
@article{Sh:938,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:0905.3021 },
pages = {1--40},
title = {{PCF arithmetic without and with choice}},
volume = {191},
year = {2012},
abstract = {We deal with relatives of GCH which are provable.
In particular we deal with rank version of the revised GCH.
Our motivation was to find such results when only weak versions of
the axiom of choice are assumed but some of the results gives
us additional information even in ZFC.},
},
@article{KrSh:939,
author = {Kellner, Jakob and Shelah, Saharon},
trueauthor = {Kellner, Jakob and Shelah, Saharon},
fromwhere = {A,IL},
journal = {Archive Math Logic},
note = { arxiv:0905.3913 },
pages = {477-501},
title = {{More on the pressing down game}},
volume = {50},
year = {2011},
abstract = {We compare the pressing down and the Banach Mazur games and
show: Consistently relative to a supercompact there is a nowhere
precipitous normal ideal $I$ on $\aleph_2$ such that for every
$I$-positive set $A$ nonempty wins the pressing down game of length
$\aleph_1$ on $I$ starting with $A$.},
},
@article{JaSh:940,
author = {Jarden, Adi and Shelah, Saharon},
trueauthor = {Jarden Adi and Shelah, Saharon},
fromwhere = {IL, IL},
journal = {Notre Dame Journal of Formal Logic},
title = {{Non forking good frames minus local character}},
volume = {submitted},
abstract = {We prove that if $s$ is an almost good $\lambda$-frame (i.e.
$s$ is a good $\lambda$-frame except that it satisfies just a weak
version of local character), then we can complete the frame s.t. it
will satisfy local character too. This theorem has an important
application. In \cite{she46} it has proved that (under mild set
theoretic assumptions) Categoricity in $\lambda,\lambda^+$ and
intermediate number of models in $K_{\lambda^{++}}$ implies existence
of an almost good $\lambda$-frame. So by our theorem, we can get the
local character too. So by categoricity assumptions
in $\lambda,\lambda^+, \lambda^{++}$ we can get existence of a
good $\lambda$-frame. Combining this with \cite{sh600}, we
conclude that the function $\lambda \rightarrow I(\lambda,K)$,
which correspond to each cardinal $\lambda$, the number of models in
K of cardinality $\lambda$, is not arbitrary.},
},
@article{RShS:941,
author = {Roslanowski, Andrzej and Shelah, Saharon and Spinas, Otmar},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon and Spinas,
Otmar},
fromwhere = {1,IL,CH},
journal = {Bulletin of the London Mathematical Society},
note = { arxiv:0905.0526 },
pages = {299--310},
title = {{Nonproper Products}},
volume = {44},
year = {2012},
abstract = {We show that there exist two proper creature forcings having
a simple (Borel) definition, whose product is not proper. We also give
a new condition ensuring properness of some forcings with norms.},
},
@article{RoSh:942,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:1105.6049 },
pages = {603--629},
title = {{More about $\lambda$--support iterations
of $({<}\lambda)$--complete forcing notions}},
volume = {52},
year = {2013},
abstract = {This article continues Ros{\l}anowski and
Shelah \cite{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 $\lambda$). },
},
@article{GbHSh:943,
author = {Goebel, Ruediger and Herden, Daniel and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Herden, Daniel and Shelah,
Saharon},
fromwhere = {D,D,IL},
journal = {Journal of the European Mathematical Society},
pages = {845--901},
title = {{Skeletons, bodies and generalized $E(R)$-algebras}},
volume = {11},
year = {2009},
abstract = {fill in},
},
@article{Sh:944,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Sarajevo Journal of Mathematics},
note = { arxiv:0901.1499 },
title = {{Models of PA: Standard Systems without Minimal Ultrafilters}},
volume = {submitted},
abstract = {We prove that $\Bbb N$ has an uncountable
elementary extension $N$ such that there is no ultrafilter on
the Boolean Algebra of subsets of $\Bbb N$ represented in $N$ which is
so called minimal.},
},
@article{Sh:945,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:0904.0817 },
title = {{On CON(${\mathfrak d \/}_\lambda >$ cov$_\lambda$(meagre))}},
volume = {submitted},
abstract = {We prove the consistency of: for suitable
strongly inaccessible cardinal $\lambda$ the dominating number, i.e.
the cofinaty of ${}^\lambda \lambda$ is strictly bigger than
cov(meagre$_\lambda$), i.e. the minimal number of no-where-dense
subsets of ${}^\lambda 2$ needed to cover it. This answers a question
of Matet.},
},
@article{KpSh:946,
author = {Kaplan, Itay and Shelah, Saharon},
trueauthor = {Kaplan, Itay and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Journal of Symbolic Logic},
pages = {585--619},
title = {{Examples in dependent theories}},
volume = {49},
year = {2014},
abstract = {We show a counterexample to a conjecture by Shelah regarding
the existence of indiscernible sequences in dependent theories},
},
@article{LrNeSh:947,
author = {Larson, Paul and Neeman, Itay and Shelah, Saharon},
trueauthor = {Larson, Paul and Neeman, Itay and Shelah, Saharon},
fromwhere = {1,1,IL},
journal = {Fundamenta Mathematicae},
pages = {173--192},
title = {{Universally measurable sets in generic extensions}},
volume = {208},
year = {2010},
abstract = { A subset of a topological space is said to be
universally measurable if it is measurable with respect to every
complete, countably additive $\sigma$-finite measure on the space,
and universally null if it has measure zero for each such
atomless measure. In 1934, Hausdorff proved that there
exist universally null sets of cardinality $\aleph_1$, and thus
that there exist a least $2^{\aleph_1}$ such sets. Laver showed in the
1970's that consistently there are just continuum many universally null
sets. The question of whether there exist more than continuum many
universally measurable sets was asked by Mauldin no later than 1984.
We show that consistently there exist only continuum many universally
measurable sets.},
},
@article{GbHeSh:948,
author = {Goebel, Ruediger and Herden, Daniel and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Herden, Daniel and Shelah,
Saharon},
fromwhere = {D,D,IL},
journal = {Advances in Mathematics},
pages = {235--253},
title = {{Absolute $E$-rings}},
volume = {226},
year = {2011},
abstract = { A ring $R$ with $1$ is called an {\em $E$-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 {\em 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 $\omega$-Erd\H{o}s 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 \cite{HS} and the construction
of absolute $E$-rings $R$ is based on an old result by Shelah \cite{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 {\em 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].},
},
@article{GaSh:949,
author = {Garti, Shimon and Shelah, Saharon},
trueauthor = {Garti, Shimon and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Journal of Symbolic Logic},
pages = {766--776},
title = {{A strong polarized relation}},
volume = {77},
year = {2012},
abstract = {We prove that the strong polarized relation
${{\mu^+}\choose {\mu}}\rightarrow {{\mu^+}\choose {\mu}}^{1,1}_2$
is consistent with ZFC.},
},
@article{Sh:950,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
note = { arxiv:1202.5795 },
title = {{Dependent dreams: recounting types}},
volume = {preprint},
abstract = {We investigate the class of models of a general
dependent theory. We continue \cite{Sh:900} in particular
investigating so called ``decomposition of types''; thesis is
that what holds for stable theory and for Th$(\Bbb Q,<)$ hold for
dependent theories. Another way to say this is: we have to look at
small enough neighborhood and use reasonably definable types to analyze
a type. We note the results understable without reading. First, a
parallel to the ``stability spectrum'', the ``recounting of types'',
that is assume $\lambda = \lambda^{< \lambda}$ is large enough, $M$ a
saturated model of $T$ of cardinality $\lambda$, let $\bold
S_{\text{aut}}(M)$ be the number of complete types over $M$ up to being
conjugate, i.e. we identify $p,q$ when some automorphism of $M$ maps
$p$ to $q$. Whereas for independent $T$ the number is $2^\lambda$, for
dependent $T$ the number is $\le \lambda$ moreover it is
$\le |\alpha|^{|T|}$ when $\lambda = \aleph_\alpha$. Second, for stable
theories ``lots of indiscernibility exists'' a ``too good indiscernible
existence theorem'' saying, e.g. that if the
type tp$(d_\beta;\{d_\beta:\beta < \alpha\})$ is increasing for $\alpha
< \kappa = \text{ cf}(\kappa)$ and $\kappa > 2^{|T|}$ {then}
$\langle d_\alpha:\alpha \in S\rangle$ is indiscernible for some
stationary $S \subseteq \kappa$. Third, for stable $T$,a model
is $\kappa$-saturated iff it is $\aleph_\varepsilon$-saturated and
every infinite indiscernible set (of elements) of cardinality $<
\kappa$ can be increased. We prove here an analog. Fourth, for $p \in
\bold S(M)$, the number of ultrafilters on the outside definable
subsets of $M$ extending $p$ has an absolute bound $2^{|T|}$.
Restricting ourselves to one $\varphi(x,\bar y)$, the number is finite,
with an absolute found (well depending on $T$ and $\varphi$). Also if
$M$ is saturated then $p$ is the average of an indiscernible sequence
inside the model. Lastly, the so-called generic pair conjecture was
proved in \cite{Sh:900} for $\kappa$ measurable, here it is
essentially proved, i.e. for $\kappa > |T| + \beth_\omega$.},
},
@article{MdSh:951,
author = {Mildenberger, Heike and Shelah, Saharon},
trueauthor = {Mildenberger, Heike and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Fundamenta Mathematicae},
pages = {1--38},
title = {{Proper translation}},
volume = {215},
year = {2011},
abstract = {We continue our work on weak diamonds \cite{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 \cite[Ch.~IV]{Sh:f}) and \mbox{$<\omega_1$}-proper
${\ensuremath{{}^\omega \omega}}$-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.},
},
@article{ShZa:952,
author = {Shelah, Saharon and Zapletal, Jindrich},
trueauthor = {Shelah, Saharon and Zapletal, Jind\v{r}ich},
fromwhere = {IL,1},
journal = {Combinatorica},
pages = {225--244},
title = {{Ramsey theorems for product of finite sets with submeasures}},
volume = {31},
year = {2011},
abstract = {We prove parametrized partition theorem on products
of finite sets equipped with submeasures, improving the results of
DiPrisco, Llopis, and Todorcevic},
},
@incollection{DoSh:953,
author = {Doron, Mor and Shelah, Saharon},
trueauthor = {Doron, Mor and Shelah, Saharon},
booktitle = {Fields of Logic and Computation: Essays dedicated to
Yuri Gurevich on the Occasion his 70th Birthday},
fromwhere = {IL,IL},
pages = {581--614},
publisher = {Springer, A. Blass, N. Dershowitz, W. Reisig (eds.)},
series = {Lecture Notes in Computer Science},
title = {{Hereditary Zero-One Laws for Graphs}},
volume = {6300},
year = {2010},
abstract = {We consider the random graph $M^n_{\bar{p}}$ on the
set $[n]$, were the probability of $\{x,y\}$ being an edge
is $p_{|x-y|}$, and $\bar{p}=(p_1,p_2,p_3,...)$ is a series of
probabilitie. We consider the set of all $\bar{q}$ derived from
$\bar{p}$ by inserting 0 probabilities to $\bar{p}$, or alternatively
by decreasing some of the $p_i$. We say that $\bar{p}$ hereditarily
satisfies the 0-1 law if the 0-1 law (for first order logic) holds in
$M^n_{\bar{q}}$ for any $\bar{q}$ derived from $\bar{p}$ in the
relevant way described above. We give a necessary and sufficient
condition on $\bar{p}$ for it to hereditarily satisfy the 0-1 law.},
},
@article{FaSh:954,
author = {Farah, Ilijas and Shelah, Saharon},
trueauthor = {Farah, Ilijas and Shelah, Saharon},
fromwhere = {3,IL},
journal = {Journal of Mathematical Logic},
pages = {45--81},
title = {{A Dichotomy for the number of ultrapowers}},
volume = {10},
year = {2010},
abstract = {We prove a strong dichotomy for the number of ultrapowers of
a given model of cardinality $\le c$ associated with nonprincipal
ultrafilters on $\mathbb{N}$. They are either all isomorphic, or else
there are $2^c$ many. We prove the analogous result for metric
structures, including $C^*$-algebras and $II_1$ factors, as well as
their relative commutants },
},
@article{Sh:955,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:1107.4625 },
pages = {185--231},
title = {{Pseudo PCF}},
volume = {201},
year = {2014},
abstract = {We continue our investigation on pcf with weak forms of the
axiom of choice. Characteristically, we assume DC + ${\mathcal P}(Y)$
when looking at $\prod\limits_{s \in Y} \delta_s$. We get more
parallels of pcf theorems.},
},
@article{GaSh:956,
author = {Garti, Shimon and Shelah, Saharon},
trueauthor = {Garti, Shimon and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Journal of the Mathematical Society of Japan},
pages = {549--559},
title = {{$(\kappa, \theta)$-weak normality}},
volume = {64},
year = {2012},
abstract = {We deal with the property of weak normality (for
non-principal ultrafilters). We characterize the situation of
$|\Pi_{i<\kappa} \lambda_i / D| = \lambda$. We have an application
ofr a question of Depth in Boolean algebras.},
},
@article{RoSh:957,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Annals of Combinatorics},
note = { arxiv:1005.2803 },
pages = {353--378},
title = {{Partition theorems from creatures and idempotent
ultrafilters}},
volume = {17},
year = {2013},
abstract = {We show a general scheme of Ramsey-type results for
partitions of countable sets of finite functions, where ``one piece is
big'' is interpreted in the language originating in creature forcing.
The heart of our proofs follows Glazer's proof of the Hindman
Theorem, so we prove the existence of idempotent ultrafilters with
respect to suitable operation. Then we deduce partition theorems
related to creature forcings.},
},
@article{BlSh:958,
author = {Baldwin, John and Shelah, Saharon},
trueauthor = {Baldwin, John and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Fundamenta Mathematica},
pages = {255-270},
title = {{A Hanf number for saturation and omission}},
volume = {213},
year = {2011},
abstract = {Suppose \mbox{\boldmath $T$} $ =(T,T_1,p)$ is a triple
of two countable theories in languages $\tau \subset \tau_1$ and a
$\tau_1$-type $p$ over the empty set. We show the Hanf number for the
property: There is a model $M_1$ of $T_1$ which omits $p$, but $M_1
\restriction \tau$ is saturated is at least the L{\"{o}}wenheim number
of second order logic.},
},
@article{BlSh:959,
author = {Baldwin, John and Shelah, Saharon},
trueauthor = {Baldwin, John and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Mathematical Logic},
pages = {19 pps.},
title = {{The stability spectrum for classes of atomic models}},
volume = {12},
year = {2012},
abstract = { We prove two results on the stability spectrum for
$L_{\omega_1,\omega}$. Here $S^m_i(M)$ denotes an appropriate notion
(${\rm at}$(+ 1 962) or ${\rm mod}$) of Stone space of $m$-types over
$M$. Theorem A. Suppose that for some positive integer $m$ and for
every $\alpha< \delta(T)$, there is an $M \in$ \mbox{\boldmath$K$} with
$|S^m_i(M)| > |M|^{\beth_\alpha(|T|)}$. Then for every $\lambda \geq
|T|$, there is an $M$ with $|S^m_i(M)| > |M|$. Theorem B. Suppose that
for every $\alpha<\delta(T)$, there is $M_\alpha \in$
\mbox{\boldmath$K$} such that $\lambda_\alpha = |M_{\alpha}| \geq
\beth_\alpha$ and $|S^m_{i}(M_\alpha)| > \lambda_\alpha$. Then for any
$\mu$ with $\mu^{\aleph_0}>\mu$, \mbox{\boldmath$K$} is not $i$-stable
in $\mu$. These results provide a new kind of sufficient condition
for the unstable case and shed some light on the spectrum of strictly
stable theories in this context. The methods avoid the use of
compactness in the theory under study. In the
Section~\label{treeindis}, we expound the construction of tree
indiscernibles for sentences of $L_{\omega_1,\omega}$.},
},
@article{Sh:960,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Sarajevo Journal of Mathematics},
note = { arxiv:1005.2802 },
title = {{Preserving old $([\omega]^{\aleph_0},\supseteq)$ is proper}},
volume = {submitted},
abstract = {We give some sufficient and necessary conditions on
a forcing notion $Q$ for preserving the forcing
notion $([\omega]^{\aleph_0},\supseteq)$ is proper. They cover
many reasonable forcing notions.},
},
@article{KrSh:961,
author = {Kellner, Jakob and Shelah, Saharon},
trueauthor = {Kellner, Jakob and Shelah, Saharon},
fromwhere = {A,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:1003.3425 },
pages = {49--70},
title = {{Creature forcing and large continuum: the joy of halving}},
volume = {51},
year = {2012},
},
@article{GaSh:962,
author = {Garti, Shimon and Shelah, Saharon},
trueauthor = {Garti, Shimon and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Annals of Combinatorics},
pages = {709--717},
title = {{Combinatorial aspects of the splitting number}},
volume = {16},
year = {2012},
abstract = {We prove that the existence of strong splitting families
is equivalent to the failure of the polarized
relation ${\mathfrak{s}\choose\omega} \rightarrow
{\mathfrak{s}\choose {\omega}^{1,1}_2}$. We show that both directions
are consistent with ZFC, and that the strong splitting number equals to
the splitting number, when exists. Consequently, we can put some
restriction on the possibility that $\mathfrak{s}$ is singular.},
},
@article{CDMMSh:963,
author = {Cummings, James and Dzamonja, Mirna and Magidor, Menachem and
Morgan, Charles and Shelah, Saharon},
trueauthor = {Cummings, James and D\v{z}amonja, Mirna and Magidor,
Menachem and Morgan, Charles and Shelah, Saharon},
fromwhere = {1,UK,IL,UK,IL},
journal = {Transactions of the AMS},
note = { arxiv:math.LO/1403.6795v1 },
title = {{A framework for forcing constructions at successors of
singular cardinals}},
volume = {accepted},
abstract = {We describe a framework for proving consistency results
about singular cardinals of arbitrary cofinality and their successors.
This framework allows the construction of models in which the Singular
Cardinals Hypothesis fails at a singular cardinal $\kappa$ of
uncountable cofinality, while $\kappa^+$ enjoys various combinatorial
properties. \endgraf As a sample application, we prove the consistency
(relative to that of ZFC plus a supercompact cardinal) of there being a
strong limit singular cardinal $\kappa$ of uncountable cofinality
where SCH fails and such that there is a collection of size less than
$2^{\kappa^+}$ of graphs on $kappa^+$ such that any graph on
$\kappa^+$ embeds into one of the graphs in the collection.},
},
@article{GaSh:964,
author = {Garti, Shimon and Shelah, Saharon},
trueauthor = {Garti, Shimon and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Annals of Combinatorics},
pages = {271--276},
title = {{Strong polarized relations for the continuum}},
volume = {16},
year = {2012},
abstract = {sufficient conditions for the bipartite $(2^\kappa, \kappa)
------> (2^\kappa , \kappa)$ when $ \mu - 2 \kappa $ is singular eg
$cf(\mu) < {\frak s} <\mu$, generalize for $\lambda$ instead
${\aleph_0}$},
},
@article{LrMaSh:965,
author = {Larson, Paul and Matteo, Nick and Shelah, Saharon},
trueauthor = {Larson, Paul and Matteo, Nicholas and Shelah, Saharon},
fromwhere = {1,1,IL},
journal = {Discrete Mathematics},
pages = {1336--1352},
title = {{Majority decisions when abstention is possible}},
volume = {312},
year = {2012},
abstract = {Suppose we are given a family of choice functions on pairs
from a given finite set. The set is considered as a set of
alternatives (say candidates for an office) and the functions as
potential ``voters.'' The question is, what choice functions agree, on
every pair, with the majority of some finite subfamily of the voters?
For the problem as stated, a complete characterization was given in
\cite{Sh:816}, but here we allow each voter to abstain. There are
four cases.},
},
@article{JrSh:966,
author = {Jarden, Adi and Shelah, Saharon},
trueauthor = {Jarden, Adi and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {preprint},
title = {{Existence of uniqueness triples without stability}},
},
@article{MdSh:967,
author = {Mildenberger, Heike and Shelah, Saharon},
trueauthor = {Mildenberger, Heike and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Journal of Symbolic Logic},
pages = {1322--1340},
title = {{The minimal cofinality of an ultrapower of $\omega$ and the
cofinality of the symmetric group can be larger than
$\mathfrak{\lowercase{b}}^+$ }},
volume = {76},
year = {2011},
abstract = {We prove the statement in the title},
},
@article{ShSm:968,
author = {Shelah, Saharon and Struengmann, Lutz},
trueauthor = {Shelah, Saharon and Str{\"{u}}ngmann, Lutz},
fromwhere = {IL,D},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/1401.5317 },
title = {{On the p-rank of {$\bf Ext(A,B)$} for countable abelian groups
$\bf A$ and $\bf B$}},
volume = {submitted},
abstract = {In this note we show that the p-rank of ${\rm Ext}(A,B)$
for countable torsion-free abelian groups $A$ and $B$ is either
countable or the size of the continuum.},
},
@article{GoKrShWo:969,
author = {Goldstern, Martin and Kellner, Jakob and Shelah, Saharon and
Wohofsky, Wolfgang},
fromwhere = {A, A, IL, A},
journal = {Transactions of the American Mathematical Society},
note = { arxiv:1105.0823 },
pages = {245--307},
title = {{Borel Conjecture and Dual Borel Conjecture}},
volume = {366},
year = {2014},
abstract = {We show that it is consistent that the Borel Conjecture and
the dual Borel Conjecture hold simultaneously.},
},
@article{GbHeSh:970,
author = {Goebel, Ruediger and Herden, Daniel and Shelah, Saharon},
trueauthor = {G{\"{o}}bel, R{\"{u}}diger and Herden, Daniel and Shelah,
Saharon},
fromwhere = {D,D,IL},
journal = {Journal of the European Mathematical Society},
title = {{Prescribing endomorphism rings of $\aleph_n$-free modules}},
volume = {accepted},
},
@article{KhSh:971,
author = {Khelif, Anatole and Shelah, Saharon},
trueauthor = {Khelif, Anatole and Shelah, Saharon},
fromwhere = {F,IL},
journal = {Comptes Rendus de l\'Academie des Sciences},
pages = {1241--1244},
title = {{Equivalence \'el\'ementaire de puissances cart\'esiennes d'un
meme groupe}},
volume = {348},
year = {2010},
abstract = {We prove that if $I$ and $J$ are infinite sets and $G$
an abelian torsion group the groups $G^I$ and $G^J$ are elementarily
equivalent for the logic $L_{\infty\omega}$. The proof is based on a
new and simple property with a Cantor-Bernstein flavour. A criterion
applying to non commutative groups allows us to exhibit various groups
(free or soluble or nilpotent or ...) $G$ such that for $I$ infinite
countable and $J$ uncountable the groups $G^I$ and $G^J$ are not even
elementarily equivalent for the $L_{\omega_I \omega}$ logic.
Another argument leads to a countable commutative group having the same
property.},
},
@article{RoSh:972,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Periodica Mathematica Hungarica},
note = { arxiv:1007.5368 },
pages = {79--95},
title = {{Monotone hulls for ${\mathcal N}\cap {\mathcal M}$}},
volume = {69},
year = {2014},
abstract = {Using the method of decisive creatures ([KrSh:872]) we show
the consistency of ``there is no increasing $\omega_2$--chain of Borel
sets and ${\rm non}({\mathcal N})= {\rm non}({\mathcal
M})=\omega_2=2^\omega$''. Hence, consistently, there are no monotone
hulls for the ideal ${\mathcal M}\cap {\mathcal N}$. This answers
Balcerzak and Filipczak. Next we use FS iteration with partial memory
to show that there may be monotone Borel hulls for the ideals
${\mathcal M}, {\mathcal N}$ even if they are not generated by
towers.},
},
@article{MdSh:973,
author = {Mildenberger, Heike and Shelah, Saharon},
trueauthor = {Mildenberger, Heike and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/1309.0196 },
pages = {573--608},
title = {{Many countable support iterations of proper forcings preserve
Souslin trees}},
volume = {165},
year = {2014},
abstract = {We show that there are many models of
$\rm{cov} {\ensuremath{{\cal M}}}= \aleph_1$ and ${\rm
cof}{\ensuremath{{\cal M}}} = \aleph_2$ in which the club principle
holds and there are Souslin trees. The proof consists of the following
main steps: \endgraf \item[1.] We give some iterable and some
non-iterable conditions on a forcing in terms of games that imply that
the forcing is $(T,Y,\mathcal{S})$-preserving. A special case
of $(T,Y,\mathcal{S})$-preserving is preserving the Souslinity of
an $\omega_1$-tree. \item[2.] We show that some tree-creature forcings
from \cite{RoSh:470} satisfy the sufficient condition for one of the
strongest games. \item[3.] Without the games, we show that some linear
creature forcings from \cite{RoSh:470} are
$(T,Y,\mathcal{S})$-preserving. There are non-Cohen preserving
examples. \item[4.] For the wider class of non-elementary proper
forcings we show that $\omega$-Cohen preserving for certain candidates
implies $(T,Y,\mathcal{S})$-preserving. \item[5.] (+ 1 978)We give a
less general but hopefully more easily readable presentation of a
result from \cite[Chapter~18, \S 3]{Sh:f}: If all iterands in a
countable support iteration are proper
and $(T,Y,\mathcal{S})$-preserving, then also the iteration
is $(T,Y,\mathcal{S})$-preserving. This is a presentation of
the so-called case A in which a division in forcings that add reals and
those who do not is not needed. \endgraf In \cite{Mi:clubdistr} we
showed: Many proper forcings from \cite{RoSh:470} with finite or
countable ${\rm{H}}(n)$ (see Section 2.1) force over a ground model
with $\diamondsuit_{\omega_1}$ in a countable support iteration the
club principle. After $\omega_1$ iteration steps the diamond holds
anyway.},
},
@article{GaSh:974,
author = {Garti, Shimon and Shelah, Saharon},
trueauthor = {Garti, Shimon and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Houston Journal of Mathematics},
title = {{${\rm DEPTH}^+$ and ${\rm LENGTH}^+$ of Boolean Algebras}},
volume = {submitted},
abstract = { Suppose $\kappa=\cf(\kappa), \lambda>\cf(\lambda)=\kappa^+$
and $\lambda=\lambda^\kappa$. We prove that there exist a sequence
$\langle{\mathbf{B}}_i:i<\kappa\rangle$ of Boolean algebras and an
ultrafilter $D$ on $\kappa$ so that
$\lambda=\prod\limits_{i<\kappa} {\rm Depth}^+({\mathbf{B}}_i)/D< {\rm
Depth}^+(\prod\limits_{i< \kappa}{\mathbf B}_i/D)= \lambda^+$. An
identical result holds also for ${\rm Length}^+$. The proof is carried
in ZFC and it holds even above large cardinals.},
},
@article{KpSh:975,
author = {Kaplan, Itay and Shelah, Saharon},
trueauthor = {Kaplan, Itay and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
pages = {45 pps.},
title = {{A dependent theory with few indiscernibles}},
volume = {accepted},
year = {2014},
abstract = {We continue the work started in \cite{KpSh:946}, and
prove the following theorem: For every $\theta$ there is a
dependent theory $T$ of size $\theta$ such that for all $\kappa$
and $\delta$, $\kappa\to\left(\delta\right)_{T,1}$ iff
$\kappa\to\left(\delta\right)_{\theta}^{<\omega}$.},
},
@article{ShSm:976,
author = {Shelah, Saharon and Struengmann, Lutz},
trueauthor = {Shelah, Saharon and Str{\"{u}}ngmann, Lutz},
fromwhere = {IL,D},
journal = {Bulletin of the London Mathematical Society},
pages = {1198--1204},
title = {{Kulikov's problem on universal torsion-free abelian
groups revisited }},
volume = {43},
year = {2011},
abstract = {Let $T$ be a torsion abelian group. We consider the class of
all torsion-free abelian groups $G$ satisfying ${\rm Ext}(G,T)=0$
and search for $\lambda$-universal objects in this class. We show
that for certain $T$ there is no $\omega$-universal group. However, for
uncountable cardinals $\lambda$ there is always a $\lambda$-universal
group if we assume $(V=L)$. Together with results by the second author
this solves completely a problem by Kulikov.},
},
@article{Sh:977,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {(Groups and Model theory) Contemporary Mathematics},
pages = {305--316},
title = {{Modules and Infinitary Logics}},
volume = {576},
year = {2012},
abstract = {We prove that the theory of abelian groups and $R$-modules
even in infinitary logic is stable and understood to some extent. },
},
@article{MiSh:978,
author = {Malliaris, Maryanthe and Shelah, Saharon},
trueauthor = {Malliaris, Maryanthe and Shelah, Saharon},
fromwhere = {1, IL},
journal = {Transactions of the American Mathematical Society},
pages = {1551--1585},
title = {{Regularity lemmas for stable graphs}},
volume = {366},
year = {2014},
abstract = {Let $G$ be a finite graph with the non-$k_*$-order property
(essentially, a uniform finite bound on the size of an
induced sub-half-graph). The main result of the paper
applies model-theoretic arguments to obtain a stronger version
of Szemer\'edi's regularity lemma for such graphs: \endgraf Theorem:
Let $k_*$ and therefore $k_{**}$ (a constant $\leq 2^{k_*+2}$) be
given. Let $G$ be a graph with the non-$k_*$-order property. Then for
any $\epsilon>0$ there exists $m=m(\epsilon)$ such that if $G$ is
sufficiently large, there is a partition $\langle A_i : i*
\aleph_0$ is measurable, we construct a regular ultrafilter on
$\lambda \geq 2^\kappa$ which is flexible (thus: ok) but not good, and
which moreover has large ${\mathrm{lcf}} (\aleph_0)$ but does not even
saturate models of the random graph. This implies (a) that flexibility
alone cannot characterize saturation of any theory, however (b) by
separating flexibility from goodness, we remove a main obstacle to
proving non-low does not imply maximal, and (c) from a set-theoretic
point of view, consistently, ok need not imply good, answering a
question from Dow 1985. Under no additional assumptions, we prove that
there is a loss of saturation in regular ultrapowers of unstable
theories, and give a new proof that there is a loss of saturation in
ultrapowers of non-simple theories. More precisely, for $\mathcal{D}$
regular on $\kappa$ and $M$ a model of an unstable theory,
$M^\kappa/ \mathcal{D}$ is not $(2^\kappa)^+$-saturated; and for $M$ a
model of a non-simple theory and $\lambda = \lambda^{<\lambda}$,
$M^\lambda/\mathcal{D}$ is not $\lambda^{++}$-saturated. Finally, we
investigate realization and omission of symmetric cuts, significant
both because of the maximality of the strict order property in
Keisler's order, and by recent work of the authors on $SOP_2$. We
prove that if $\mathcal{D}$ is a $\kappa$- complete ultrafilter on
$\kappa$, any ultrapower of a sufficiently saturated model of linear
order will have no $(\kappa, \kappa)$-cuts, and that if $\mathcal{D}$
is also normal, it will have a $(\kappa^+, \kappa^+)$-cut. We apply
this to prove that for any $n < \omega$, assuming the existence
of $n$ measurable cardinals below $\lambda$, there is a regular
ultrafilter $D$ on $\lambda$ such that any $D$-ultrapower of a
model of linear order will have $n$ alternations of cuts, as defined
below. Moreover, $D$ will $\lambda^+$-saturate all stable theories but
will not $(2^{\kappa})^+$-saturate any unstable theory, where $\kappa$
is the smallest measurable cardinal used in the construction.},
},
@article{MiSh:997,
author = {Malliaris, Maryanthe and Shelah, Saharon},
trueauthor = {Malliaris, Maryanthe and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of Symbolic Logic},
pages = {103--134},
title = {{Model-theoretic properties of ultrafilters built by
independent families of functions}},
volume = {79},
year = {2014},
abstract = {In this paper, the second of two, we continue our
investigations of model-theoretic properties of
ultrafilters: $\mu(\mathcal{D})$, the minimum size of a product of an
unbounded sequence of natural numbers modulo
$\mathcal{D}$; $\mathrm{lcf}(\aleph_0, \mathcal{D})$ the lower
cofinality (coinitiality) of $\aleph_0$ modulo
$\mathcal{D}$;flexibility, discussed extensively in the paper;
realization of symmetric cuts; and goodness. We work in ZFC except
where noted. Our main results are as follows. First, we prove that any
ultrafilter $\mathcal{D}$ which is $\lambda$-flexible (thus:
$\lambda^+$-o.k.) must have $\mu(\mathcal{D}) = 2^\lambda$. Thus, a
fortiori, $\mathcal{D}$ will saturate any stable theory. This is the
strongest possible statement about the saturation power of flexibility
alone, in light of our proof in the companion paper (I) that
consistently, flexibility does not imply saturation of the random
graph. In the remainder of the paper, we focus on the method of
constructing ultrafilters via families of independent functions. Our
second result is a constraint, that is, a tool for building
ultrafilters which are not flexible. Specifically, we prove that if,
at any point in a construction by independent functions the
cardinality of the range of the remaining independent family is
strictly smaller than the index set, then essentially no subsequent
ultrafilter can be flexible. This is a useful point of leverage since
any ultrafilter which is not flexible will fail to saturate any
non-low theory. The third and fourth results are ultrafilter
constructions. Third, assuming the existence of a measurable cardinal
$\kappa$ (to obtain a $\kappa$-complete ultrafilter), we prove that
on any $\lambda \geq \kappa^+$ there is a regular ultrafilter which is
flexible but not good. This gives a second proof, of independent
interest, of a question from Dow [1985], complementing the proof in
the companion paper (I). Fourth, assuming the existence of a weakly
compact cardinal $\kappa$, we prove that for $\aleph_0 < \theta =
\cf(\theta) < \kappa \leq \lambda$ there is a regular ultrafilter
$\mathcal{D}$ on $\lambda$ such that $\mathrm{lcf}(\aleph_0,
\mathcal{D}) = \theta$ but $(\mathbb{N}, <)^\lambda/\mathcal{D}$ has no
$(\kappa, \kappa)$-cuts. This appears counter to model-theoretic
intuition, since it shows some families of cuts in linear order can
be realized without saturating the minimum unstable theory. We give
several extensions of this last result, and show how to eliminate the
large cardinal hypothesis in the case of asymmetric cuts.},
},
@article{MiSh:998,
author = {Malliaris, Maryanthe and Shelah, Saharon},
trueauthor = {Malliaris, Maryanthe and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Journal of the American Mathematical Society},
note = { arxiv:1208.5424 },
pages = {237--297},
title = {{Cofinality spectrum theorems in model theory, set theory and
general topology}},
volume = {29},
year = {2016},
abstract = {We solve a set-theoretic problem ($p=t$) and show it has
consequences for the classification of unstable theories. That is, we
consider what pairs of cardinals $(\kappa_1, \kappa_2)$ may appear as
the cofinalities of a cut in a regular ultrapower of linear order,
under the assumption that all symmetric pre-cuts of cofinality no more
than the size of the index set are realized. We prove that the only
possibility is $(\kappa, \kappa^+)$ where $\kappa$ is regular and
$\kappa^+$ is the cardinality of the index set $I$. This shows that
unless $|I|$ is the successor of a regular cardinal, any such
ultrafilter must be $|I|^+$-good. We then connect this work to the
problem of determining the boundary of the maximum class in Keisler's
order. Currently, $SOP_3$ is known to imply maximality. Here, we show
that the property of realizing all symmetric pre-cuts characterizes
the existence of paths through trees and thus realization of types
with $SOP_2$ (it was known that realizing {all} pre-cuts characterizes
realization of types with $SOP_3$). Thus whenever $\lambda$ is not the
successor of a regular cardinal, $SOP_2$ is $\lambda$-maximal in
Keisler's order. Moreover, the question of the full maximality of
$SOP_2$ is reduced to either constructing a regular ultrafilter
admitting the single asymmetric cut described, or showing one cannot
exist.},
},
@article{MiSh:999,
author = {Malliaris, Maryanthe and Shelah, Saharon},
trueauthor = {Malliaris, Maryanthe and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Advances in Mathematics},
pages = {250--288},
title = {{A dividing line within simple unstable theories}},
volume = {249},
year = {2013},
abstract = {We give the first (ZFC) dividing line in Keisler's order
among the unstable theories, specifically among the simple unstable
theories. That is, for any infinite cardinal $\lambda$ for which there
is $\mu < \lambda \leq 2^\mu$, we construct a regular ultrafilter
$\mathcal{D}$ on $\lambda$ so that (i) for any model $M$ of a stable
theory or of the random graph, $M^\lambda/\mathcal{D}$ is
$\lambda^+$-saturated but (ii) if $Th(N)$ is not simple or not low
then $N^\lambda/\mathcal{d}$ is not $\lambda^+$-saturated. The
non-saturation result relies on the notion of flexible ultrafilters. To
prove the saturation result we develop a property of a class of simple
theories, called $Qr_1$, generalizing the fact that whenever $B$ is a
set of parameters in some sufficiently saturated model of the random
graph, $|B| = \lambda$ and $\mu < \lambda \leq 2^\mu$, then there is a
set $A$ with $|A| = \mu$ so that any nonalgebraic $p \in S(B)$ is
finitely realized in $A$. In addition to giving information about
simple unstable theories, our proof reframes the problem of saturation
of ultrapowers in several key ways. We give a new characterization of
good filters in terms of ``excellence,'' a measure of the accuracy of
the quotient Boolean algebra. We introduce and develop the notion of
{moral} ultrafilters on Boolean algebras. We prove a so-called
``separation of variables'' result which shows how the problem of
constructing ultrafilters to have a precise degree of saturation may be
profitably separated into a more set-theoretic stage, building an
excellent filter, followed by a more model-theoretic stage: building
so-called moral ultrafilters on the quotient Boolean algebra, a
process which highlights the complexity of certain patterns, arising
from first-order formulas, in certain Boolean algebras.},
},
@article{Sh:1000,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
title = {{SAVED}},
},
@article{RoSh:1001,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:1406.4217 },
title = {{The last forcing standing with diamonds}},
volume = {submitted},
abstract = {This article continues Ros{\l}anowski and Shelah
\cite{RoSh:655}, \cite{RoSh:942}. We introduce here yet another
property of $({<}\lambda)$--strategically complete forcing notions
which implies that their $\lambda$--support iterations do not collapse
$\lambda^+$.},
},
@article{GaSh:1002,
author = {Garti, Shimon and Shelah, Saharon},
trueauthor = {Garti, Shimon and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Acta Mathematica Hungarica},
pages = {296--301},
title = {{The ultrafilter number for singular cardinals}},
volume = {137},
year = {2012},
abstract = {We prove the consistency of a singular cardinal $\lambda$
with small value of the ultrafilter number $\mathfrak{u}_\lambda$, and
arbitrarily large value of $2^\lambda$.},
},
@article{BlLrSh:1003,
author = {Baldwin, John T. and Larson, Paul B. and Shelah, Saharon},
trueauthor = {Baldwin, John T. and Larson, Paul B. and Shelah, Saharon},
fromwhere = {1,1,IL},
journal = {Journal of Symbolic Logic },
title = {{Almost Galois $\omega$-Stable classes}},
volume = {accepted modulo corrections},
abstract = { {\bf Theorem.} Suppose that an $\aleph_0$-presentable
Abstract Elementary Class (AEC), $\boldmath {K}$, has the joint
embedding and amalgamation properties in $\aleph_0$ and
$<2^{\aleph_1}$ models in $\aleph_1$. If $\boldmath {K}$ has only
countably many models in $\aleph_1$, then all are small. If, in
addition, $\boldmath{K}$ is almost Galois $\omega$-stable then
$\boldmath{K}$ is Galois $\omega$-stable.},
},
@article{Sh:1004,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:1202.5799 },
title = {{A parallel to the null ideal for inaccessible $\lambda$}},
volume = {submitted},
abstract = {It is well known how to generalize the meagre ideal
replacing $\aleph_0$ by a (regular) cardinal $\lambda > \aleph_0$ and
requiring the ideal to be $\lambda^+$-complete. But can we
generalize the null ideal? In terms of forcing, this means finding a
forcing notion similar to the random real forcing, replacing $\aleph_0$
by $\lambda$. So naturally, to call it a generalization we require it
to be $(< \lambda)$-complete and $\lambda^+$-c.c. and more. Of course,
we would welcome additional properties generalizing the ones of the
random real forcing. Returning to the ideal (instead of forcing) we
may look at the Boolean Algebra of $\lambda$-Borel sets modulo
the ideal. Common wisdom have said that there is no such thing, but
here surprisingly \underline{first} we get a positive = existence
answer for $\lambda$ a ``mild'' large cardinal: the weakly compact
one. Second, we try to show that this together with the meagre ideal
(for $\lambda$) behaves as in the countable case. In particular,
consider the classical Cicho\'n diagram, which compare several
cardinal characterizations of those ideals. Last but not least,
we apply this to get consistency results on cardinal invariants for
such $\lambda$'s. We shall deal with other cardinals, and with more
properties related forcing notions in a continuation.},
},
@article{Sh:1005,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:math.LO/1411.7164 },
pages = {239--294},
title = {{ZF + DC + AX$_4$}},
volume = {55},
year = {2016},
abstract = {We consider mainly the following version of set theory:``ZF
+ DC and for every $\lambda,\lambda^{\aleph_0}$ is well ordered'',
our thesis is that this is a reasonable set theory, e.g. much can be
said. In particular, we prove that for a sequence $\bar\delta =
\langle \delta_s:s \in Y\rangle,\cf(\delta_s)$ large enough compared
to $Y$, we can prove the pcf theorem with minor changes (using true
cofinalities not the pseudo one). We then deduce the existence of
covering numbers and define and prove existence of truely successor
cardinals. From this we show that some diagonalization arguments (more
specifically some black boxes and consequence) on Abelian groups. We
end but showing some such consequences hold in ZF above.},
},
@article{Sh:1006,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Acta Mathematica Hungarica},
note = { arxiv:math.LO/1205.0064 },
pages = {363--371},
title = {{On incompactness for chromatic number of graphs}},
volume = {139(4)},
year = {2013},
abstract = {We deal with incompactness. Assume the existence
of non-reflecting stationary set of cofinality $\kappa$. We prove that
one can define a graph $G$ whose chromatic number is $>\kappa$, while
the chromatic number of every subgraph $G' \subseteq G,|G'| < |G|$ is
$\le \kappa$. The main case is $\kappa = \aleph_0$.},
},
@article{CeKpSh:1007,
author = {Chernikov, Artem and Kaplan, Itay and Shelah, Saharon},
trueauthor = {Chernikov, Artem and Kaplan, Itay and Shelah, Saharon},
fromwhere = {R,IL,IL},
journal = {Journal of the European Mathematical Society},
note = { arxiv:math.LO/1205.3101 },
pages = {2821--2848},
title = {{On non-forking spectra}},
volume = {18},
year = {2016},
abstract = {Non-forking is one of the most important notions in modern
model theory capturing the idea of a generic extension of a type
(which is a far-reaching generalization of the concept of a generic
point of a variety). \endgraf To a countable first-order theory we
associate its \emph{non-forking spectrum} --- a function of two
cardinals $\kappa$ and $\lambda$ giving the supremum of the possible
number of types over a model of size $\lambda$ that do not fork over a
sub-model of size $\kappa$. This is a natural generalization of the
stability function of a theory. \endgraf We make progress towards
classifying the non-forking spectra. On the one hand, we show that the
possible values a non-forking spectrum may take are quite limited. On
the other hand, we develop a general technique for constructing
theories with a prescribed non-forking spectrum, thus giving a number
of examples. In particular, we answer negatively a question of Adler
whether NIP is equivalent to bounded non-forking. \endgraf In
addition, we answer a question of Keisler regarding the number of cuts
a linear order may have. Namely, we show that it is possible that
${ded}\kappa<\left({ded}\kappa\right)^{ \omega}$. },
},
@article{Sh:1008,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Acta Mathematica Hungarica},
note = { arxiv:1206.2048 },
pages = {11--35},
title = {{Non-reflection of the bad set for $\check I_\theta[\lambda]$
and pcf}},
volume = {141},
year = {2013},
abstract = {We reconsider here the following related pcf questions and
make some advances: (Q1) concerning the ideal $\check
I_\kappa[\lambda]$ how much reflection do we have for the bad
set $S^{bd}_{\lambda,\kappa}\subseteq
\{\delta<\lambda: cf(\delta)=\kappa\}$ assuming it is well
defined? (Q2) for an ideal $J$ on $\kappa$ how large are
$S^{bd}_J[\bar f],S^{ch}_J[\bar f]$ for $\bar f=\langle f_\alpha:
\alpha < \lambda\rangle$ which is $<_J$-increasing and cofinal
in $(\prod\limits_{i<\kappa}\lambda_i,<_J)$? (Q3) are there somewhat
free black boxes?},
},
@incollection{MiSh:1009,
author = {Malliaris, Maryanthe and Shelah, Saharon},
trueauthor = {Malliaris, Maryanthe and Shelah, Saharon},
booktitle = {Logic without borders},
fromwhere = {1,IL},
note = { arxiv:math.LO/1208.5585 },
pages = {319--337},
publisher = {Berlin, Boston: De Gruyter},
series = {Ontos Mathematical Logic, vol. 5 Roman Kossak is organizing a
volume in honor of Jouko Vaananen's 60th birthday},
title = {{Saturating the random graph with an independent family of
small range}},
year = {2015},
abstract = {Motivated by Keisler's order, a far-reaching program of
understanding basic model-theoretic structure through the lens of
regular ultrapowers, we prove that for a class of regular filters
$\mathcal{D}$ on $I$, $|I| = \lambda > \aleph_0$, the fact that
${\mathcal{P}}(I)/ \mathcal{D}$ has little freedom (as measured by the
fact that any maximal antichain is of size $<\lambda$, or even
countable) does not prevent extending $\mathcal{D}$ to an ultrafilter
$\mathcal{D}1$ on $I$ which saturates ultrapowers of the random
graph. ``Saturates'' means that $M^I/\mathcal{D}_1$ is
$\lambda^+$-saturated whenever $M \models T_{\mathbf{rg}}$. This was
known to be true for stable theories, and false for non-simple and
non-low theories. This result and the techniques introduced in the
proof have catalyzed the authors' subsequent work on Keisler's
order for simple unstable theories. The introduction, which includes a
part written for model theorists and a part written for set theorists,
discusses our current program and related results.},
},
@article{ShUb:1010,
author = {Shelah, Saharon and Usuba, Toshimichi},
trueauthor = {Shelah, Saharon and Usuba, Toshimichi},
fromwhere = {IL, J},
journal = {Fundamenta Mathematicae},
title = {{$\omega_1$-Stationary preserving $\sigma$-Baire posets of
size $\aleph_1$ }},
volume = {submitted},
},
@article{KeShVa:1011,
author = {Kennedy, Juliette and Shelah, Saharon and Vaananen, Jouko},
trueauthor = {Kennedy, Juliette and Shelah, Saharon and
V{\"{a}}{\"{a}}n{\"{a}}nen},
fromwhere = {F,IL,SF},
journal = {Notre Dame Journal of Formal Logic},
note = { arxiv:math.LO/1307.6396 },
pages = {417--428},
title = {{Regular Ultrapowers at Regular Cardinals}},
volume = {56},
year = {2015},
abstract = {ADD},
},
@article{GaSh:1012,
author = {Garti, Shimon and Shelah, Saharon},
trueauthor = {Garti, Shimon and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Fundamenta Mathematicae},
pages = {14pps},
title = {{Open and solved problems concerning polarized partition
relations}},
volume = {234},
year = {2016},
abstract = {We list some open problems, concerning the polarized
partition relation. We solve a couple of them, by showing that for
every singular cardinal $\mu$ there exists a forcing notion
$\mathbb{P}$ such that the strong polarized relation ${\mu^+ \choose
\mu} \rightarrow {\mu^+ \choose \mu}^{1,1}_2$ holds in ${\rm\bf
V}^{\mathbb{P}}$.},
},
@article{GiSh:1013,
author = {Gitik, Moti and Shelah, Saharon},
trueauthor = {Gitik, Moti and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/1307.5977 },
pages = {855--865},
title = {{Applications of pcf for mild large cardinals to
elementary embeddings}},
volume = {164},
year = {2013},
abstract = { The following pcf results are proved: 1. Assume that
$\kappa>\aleph_0$ is a weakly compact cardinal. Let $\mu>2^\kappa$ be a
singular cardinal of cofinality $\kappa$. Then for every regular
$\lambda<{\rm pp}^+_{\Gamma(\kappa)}(\mu)$ there is an increasing
sequence $\langle \lambda_i \mid i<\kappa \rangle$ of regular
cardinals converging to $\mu$ such that $\lambda= {\rm
tcf}(\prod_{i<\kappa} \lambda_i, <_{J^{bd}_{\kappa}})$. 2. Let $\mu$ be
a strong limit cardinal and $\theta$ a cardinal above $\mu$. Suppose
that at least one of them has an uncountable cofinality. Then there
is $\sigma_*<\mu$ such that for every $\chi<\theta$ the following
holds: $$ \theta> {\rm sup}\{ {\rm sup} {\rm pcf}_{\sigma_{*}-{ \rm
complete}}(\frak{a}) \mid \frak{a}\subseteq {\rm Reg} \cap (\mu^+,\chi)
\text{ and } | \frak{a}|<\mu \}.$$ As an application we show that:
\endgraf if $\kappa$ is a measurable cardinal and $j:V \to M$ is
the elementary embedding by a $\kappa$--complete ultrafilter over
a measurable cardinal $\kappa$, then for every $\tau$ the
following holds: \begin{enumerate} \item if $j(\tau)$ is a cardinal
then $j(\tau)=\tau$; \item $|j(\tau)|=|j(j(\tau))|$; \item for any
$\kappa$--complete ultrafilter $W$ on
$\kappa$,\quad $|j(\tau)|=|j_W(\tau)|$. \end{enumerate} The first two
items provide affirmative answers to questions from \cite{G-Sh} and the
third to a question of D. Fremlin. },
},
@article{LcSh:1014,
author = {Luecke, Philipp and Shelah, Saharon},
trueauthor = {L{\"{u}}cke, Philipp and Shelah, Saharon},
fromwhere = {D, IL},
journal = {Forum of Mathematics, Sigma},
note = { arxiv:1211.6891 },
pages = {18 pps},
title = {{Free groups and automorphism groups of infinite structures}},
volume = {2},
year = {2014},
abstract = {Let $\lambda$ be a cardinal with
$\lambda=\lambda^{\aleph_0}$ and $p$ be either $0$ or a prime number.
We show that there are fields $K_0$ and $K_1$ of cardinality $\lambda$
and characteristic $p$ such that the automorphism group of $K_0$ is a
free group of cardinality $2^\lambda$ and the automorphism group of
$K_1$ is a free abelian group of cardinality $2^\lambda$. This
partially answers a question from \cite{MR1736959} and complements
results from \cite{MR1934424}, \cite{MR2773054} and \cite{MR1720580}.
The methods developed in the proof of the above statement also allow
us to show that the above cardinal arithmetic assumption is
consistently not necessary for the existence of such fields and that
it is necessary to use large cardinal assumptions to construct a model
of set theory containing a cardinal $\lambda$ of uncountable
cofinality with the property that no free group of cardinality greater
than $\lambda$ is isomorphic to the automorphism group of a field of
cardinality $\lambda$.},
},
@article{KsSh:1015,
author = {Koszmider, Piotr and Shelah, Saharon},
trueauthor = {Koszmider, Piotr and Shelah, Saharon},
fromwhere = {P, IL},
journal = {Algebra Universalis},
note = { arxiv:math.LO/1209.0177 },
pages = {305--312},
title = {{Independent families in Boolean algebras with some separation
properties}},
volume = {69},
year = {2013},
abstract = {We prove that any Boolean algebra with the subsequential
completeness property contains an independent family of size
continuum. This improves a result of Argyros from the 80ties
which asserted the existence of an uncountable independent family. In
fact we prove it for a bigger class of Boolean algebras satisfying
much weaker properties. It follows that the Stone spaces $K_{\mathcal
A}$ of all such Boolean algebras $\mathcal A$ contains a copy of
the \v Cech-Stone compactification of the integers $\beta\mathbb{N}$
and the Banach space $C(K_{\mathcal A})$ has $l_\infty$ as a quotient.
Connections with the Grothendieck property in Banach spaces are
discussed.},
},
@article{LwSh:1016,
author = {Laskowski, Michael C. and Shelah, Saharon},
trueauthor = {Laskowski, Michael C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/1211.0558 },
pages = {1--46},
title = {{Borel completeness of some $aleph_0$-stable theories}},
volume = {229},
year = {2015},
abstract = {We study $\aleph_0$-stable theories, and prove that if $T$
either has eni-DOP or is eni-deep, then its class of countable models
is Borel complete. We introduce the notion of $\lambda$-Borel
completeness and prove that such theories are $\lambda$-Borel
complete. Using this, we conclude that an $\aleph_0$-stable theory
satisfies $I_{\infty, \aleph_0}(T, \lambda) = 2^\lambda$ for all
cardinals $\lambda$ if and only if $T$ either has eni-DOP or is
eni-deep.},
},
@article{Sh:1017,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Geombinatorics},
pages = {108--126},
title = {{Ordered black boxes: existence}},
volume = {23},
year = {2014},
abstract = {We defined ordered black boxes in which for a partial $J$
we try to predict just a bound in $J$ to a function restricted to
$C_\alpha$. The existence results are closely related to pcf,
propagating downward. We can start with trivial cases.},
},
@article{Sh:1018,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
title = {{Compactness of chromatic number II}},
abstract = {At present true but not clear how more interesting than
[1006]; rethink about this; see F1296},
},
@article{Sh:1019,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:new },
title = {{Model theory for a compact cardinal}},
abstract = {We like to develop model theory for $T$, a complete theory
in $\mathbb{L}_{\theta,\theta}(\tau)$ when $\theta$ is a
compact cardinal. By \cite{Sh:300a} we have bare
bones stability and it seemed we can go no further. Dealing with
ultrapowers (and ultraproducts) we restrict ourselves to ``$D$ a
$\theta$-complete ultrafilter on $I$, probably $(I,\theta)$-regular''.
The basic theorems work, but can we generalize deeper parts of model
theory? In particular can we generalize stability enough to generalize
\cite[Ch.VI]{Sh:c}? We prove that at least we can characterize the
$T$'s which are minimal under Keisler's order, i.e. such that $\{D:D$
is a regular ultrafilter on $\lambda$ and $M \models T \Rightarrow
M^\lambda/D$ is $\lambda$-saturated$\}$. Further we succeed to connect
our investigation with the logic $\mathbb{L}^1_{< \theta}$ introduced
in \cite{Sh:797}: two models are $\mathbb{L}^1_{< \theta}$-equivalent
iff \, for some $\omega$- sequence of$\theta$-complete ultrafilters,
the iterated ultra-powers by it of those two models are
isomorphic. 2013.11.14 Doron will read it for a grade, have to finish
till 2013.12.31. Updates 2013.12.27 Have update, revise \S2 strongly
now have both versions of minimality- different characterization;},
},
@article{ShUs:1020,
author = {Shelah, Saharon and Usvyatsov, Alex},
trueauthor = {Shelah, Saharon and Usvyatsov, Alex},
fromwhere = {IL,IL},
journal = {Advances in Mathematics},
note = { arxiv:math.LO/1402.6513 },
title = {{Minimal types in the stable Banach spaces}},
volume = {submitted},
year = {2008-09-25},
abstract = {We prove existence of wide types in a continuous theory
expanding a Banach space, and density of minimal wide types among
stable types in such a theory. We show that every minimal wide stable
type is ``generically'' isometric to an $\ell_2$ space. We conclude
with a proof of the following formulation of Henson's Conjecture:
every model of an uncountably categorical theory expanding a Banach
space is prime over a spreading model, isometric to the standard basis
of a Hilbert space.},
},
@article{MdSh:1021,
author = {Mildenberger, Heike and Shelah, Saharon},
trueauthor = {Mildenberger, Heike and Shelah, Saharon},
fromwhere = {D,IL},
journal = {Journal of Symbolic Logic},
title = {{The cofinality of the symmetric group and the cofinality of
ultrapowers}},
volume = {submitted},
abstract = { We prove that ${\mathfrak{\lowercase{mcf}}} < {\rm cf}
({\rm{Sym}(\omega)})$ and ${\mathfrak{\lowercase{mcf}}} > {\rm
cf}({\rm{Sym}(\omega)})= {\mathfrak{\lowercase{b}}}$ are both
consistent relative to {\sf ZFC}. This answers a question
from \cite{BanakhRepovsZdomskyy} and from \cite{MdSh:967}.},
},
@article{RoSh:1022,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Colloquium Mathematicum},
note = { arxiv:1304.5683 },
pages = {211--225},
title = {{Around {\tt cofin}}},
volume = {134},
year = {2014},
abstract = {We show the consistency of ``there is a nice
$\sigma$--ideal $I$ on the reals with $add(I)=\omega_1$ which cannot be
represented as the union of a strictly increasing sequence of length
$\omega_1$ of $\sigma$-subideals''. This answers a question
by Borodulin--Nadzieja and G{\l}\c{a}b.},
},
@article{Sh:1023,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
note = { arxiv:math.LO/1308.2394 },
title = {{Indecomposable explicit abelian group, Withdrawn}},
abstract = {For every $\lambda$ we give an explicit construction of
an Abelian group with no non-trivial automorphisms. In particular the
group absolutely has no non-trivial automorphisms, hence is
absolutely indecomposable. In another direction the construction does
not use the axiom of choice. 2014.7.26 At present the proof fail, read
877!},
},
@article{KuKuSh:1024,
author = {Kuhlmann, Katarzyna and Kuhlmann, Franz-Viktor and Shelah,
Saharon},
trueauthor = {Kuhlmann, Katarzyna and Kuhlmann, Franz-Viktor and Shelah,
Saharon},
fromwhere = {C,C,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/1308.0780 },
pages = {261--290},
title = {{Symmetrically complete ordered sets, abelian groups and
fields}},
volume = {208},
year = {2015},
abstract = {We characterize and construct linealy ordered sets, abelian
groups and fields that are {\emph symmetrically complete},
meaning that the intersection over any chain of closed bounded
intervals is nonempty. Such ordered abelian groups and fields are
important because generalizations of Banach's Fixed Point Theorem can
be proved for them. We prove that symmetrically complete ordered
abelian groups and fields are divisible Hahn products and real closed
power series fields, respectively. We show how to extend any given
ordered set, abelian group or field to one that is symmetrically
complete. A main part of the paper establishes a detailed study of the
cofinalities in cuts.},
},
@article{JuSh:1025,
author = {Juhasz, Istvan and Shelah, Saharon},
trueauthor = {Juh\'asz, Istv\'an and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Proceedings of the AMS},
note = { arxiv:math.LO/1307.1989 },
pages = {2241--2247},
title = {{Strong colorings yield $\kappa$-bounded spaces with discretely
untouchable points}},
volume = {143},
year = {2015},
abstract = {It is well-known that every non-isolated point in a compact
Hausdorff space is the accumulation point of a discrete subset.
Answering a question raised by Z. Szentmikl\'ossy and the first
author. We show that this statement fails for countably compact
regular spaces, and even for $\omega$-bounded regular spaces. In
fact, there are $\kappa$-bounded counterexamples for every infinite
cardinal $\kappa$. The proof makes essential use of the so-called {\em
strong colorings} that were invented by the second author.},
},
@article{Sh:1026,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Acta Mathematica Hungarica},
note = { arxiv:new },
title = {{The spectrum of ultraproducts of finite cardinals for
ultrafilter}},
volume = {submitted},
abstract = {We complete the characterization of the possible spectrum of
regular ultrafilters $D$ on a set $I$, where the spectrum is the set
of infinite cardinals which are ultra-products of finite
cardinals modulo $D$},
},
@article{Sh:1027,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Acta Mathematica Hungarica},
note = { arxiv:new },
title = {{The coloring existence theorem revisited}},
volume = {submitted},
abstract = {We prove a better colouring theorem for $\aleph_3$},
},
@article{Sh:1028,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/1404.2775 },
title = {{Quite free complicated abelian group, pcf and BB}},
volume = {accepted},
},
@article{Sh:1029,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Forum Mathematicum},
note = { arxiv:math.LO/1311.4997 },
pages = {573--585},
title = {{No universal group in a cardinal}},
volume = {28},
year = {2016},
abstract = {For many classes of models there are universal members in
any cardinal $\lambda$ which ``essentially satisfied GCH'', i.e.
$\lambda = 2^{\le \lambda}$. But if the class is ``complicated
enough'', e.g. the class of linear orders, we know that if $\lambda$
is ``regular and not so close to satisfying GCH'' then there is no
universal member. Here we find new sufficient conditions (which we
call the olive property), not covered by earlier cases (i.e. fail the
so-called SOP$_4$). The advantage of those conditions is witnessed by
proving that the class of groups satisfies one of those conditions.},
},
@article{MiSh:1030,
author = {Malliaris, Maryanthe and Shelah, Saharon},
trueauthor = {Malliaris, Maryanthe and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Advances in Mathematics},
note = { arxiv:math.LO/1404.2919 },
pages = {614--681},
title = {{Existence of optimal ultrafilters and the fundamental
complexity of simple theories}},
volume = {290},
year = {2016},
abstract = {We characterize the simple theories in terms of saturation
of ultrapowers. This gives a dividing line at simplicity in Keisler's
order and gives a true outside definition of simple theories.
Specifically, for any $\lambda \geq \mu \geq \theta \geq \sigma$ such
that $\lambda = \mu^+$, $\mu = \mu^{<\theta}$ and $\sigma$ is
uncountable and compact (natural assumptions given our prior work,
which allow us to work directly with models), we define a family of
regular ultrafilters on $\lambda$ called \emph{optimal}, prove that
such ultrafilters exist and prove that for any $\mathcal{D}$ in this
family and any $M$ with countable signature, $M^\lambda/\mathcal{D}$
is $\lambda^+ $-saturated if $Th(M)$ is simple and
$M^\lambda/ \mathcal{D}$ is not $\lambda^+ $-saturated if $Th(M)$ is
not simple. The proof lays the groundwork for a stratification of
simple theories according to the inherent complexity of coloring, and
gives rise to a new division of the simple theories: $(\lambda, \mu,
\theta) $-explicitly simple.},
},
@article{FRSh:1031,
author = {Filipczak, Tomasz and Roslanowski, Andrzej and Shelah,
Saharon},
trueauthor = {Filipczak, Tomasz and Ros{\l}anowski, Andrzej and
Shelah, Saharon},
fromwhere = {1,IL},
journal = {Real Analysis Exchange},
note = { arxiv:1308.3749 },
pages = {129-140},
title = {{On Borel hull operations}},
volume = {40},
year = {2015},
abstract = {We show that if the meager ideal admits a monotone Borel
hull operation, then there is also a monotone Borel hull operation on
the $\sigma$--algebra of sets with the property of Baire.},
},
@article{MaShTs:1032,
author = {Machura, Michal and Shelah, Saharon and Tsaban, Boaz},
trueauthor = {Machura, Michal and Shelah, Saharon and Tsaban, Boaz},
fromwhere = {P,IL,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/1404.2239 },
pages = {15--40},
title = {{The linear refinement number and selection theory}},
volume = {234},
year = {2016},
abstract = {The linear refinement number $\mathfrak{lr}$ is a
combinatorial cardinal characteristic of the continuum. This number,
which is a relative of the pseudointersection number $\mathfrak{p}$,
showed up in studies of selective covering properties, that in turn
were motivated by the tower number $\mathfrak{t}$. \endgraf It was
long known that $\mathfrak{p}=\min\{\mathfrak{t},\mathfrak{lr}\}$
and that $\mathfrak{lr}\le\mathfrak{d}$. We prove that if
$\mathfrak{lr}=\mathfrak{d}$ in all models where the continuum is
$\aleph_2$, and that $\mathfrak{lr}$ is not provably equal to any
classic combinatorial cardinal characteristic of the
continuum. \endgraf These results answer several questions from the
theory of selection principles.},
},
@article{ChSh:1033,
author = {Cherlin, Gregory and Shelah, Saharon},
trueauthor = {Cherlin, Gregory and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Combinatorica},
note = { arxiv:math.LO/1404.5757 },
pages = {249--264},
title = {{Universal graphs with a forbidden subgraph: Block path
solidity}},
volume = {36},
year = {2016},
abstract = {Let $C$ be a finite connected graph for which there is a
countable universal $C$-free graph, and whose tree of blocks is a
path. Then the blocks of $C$ are complete. This generalizes a result of
F{\"{u}}redi and Komj\'ath, and fits naturally into a set of
conjectures regarding the existence of countable $C$-free graphs, with
$C$ an arbitrary finite connected graph.},
},
@article{ShVaVe:1034,
author = {Shelah, Saharon and Vaananen, Jouko and Velickovic, Boban},
trueauthor = {Shelah, Saharon and V{\"{a}}{\"{a}}n{\"{a}}nen, Jouko and
Veli\v{c}kovi\'c, Boban},
fromwhere = {IL,SF,},
journal = {Journal of Symbolic Logic},
pages = {285--300},
title = {{Positional Strategies in long Ehrenfeucht-Fra{\"{i}}ss\'e
games}},
volume = {80},
year = {2015},
abstract = {We prove that it is relatively consistent with ${\rm ZF +
CH}$ that there exist two models of cardinality $\aleph_2$ such that
the second player has a winning strategy in the
Ehrenfeucht-Fra{\"\i}ss\'e-game of length $\omega_1$ but there is no
$\sigma$-closed back-and-forth set for the two models. If ${\rm CH}$
fails, no such pairs of models exist.},
},
@article{CeSh:1035,
author = {Chernikov, Artem and Shelah, Saharon},
trueauthor = {Chernikov, Artem and Shelah, Saharon},
fromwhere = {R,IL},
journal = {Journal of the Institute of Mathematics of Jussieu},
note = { arxiv:math.LO/1308.3099 },
pages = {771--784},
title = {{On the number of Dedekind cuts and two-cardinal models of
dependent theories}},
volume = {15},
year = {2016},
abstract = {For an infinite cardinal $\kappa$, let ${\rm
ded}\kappa$ denote the supremum of the number of Dedekind cuts in
linear orders of size $\kappa$. It is known that $\kappa<{\rm
ded}\kappa \leq2^{\kappa}$ for all $\kappa$ and that ${\rm
ded}\kappa< 2^{\kappa}$ is consistent for any $\kappa$ of uncountable
cofinality. We prove however that $2^{\kappa}\leq{\rm ded}\left( {\rm
ded}\left({\rm ded}\left({\rm ded} \kappa\right)\right)\right)$ always
holds. Using this result we calculate the Hanf numbers for the
existence of two-cardinal models with arbitrarily large gaps and for
the existence of arbitrarily large models omitting a type in the class
of countable dependent first-order theories. Specifically, we show that
these bounds are as large as in the class of all countable theories.},
},
@article{Sh:1036,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
note = { arxiv:math.LO/1310.4042 },
title = {{Forcing axioms for $ \lambda $-complete $\mu ^+ $-c.c.}},
abstract = {We note that some form of the condition ``$p_1, p_2$ have a
$\leq_{\mathbb{Q}}$-lub in $\mathbb{Q}$'' is necessary in some forcing
axiom for $\lambda$-complete $\mu^+$-c.c. forcing notions. We also
show some versions are really stronger than others, a strong way to
answer Alexie's question of having $\mathbb{P}$ satisfying one
condition but no $\mathbb{P}'$ equivalent to $\mathbb{P}$ satisfying
another. We have not looked systematically whether any such question
(of interest) is open. [Ask Ashutosh to read in Aug 2014]; B. check
counterexmple in [SHSt:154, page 235], and one in Ap \S2 or 3 of
[SH:f]; check the paper with Otmar- have another variant},
},
@article{BaLaSh:1037,
author = {Baldwin, John and Laskowski, Chris and Shelah, Saharon},
trueauthor = {Baldwin, John and Laskowski, Chris and Shelah, Saharon},
fromwhere = {1,1,IL},
journal = {Journal of Symbolic Logic},
pages = {1142--1162},
title = {{Constructing many atomic models in $\aleph_1$}},
volume = {81},
year = {2016},
},
@article{ShSi:1038,
author = {Shelah, Saharon and Spinas, Otmar},
trueauthor = {Shelah, Saharon and Spinas, Otmar},
fromwhere = {IL,CH},
journal = {Journal of Symbolic Logic},
note = { arxiv:math.LO/1402.5616 },
pages = {901--916},
title = {{Mad spectra}},
volume = {80},
year = {2015},
abstract = {The mad spectrum is the set of all cardinalities of infinite
maximal almost disjoint families on $\omega$. We treat the problem to
characterize those sets $\mathcal A$ which, in some forcing extension
of the universe, can be the mad spectrum. We solve this problem to
some extent. What remains open is the possible values of min$({\mathcal
A})$ and max$({\mathcal A})$.},
},
@article{GnSh:1039,
author = {Greenberg, Noam and Shelah, Saharon},
trueauthor = {Greenberg, Noam and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/1309.3938 },
pages = {1557--1576},
title = {{Models of Cohen measurability}},
volume = {165},
year = {2014},
abstract = {We show that in contrast with the Cohen version of Solovay's
model, it is consistent for the continuum to be Cohen-measurable and
for every function to be continuous on a non-meagre set.},
},
@article{GaMaSh:1040,
author = {Garti, Shimon and Magidor, Menachem and Shelah, Saharon},
trueauthor = {Garti, Shimon and Magidor, Menachem and Shelah, Saharon},
fromwhere = {IL,IL,IL},
journal = {Notre Dame Journal of Formal Logic},
note = { arxiv:math.LO/1601.01409 },
title = {{On the spectrum of characters of ultrafilters}},
volume = {accepted},
abstract = {We show that the character spectrum ${\rm Sp}_\chi(\lambda)$
(for a singular cardinal $\lambda$ of countable cofinality) may
include any prescribed set of regular cardinals between $\lambda$ and
$2^\lambda$. \newline Nous prouvons que ${\rm Sp}_\chi(\lambda)$ (par
un cardinal singulier $\lambda$ avec cofinalit\`e nombrable) peut
comporter tout l'ensemble prescrit de cardinaux reguliers entre
$\lambda$ et $2^\lambda$.},
},
@article{BgSh:1041,
author = {Bagaria, Joan and Shelah, Saharon},
trueauthor = {Bagaria, Joan and Shelah, Saharon},
fromwhere = {SP,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:math.LO/1404.2776 },
pages = {181--197},
title = {{On partial orderings having precalibre-$\aleph_1$ and
fragments of Martin's axiom}},
volume = {232},
year = {2016},
abstract = {We define a countable antichain condition (ccc) property for
partial orderings, weaker than precalibre-$\aleph_1$, and show
that Martin's axiom restricted to the class of partial orderings that
have the property does not imply Martin's axiom for $\sigma$-linked
partial orderings. This yields a new solution to an old question of
the first author about the relative strength of Martin's axiom for
$\sigma$-centered partial orderings together with the assertion that
every Aronszajn tree is special. We also answer a question of J.
Steprans and S. Watson (1988) by showing that, by a forcing that
preserves cardinals, one can destroy the precalibre-$\aleph_1$
property of a partial ordering while preserving its ccc-ness.},
},
@article{FaSh:1042,
author = {Farah, Ilijas and Shelah, Saharon},
trueauthor = {Farah, Ilijas and Shelah, Saharon},
fromwhere = {3,IL},
journal = {Journal of Institute of Mathematics at Jussieu},
note = { arxiv:math.LO/1401.6689 },
pages = {1--28},
title = {{Rigidity of continuous quotients}},
volume = {15},
year = {2016},
abstract = {The assertion that the \v Cech--Stone remainder of the
half-line has only trivial automorphisms is independent from ZFC.
The consistency of this statement follows from Proper Forcing
Axiom and this is the first known example of a connected space with
this property. The existence of $2^{\aleph_1}$ autohomeomorphisms
under the Continuum Hypothesis follows from a general model-theoretic
fact. We introduce continuous fields of metric models and prove
countable saturation of the corresponding reduced products. We also
provide an (overdue) proof that the reduced products of metric models
corresponding to the Fr\'echet ideal are countably saturated. This
provides an explanation of why the asymptotic sequence C*-
algebras and the central sequence C*-algebras are as useful as
ultrapowers.},
},
@article{Sh:1043,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Mathematical Logic Quarterly},
note = { arxiv:math.LO/1412.0421 },
pages = {140--154},
title = {{Superstable theories and representation}},
volume = {62},
year = {2016},
abstract = {In this paper we give characterizations of the super-stable
theories, in terms of an external property called representation. In
the sense of the representation property, the mentioned class of
first-order theories can be regarded as ``not very complicated''},
},
@article{FiGoKrSh:1044,
author = {Fischer, Arthur and Goldstern, Martin and Kellner, Jakob and
Shelah, Saharon},
fromwhere = {CA,AT,AT,IL},
journal = {Archive for Mathematical Logic},
note = { arxiv:1402.0367 },
pages = {59 pps},
title = {{Creature forcing and five cardinal characteristics of the
continuum}},
volume = {on-line},
abstract = {We use a (countable support) creature construction to show
that consistently $\mathfrak{d} = \aleph_1 = cov(\mathcal{N}) < non
(\mathcal{M}) < non (\mathcal{N}) < cof(\mathcal{N}) < 2^{aleph_0}$.},
},
@article{Sh:1045,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
title = {{Quite free Abelian groups with prescribed endomorphism ring}},
abstract = {In \cite{Sh:1028} we like to build Abelian groups (or
$R$-modules) which on the one hand are quite free, say $\aleph_{\omega
+1}$-free, and on the other hand, are complicated in suitable sense.
We choose as our test problem having no non-trivial homomorphism
to $\mathbb Z$ (known classically for $\aleph_1$-free, recently
for $\aleph_n$-free). We get even $\aleph_{\omega_1 \cdot n}$-free.
Other applications were delayed to the present work. \endgraf The
construction (there and here) requires building $n$-dimensional black
boxes, which are quite free. Here we continue \cite{Sh:1028} (in some
ways). In particular, we consider building quite free Abelian groups
with a pre-assigned ring of endomorphism. \endgraf In \S2 we try to
elaborate \cite[\S(2B)]{Sh:1028}, on ``minimal'' ${\rm Hom}(G,{}_R R)$,
e.g. for separable $p$-groups. In \S(3C), \S(3D) are some old
continuation of \cite[\S3]{Sh:1028}, so of unclear status. In \S(4A) we
control ${\rm End}(G,R,+)$, e.g. $=R$. In \S(4B), done when we have a
1-witness; seems nearly done. In \S(4C) we intend to fill the 4-witness
case. \endgraf Presently (2014.1.13) it seems that \S(4A), \S(4B) do
something, construct from a 1-witness, which \S(4C) has to be
completed. Also \S(2A) do something, not clear how final},
},
@article{KmSh:1046,
author = {Kumar, Ashutosh and Shelah, Saharon},
trueauthor = {Kumar, Ashutosh and Shelah, Saharon},
fromwhere = {IN, IL},
journal = {Journal of Symbolic Logic},
title = {{RVM, RVC revisited: Clubs and Lusin sets}},
volume = {submitted},
abstract = {A cardinal $\kappa$ is Cohen measurable (RVC) if for
some $\kappa$-additive ideal $\cal{I}$ over $\kappa$,
$\cal{P}(\kappa) \slash \cal{I}$ is forcing isomorphic to adding
$\lambda$ Cohen reals for some $\lambda$. Such cardinals can be
obtained by starting with a measurable cardinal $\kappa$ and adding at
least $\kappa$ Cohen reals. We construct various models of RVC with
different properties than this model.Our main results are: (1) $\kappa
= 2^{\omega}$ is RVC does not decide $\club_S$ for various stationary
$S \subseteq \kappa$. (2) $\kappa \leq \lambda = cf(\lambda) <
2^{\omega}$ does not decide $\club_S$ for various stationary $S
\subseteq \lambda$. (3) $\kappa = 2^{\omega}$ is RVC does not decide
the existence of a Lusin set of size $\kappa$.},
},
@article{GaSh:1047,
author = {Garti, Shimon and Shelah, Saharon},
trueauthor = {Garti, Shimon and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Bulletin of the London Mathematical Society},
note = { arxiv:math.LO/1609.00242 },
title = {{How you are with $\mathfrak{s}$ and $\mathfrak{r}$}},
volume = {accepted},
abstract = {We prove that if $\mathfrak{r}=\mathfrak{c}$ then
$\mathfrak{s}\leq\cf(\mathfrak{c})$, and conclude that if
$\mathfrak{r}=\mathfrak{s}=\mathfrak{c}$ then $\mathfrak{s}$ is a
regular cardinal. Nous prouvons que
$\mathfrak{s}\leq\cf(\mathfrak{c})$ si $\mathfrak{r}=\mathfrak{c}$, et
conclurons que $\mathfrak{s}$ est un cardinal regulier si
$\mathfrak{r}=\mathfrak{s}= \mathfrak{c}$.},
},
@article{Sh:1048,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:new },
title = {{Hanf number for the strictly stable cases}},
volume = {submitted},
abstract = {Suppose $\bold {t} = (T, T_1, p)$ is a triple of two
theories in vocabularies $\tau \subset \tau_1$ of cardinality
$\lambda$ and a $\tau_1$-type $p$ over the empty set: here we fix $T$
and assume it is stable. We show the Hanf number for the property:
``there is a model $M_1$ of $T_1$ which omits $p$, but
$M_1\restriction \tau$ is saturated'' is larger than the Hanf number
of $L_{\lambda^+, \kappa}$ but smaller than the Hanf number of
$L_{(2^\lambda)^+, \kappa}$ when $T$ is stable with $\kappa =
\kappa(T)$.},
},
@article{HeSh:1049,
author = {Herden, Daniel and Shelah, Saharon},
trueauthor = {Herden, Daniel and Shelah, Saharon},
fromwhere = {G,IL},
journal = {preprint},
title = {{On group well represented as automorphic groups of groups}},
abstract = {Assuming (less than) $\mathbf V = \mathbf L$,
we characterize group GL such that there are arbitrarily large group
$H$ such that $\rm{aut}(H) \cong L$. In particular it suffies to have
one of cardinal $>|L|^{\aleph_0}$. In addition, if $|L| <
\aleph_\omega$ no need for $\mathbf V = \mathbf L$ (full). Similarly
for End$(G) \cong L$ so $L$ is semi-group (fill)},
},
@article{MiSh:1050,
author = {Malliaris, Maryanthe and Shelah, Saharon},
trueauthor = {Malliaris, Maryanthe and Shelah, Saharon},
fromwhere = {1, IL},
journal = {Israel Journal of Mathematics},
title = {{Keisler's order has infinitely many classes}},
volume = {accepted},
abstract = {We prove, in ZFC, that there is an infinite strictly
descending chain of classes of theories in Keisler's order. Thus
Keisler's order is infinite and not a well order. Moreover, this chain
occurs within the simple unstable theories, considered
model-theoretically tame. Keisler's order is a large scale
classification program in model theory, introduced in the 1960s,
which compares the complexity of theories. Prior to this paper, it
was thought to have finitely many classes, linearly ordered. The
model-theoretic complexity we find is witnessed by a very natural
class of theories, the $n$-free $k$-hypergraphs studied by
Hrushovski. Notably, this complexity reflects the difficulty of
amalgamation and appears orthogonal to forking.},
},
@article{MiSh:1051,
author = {Maryanthe Malliaris and Shelah, Saharon},
trueauthor = {Maryanthe Malliaris and Shelah, Saharon},
fromwhere = {1, IL},
journal = {Israel Journal of Mathematics},
pages = {67 pps},
title = {{Model-theoretic applications of cofinality spectrum
problems}},
volume = {appeared online},
abstract = {We apply the recently developed technology of cofinality
spectrum problems to prove a range of theorems in model theory. First,
we prove that any model of Peano arithmetic is $\lambda$-saturated iff
it has cofinality $\geq \lambda$ and the underlying order has no
$(\kappa, \kappa)$-gaps for regular $\kappa < \lambda$. We also answer
a question about balanced pairs of models of PA. Second, assuming
instances of GCH, we prove that $SOP_2$ characterizes maximality in
the interpretability order $\triangleleft^*$, settling a prior
conjecture and proving that $SOP_2$ is a real dividing line. Third,
we establish the beginnings of a structure theory for $NSOP_2$,
proving that $NSOP_2$ can be characterized by the existence of few
so-called higher formulas. In the course of the paper, we show that
$\mathfrak{p}_{\mathbf{s}} = \mathfrak{t}_{\mathbf{s}}$ in \emph{any}
weak cofinality spectrum problem closed under exponentiation
(naturally defined). We also prove that the local versions of these
cardinals need not coincide, even in cofinality spectrum problems
arising from Peano arithmetic. *2015.02.12 The new note from Feb on
the delayed lemma make no sense:- the finite satisfiability ensure the
formulas are compatible. Look back at the old.. The second theorem
which is delayed (union of few pairwise non-contradictory) - a debt,
look again},
},
@article{Sh:1052,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Proceedings of the American Mathematical Society},
note = { arxiv:math.LO1503.02423 },
pages = {5371--5383},
title = {{Lower bounds on coloring numbers from hardness hypotheses in
PCF theory}},
volume = {144},
year = {2016},
abstract = {We prove that the statement ``for every infinite cardinal
$\nu$, every graph with list chromatic $\nu$ has coloring number at
most $\beth_\om(\nu)$'' proved by Kojman \cite{koj} using the RGCH
theorem \cite{sh:460} implies the RGCG theorem via a short forcing
argument. By the same method, a better upper bound than
$\beth_\om(\nu)$ in this statement implies stronger forms of the
RGCH theorem whose consistency as well as the consistency of their
negations are wide open. Thus, the optimality of Kojman's upper
bound is a purely cardinal arithmetic problem, which, as discussed
below, may be quite hard to decide.},
},
@article{ShWo:1053,
author = {Shelah, Saharon and Wohofsky, Wolfgang},
trueauthor = {Shelah, Saharon and Wohofsky, Wolfgang},
fromwhere = {IL,AT},
journal = {Mathematical Logic Quarterly},
pages = {434--438},
title = {{There are no very meager sets in the model in which both the
Borel Conjecture and the dual Borel Conjecture are true}},
volume = {62},
year = {2016},
abstract = {We show that the model in~\cite{GoKrShWo:969} (for the
simultaneous consistency of the Borel Conjecture and the dual Borel
Conjecture) actually satisfies a stronger version of the dual Borel
Conjecture: there are no uncountable very meager sets.},
},
@article{KpSh:1054,
author = {Kaplan, Itay and Shelah, Saharon},
trueauthor = {Kaplan, Itay and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Mathematical Logic Quarterly},
pages = {530--546},
title = {{Forcing a countable structure to belong to the ground model}},
volume = {62},
year = {2016},
abstract = {Suppose that $P$ is a forcing notion, $L$ is a language (in
$\mathbb{V}$), $\dot{\tau}$ a $P$-name such that $P$
forces ``$\dot{\tau}$ is a countable $L$-structure''. In the product
$P\times P$, there are names $\dot{\tau_{1}},\dot{\tau_{2}}$ such
that for any generic filter $G=G_{1}\times G_{2}$ over $P\times P$,
$\dot{\tau}_{1}\left[G\right]= \dot{\tau}\left[G_{1}\right]$ and
$\dot{\tau}_{2}\left[G\right]=\dot{\tau}\left[G_{2}\right]$. Zapletal
asked whether or not $P\times P$ forces
$\dot{\tau}_{1} \cong\dot{\tau}_{2}$ implies that there is some
$M\in\mathbb{V}$ such that $P$ forces $\dot{\tau}\cong\check{M}$. We
answer this negatively and discuss related issues.},
},
@article{KpLaSh:1055,
author = {Kaplan, Itay and Lavi, Noa and Shelah, Saharon},
trueauthor = {Kaplan, Itay and Lavi, Noa and Shelah, Saharon},
fromwhere = {IL,IL,IL},
journal = {Israel Journal of Mathematics},
pages = {259--287},
title = {{The generic pair conjecture for dependent finite diagrams}},
volume = {212},
year = {2016},
abstract = {We consider a $(\bold D,\bar\kappa)$-homomorphism, $\bold
D$ a finite diagram. Another case is fixing a compact cardinal
$\theta$, we look at $(\bar\kappa,\mathcal{L})$-saturated monster
$\mathfrak{C}$. Our intention is to generalize \cite{Sh:900}.},
},
@article{BrLrSh:1056,
author = {Bartoszynski, Tomek and Larson, Paul and Shelah, Saharon},
trueauthor = {Bartoszy\'nski, Tomek and Larson, Paul and Shelah,
Saharon},
fromwhere = {1,1,IL},
journal = {Fundamenta Mathematicae},
pages = {101--125},
title = {{Closed sets which consistently have few translations covering
the line}},
volume = {237},
year = {2017},
abstract = {We characterize those compact subsets $K$ of $2^{\omega}$
for which one can force the existence of a set $X$ of cardinality
less than the continuum such that $K+X=2^{\omega}$.},
},
@article{DwSh:1057,
author = {Dow, Alan and Shelah, Saharon},
trueauthor = {Dow, Alan and Shelah, Saharon},
fromwhere = {1, IL},
journal = {preprint},
title = {{Asymmetric tie-points on almost clopen subsets of
$\mathbb{N}^*$}},
abstract = {A tie-point of compact space is analogous to a cut-point:
the complement of the point falls apart into two relatively clopen
non-compact subsets. We review some of the many consistency results
that have depended on the construction of tie-points of
$\mathbb{N}^*$. One especially important application, due to
Velickovic, was to the existencce of non-trivial involutions on
$\mathbb{N}^*$. A tie-point $\mathbb{N}^*$ has been called symmetric
if it is the unique fixed point of an involution. We define the
notion of almost clopen set to be the closure of one of the proper
relatively clopen subsets of the complement of a tie-point. We
explore asymmetries of almost clopen subsets of $\mathbb{N}^*$ in the
sense of how may an almost clopen set differ from its natural
complementary almost clopen set.},
},
@article{RaSh:1058,
author = {Raghavan, Dilip and Shelah, Saharon},
trueauthor = {Raghavan, Dilip and Shelah, Saharon},
fromwhere = {3,IL},
journal = {Transactions of the American Mathematical Society},
title = {{On embedding certain partial orders into the P-points under
RK and Tukey reducibility}},
volume = {appeared electronically},
year = {2017},
abstract = {The study of the global structure of ultrafilters on the
natural numbers with respect to the quasi-orders of Rudin-Keisler and
Rudin-Blass reducibility was initiated in the 1970s by Blass,
Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special
class of P-points under the quasi-ordering of Rudin-Keisler
reducibility. He asked what partially ordered sets can be embedded into
the P-points when the P-points are equipped with this ordering. This
question is of most interest under some hypothesis that guarantees the
existence of many P-points, such as Martin's axiom
for $\sigma$-centered posets. In his 1973 paper he showed under this
assumption that both ${\omega}_{1}$ and the reals can be
embedded. This result was later repeated for the coarser notion of
Tukey reducibility. We prove in this paper that Martin's axiom for
$\sigma$-centered posets implies that every partial order of size at
most continuum can be embedded into the P-points both under
Rudin-Keisler and Tukey reducibility.},
},
@article{HbSh:1059,
author = {Haber, Simi and Shelah, Saharon},
trueauthor = {Haber, Simi and Shelah, Saharon},
fromwhere = {IL, IL},
note = { arxiv:math.LO/1510.06581 },
pages = {226--236},
series = {Lecture Notes in Computer Science},
title = {{An extension of the Ehrenfeucht-Fra{\"{\i}}sse game for
\mbox{first order} logics augmented with Lindstr\"{o}m quantifiers}},
volume = {9300},
year = {2015},
abstract = {We propose an extension of the Ehrenfeucht-Fra\"{\i}sse game
able to deal with logics augmented with Lindstr\"{o}m quantifiers. We
describe three different games with varying balance between simplicity
and ease of use.},
},
@article{RaSh:1060,
author = {Raghavan, Dilip and Shelah, Saharon},
trueauthor = {Raghavan, Dilip and Shelah, Saharon},
fromwhere = {3,IL},
journal = {Fundamenta Mathematicae},
note = { arxiv:LO/1505.06296 },
title = {{Two inequalities between cardinal invariants}},
volume = {accepted},
abstract = {We prove two ZFC inequalities between cardinal invariants.
The first inequality involves cardinal invariants associated with an
analytic P-ideal, in particular the ideal of subsets of $\omega$ of
aymptotic density $0$. We obtain an upper bound on the $\ast$-covering
number, sometimes also called the weak covering number, of this ideal
by proving in Section 2 that cov${\ast} (\mathcal{Z}_0) \leq
\mathfrak{d}$. \endgraf In Section 3 we investigate the relationship
between the bounding and splitting numbers at regular uncountable
cardinals. We prove in sharp contrast to the case when $\kappa =
\omega$, that if $\kappa$ is any regular uncountable cardinal, then
$\mathfrak{s}_\kappa \leq \mathfrak{b}_\kappa$.},
},
@article{Sh:1061,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
pages = {293--296},
series = {Lecture Notes in Computer Science},
title = {{On failure of 0-1 laws}},
volume = {9300},
year = {2015},
abstract = {Let $\alpha \in (0,1)_{\mathbb{R}}$ be irrational and $G_n
= G_{n,1/n^\alpha}$ be the random graph with edge
probability $1/n^\alpha$; we know that it satisfies the 0-1 law for
first order logic. We deal with the failure of the 0-1 law for
stronger logics: $\mathbb{L}_{\infty,k},k$ large enough and the
inductive logic.},
},
@article{Sh:1062,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
note = { arxiv:math.LO/1706.01226 },
title = {{Failure of 0-1 law for sparse random graph in strong logics}},
volume = {Beyond First Order Model Theory},
abstract = {Let $\alpha \in (0,1)_{\mathbb{R}}$ be irrational and $G_n
= G_{n,1/n^\alpha}$ be the random graph with edge
probability $1/n^\alpha$; we know that it satisfies the 0-1 law for
first order logic. We deal with the failure of the 0-1 law for
stronger logics: $\mathbb{L}_{\infty,k},k$ large enough and the
inductive logic.},
},
@article{KmSh:1063,
author = {Kumar, Ashutosh and Shelah, Saharon},
trueauthor = {Kumar, Ashutosh and Shelah, Saharon},
fromwhere = {IN, IL},
journal = {Mathematical Logic Quarterly},
title = {{Clubs on quasi measurable cardinals}},
volume = {accepted},
abstract = {We construct a model satisfying $\kappa < 2^{\aleph_0} +
\club_{\kappa} +\kappa$ is quasi measurable. Here, we call $\kappa$
quasi measurable if there is an $\aleph_1$-saturated
$\kappa$-additive ideal $\cal{I}$ over $\kappa$. We also show that, in
this model, forcing with $\cal{P}(\kappa) \slash \cal{I}$ adds one but
not $\kappa$ Cohen reals. We introduce a weak club principle and use
it to show that, consistently, for some $\aleph_{\cal{1}}$-saturated
$\kappa$-additive ideal $\cal{I}$ over $\kappa$, forcing with
$P(\kappa) \slash I$ adds one but not $\kappa$ random reals.},
},
@article{Sh:1064,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
note = { arxiv:math.LO/1601.04824 },
title = {{Atomic saturation of reduced powers}},
abstract = {2014.1.24 We continue [1019] looking at infinitary logics
but now for filters instead ultrafilters. Earlier this month has
written \S1 which generalize [Sh:c,VI,2.6], so do not depend on $
\theta $ being a compact cardinal, (this apply also to $ \theta =
{\aleph_0} $ but this is marginal here) In \S2 we deal with how to
generalize ``good ultrafilter'' Intentions: (A) we hope to sort out
the versions of cp implicit in \S2. (B) Even assume T has $\theta
$-cp, can we sort out the maximal $T$? (C) Recall the problem of
building a good filter of $ \lambda $ such that the quotient is a
complete BA free enough failing chain condition 2014.1.26 [here or
F1396] Noted (Friday) that if a model is an ultrapower by a normal
ultrafilter on $ \theta $, then looking at all sequences of $ \theta $
formulas, we can colour by $ 2^ \theta $ colours, such that the colour
of a branch determine if the type is realized. THIS is one side on the
other SIDE we start with a strong limit $ \mu $ of cofinality $ \theta
$ and a model of cardinality $ \mu $ which embed much, then build a
directed system of sub-models, indexed by subsets of $ \lambda $ each
of cardinality at most $ \theta $ (<)-directed, bounded below all
members of a given stationary subset of $ \lambda $ of cofinality $
\theta $. Sort out. 2015-07-24 Add f1433 sec 1,2?},
},
@article{Sh:1065,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
title = {{On many simple $T$'s for $SP^{(*)}$}},
abstract = {We return to the problem of classifying simple $T$'s
by ${\rm SP}_T(\lambda,\mu)$ and ${\rm SP}^2_T(\lambda,\mu)$ (or
with $\theta$ - not central here). Presently (2014.1.30) it seems that
we can prove: that at least consistently there are many classes of
such (simple) $T$'s. See also \cite{Sh:F1370}. (2015.5.03) We have
also tried in some instances to use freeness as in \cite{KoSh:645},
but it seems now as in \S2 to just force for some $T$, starting
with providing examples for some clear $T$'s and then iterating giving
a ``solution'' for every ``problem'', i.e. ${mathscr M},|M| = \lambda$
for some other family of $T$'s. We may fix $R$ but vary the $S^N$'s
so the $n_{\bold p}$'s.},
},
@article{GoMeSh:1066,
author = {Goldstern, Martin and Mejia, Diego and Shelah, Saharon},
trueauthor = {Goldstern, Martin and Mej\'{\i}a, Diego A. and Shelah,
Saharon},
fromwhere = {AT,AT,IL},
journal = {Proceedings of the AMS},
note = { arxiv:math.LO/1504.04192 },
pages = {4025--4042},
title = {{The left side of Cicho\'n's Diagram}},
volume = {144},
year = {2016},
abstract = {Using a finite support iteration of ccc forcings, we
construct a model of $\aleph_1<\mbox{\rm
add}({\mathcal{N}})< \mbox{\rm cov}
({\mathcal{N}})<\mathfrak{b}<\mbox{\rm non} (\mathcal{M})<\mbox{\rm
cov}(\mathcal{M}) =\mathfrak{c}$.},
},
@article{HwSh:1067,
author = {Horowitz, Haim and Shelah, Saharon},
trueauthor = {Horowitz, Haim and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {preprint},
title = {{Saccharinity with ccc}},
abstract = {Consistency of saccharinity for a nice ccc ideal but only
for ZF, using inverse limit of finite iterations},
},
@article{KmSh:1068,
author = {Kumar, Ashutosh and Shelah, Saharon},
trueauthor = {Kumar, Ashutosh and Shelah, Saharon},
fromwhere = {IN,IL},
journal = {Advances in Mathematics},
title = {{A transversal of full outer measure}},
volume = {submitted},
abstract = {We prove that for any set of real and equivalence relations
on it such that every equivalence class is countable, there is a
transversal (= a set of representations of the equivalence relations,
i.e. having exactly one member in each equivalence class) with the same
outer measure.},
},
@article{MiSh:1069,
author = {Malliaris, Maryanthe and Shelah, Saharon},
trueauthor = {Malliaris, Maryanthe and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Contemporary Mathematics},
pages = {145--159},
title = {{Open problems on ultrafilters and some connections to the
continuum}},
volume = {690},
year = {2017},
abstract = {We discuss a range of open problems at the intersection of
set theory, model theory, and general topology, mainly around the
construction of ultrafilters. Along the way we prove uniqueness for a
weak notion of cut. },
},
@article{MiSh:1070,
author = {Malliaris, Maryanthe and Shelah, Saharon},
trueauthor = {Malliaris, Maryanthe and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Topology and its Applications},
pages = {50--79},
title = {{Cofinality spectrum problems: the axiomatic approach}},
volume = {213},
year = {2016},
abstract = {Let $X$ be a set of definable linear or partial orders in
some model. We say that $X$ has \emph{uniqueness} below some cardinal
$\mathfrak{t}_*$ if for any regular $\kappa < \mathfrak{t}_*$, any
two increasing $\kappa$-indexed sequences in any two orders of $X$
have the same co-initiality. Motivated by recent work, we investigate
this phenomenon from several interrelated points of view. We define
the lower-cofinality spectrum for a regular ultrafilter $\mathcal{D}$
on $\lambda$ and show that this spectrum may consist of a strict
initial segment of cardinals below $\lambda$ and also that it may
finitely alternate. We connect these investigations to a question of
Dow on the conjecturally nonempty (in ZFC) region of OK but not good
ultrafilters, by introducing the study of so-called `automorphic
ultrafilters' and proving that the ultrafilters which are automorphic
for some, equivalently every, unstable theory are precisely the good
ultrafilters. Finally, we axiomatize a general context of ``lower
cofinality spectrum problems'', a bare-bones framework consisting
essentially of a single tree projecting onto two linear orders. We
prove existence of a lower cofinality function in this context show
that this framework holds of theories which are substantially less
complicated than Peano arithmetic, the natural home of cofinality
spectrum problems. Along the way we give new analogues of several open
problems.},
},
@article{ShSr:1071,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
fromwhere = {IL, 3},
journal = {Fundamenta Mathematicae},
pages = {167--181},
title = {{When automorphisms of
$\mathcal{P}(\kappa)/[\kappa]^{<\aleph_0}$ are trivial off a small
set}},
volume = {235(2)},
year = {2016},
abstract = {It is shown that if $\kappa > 2^{\aleph_0}$ and $\kappa$ is
less than the first inaccessible cardinal then every automorphism of
$\mathcal{P} (\kappa)/[\kappa]^{<\aleph_0}$ is trivial outside of a
set of cardinality $2^{\aleph_0}$.},
},
@article{MhSh:1072,
author = {Mohsenipour, Shahram and Shelah, Saharon},
trueauthor = {Mohsenipour, Shahram and Shelah, Saharon},
fromwhere = {IR,IL},
journal = {Notre Dame Journal of Formal Logic},
note = { arxiv:math.LO/1510.02216 },
title = {{Set mappings on 4-tuples}},
volume = {accepted},
abstract = {In this paper we study set mappings on 4-tuples. We continue
a previous work of Komjath and Shelah by getting new finite bounds on
the size of free sets. This is obtained by an entirely different
forcing construction. Moreover we prove two ZFC results for set
mappings on 4-tuples and also as another application of our forcing
construction we give a consistency result for set mappings on
triples.},
},
@article{LrSh:1073,
author = {Larson, Paul and Shelah, Saharon},
trueauthor = {Larson, Paul and Shelah, Saharon},
fromwhere = {1,IL},
journal = {preprint},
title = {{On the absoluteness of orbital $\omega$-stability}},
abstract = {We show that orbital $\omega$-stability is upwards absolute
(but not downwards absolute) for $\aleph_0$-presented Abstract
Elementary Classes satisfying amalgamation and the joint embedding
property (each for countable models). We also show that amalgamation
does not imply upwards absoluteness of orbital $\omega$-stability by
itself. The corresponding question for the joint embedding property
remains open.},
},
@article{KpShSi:1074,
author = {Kaplan, Itay and Shelah, Saharon and Simon, Pierre},
trueauthor = {Kaplan, Itay and Shelah, Saharon and Simon, Pierre},
fromwhere = {IL,IL,F},
journal = {Journal of Mathematical Logic},
pages = {18 pps},
title = {{Exact saturation in simple and NIP theories}},
volume = {17},
year = {2017},
abstract = {A theory $T$ is said to have exact saturation at a singular
cardinal $\kappa$ if it has a $\kappa$-saturated model which is not
$\kappa^+$-saturated. We show, under some set-theoretic
assumptions, that any simple theory has exact saturation. Also, an NIP
theory has exact saturation if and only if it is not distal. This
gives a new characterization of distality.},
},
@article{GsSh:1075,
author = {Golshani, Mohammad and Shelah, Saharon},
trueauthor = {Golshani, Mohammad and Shelah, Saharon},
fromwhere = {IR, IL},
journal = {Journal of Mathematical Logic},
note = { arxiv:math.LO/1510.06278 },
pages = {34 pps},
title = {{On cuts in ultraproducts of linear orders I}},
volume = {16},
year = {2016},
abstract = {For an ultrafilter $D$ on cardinal $\kappa$, we wonder for
which pair $(\theta_1, \theta_2)$ of regular cardinals, we have: for
any $(\theta_1 + \theta_2)^+$-saturated dense linear order
$J,J^\kappa/D$ has a cut of cofinality $(\theta_1, \theta_2)$. We deal
mainly with the case $\theta_1, \theta_2 > 2^\kappa$.},
},
@article{LrSh:1076,
author = {Larson, Paul and Shelah, Saharon},
trueauthor = {Larson, Paul and Shelah, Saharon},
fromwhere = {1, IL},
journal = {Mathematical Logic Quarterly},
title = {{Coding with canonical functions}},
volume = {submitted},
abstract = {A function $f$ from $\omega_{1}$ to the ordinals is called
a canonical function for an ordinal $\alpha$ if $f$ represents $\alpha$
in any generic ultrapower induced by forcing with
$\mathcal{P}(\omega_{1})/\mathrm{NS}_{\omega_{1}}$. We introduce
here a method for coding sets of ordinals using canonical functions
from $\omega_{1}$ to $\omega_{1}$. Combining this approach with
arguments from \cite{Sh:f}, we show that for each cardinal $\kappa$
there is a forcing construction preserving cardinalities and
cofinalities forcing that every subset of $\kappa$ is in the inner
model $L(\mathcal{P}(\omega_{1}))$. },
},
@article{Sh:1077,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
note = { arxiv:math.LO/1511.05383 },
title = {{Random graph: stronger logic but with the zero one law}},
abstract = {FILL},
},
@article{KmSh:1078,
author = {Kumar, Ashutosh and Shelah, Saharon},
trueauthor = {Kumar, Ashutosh and Shelah, Saharon},
fromwhere = {IN, IL},
journal = {Fundamenta Mathematicae},
title = {{On a question about families of entire functions}},
volume = {accepted},
abstract = {We show that the existence of a continuum sized family
$mathcal{F}$ of entire functions such that for each complex number
$z$, the set $\{ f(z): f \in \mathcal{F}\}$ has size less than
continuum is undecidable in ZFC plus the negation of CH},
},
@article{KmSh:1079,
author = {Kumar, Ashutosh and Shelah, Saharon},
trueauthor = {Kumar, Ashutosh and Shelah, Saharon},
fromwhere = {IN,IL},
journal = {Fundamenta Mathematicae},
pages = {263--267},
title = {{Avoiding equal distances}},
volume = {236},
year = {2017},
abstract = {We show that it is consistent that there is a non meager set
of reals $X$ each of whose non meager subsets contains equal
distances.},
},
@article{KoSh:1080,
author = {Komjath, Peter and Shelah, Saharon},
trueauthor = {Komj\'{a}th, Peter and Shelah, Saharon},
fromwhere = {H,IL},
journal = {Israel Journal of Mathematics},
title = {{Consistently $\mathcal P (\omega_1)$ is the union of less
than $2^{\aleph_1}$ strongly independent families}},
volume = {submitted},
abstract = {It is consistent that $\mathcal{P}(\o_1)$ is the union of
less than $2^{\aleph_1}$ parts such that
if $A_0,\dots,A_{n-1},B_0,\dots,B_{m-1}$ are distinct elements of
the same part then $|A_0\cap\cdots \cap A_{n-1}\cap
(\o_1-B_0)\cap\cdots \cap(\o_1-B_{m-1})|=\aleph_1$.},
},
@article{RoSh:1081,
author = {Roslanowski, Andrzej and Shelah, Saharon},
trueauthor = {Ros{\l}anowski, Andrzej and Shelah, Saharon},
fromwhere = {1,IL},
journal = {Mathematica Slovaca},
title = {{Small--large subgroups of the reals}},
volume = {accepted},
abstract = {We are interested in subgroups of the reals that are small
in one and large in another sense. We prove that, in ZFC, there exists
a non--meager Lebesgue null subgrooup of ${\mathbb R}$, while it is
consistent there there is no non--null meager subgroup of ${\mathbb
R}$.},
},
@article{KpSh:1082,
author = {Kaplan, Itay and Shelah, Saharon},
trueauthor = {Kaplan, Itay and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Journal of Symbolic Logic},
title = {{Decidability and classification of the theory of integers
with primes}},
volume = {accepted},
abstract = {We show that under Dickson's conjecture about the
distribution of primes in the natural numbers, the theory
$Th\left(\mathbb{Z},+,1,0, Pr\right)$ where $Pr$ is a predicate for the
prime numbers and their negations is decidable, unstable and
supersimple. This is in contrast with
$Th\left(\mathbb{Z},+,0,Pr,<\right)$ which is known to be undecidable
by the works of Jockusch, Bateman and Woods.},
},
@article{GaHaSh:1083,
author = {Garti, Shimon and Hayut, Yair and Shelah, Saharon},
trueauthor = {Garti, Shimon and Hayut, Yair and Shelah, Saharon},
fromwhere = {IL,IL,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/1601.07745 },
pages = {89--102},
title = {{On the verge of inconsistency: Magidor Cardinals and Magidor
Filters}},
volume = {220},
year = {2017},
abstract = {We introduce a model-theoretic characterization of Magidor
cardinals, from which we infer that Magidor filters are beyond
ZFC-inconsistency},
},
@article{BarSh:1084,
author = {Barnea, Ilan and Shelah, Saharon},
trueauthor = {Barnea, Ilan and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
title = {{The abelianization of inverse limits of groups}},
volume = {submitted; 7 Aug. 2017 they sent ref. report},
},
@article{CnSh:1085,
author = {Cohen, Shani and Shelah, Saharon},
trueauthor = {Cohen, Shani and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {Israel Journal of Mathematics},
note = { arxiv:math.LO/1603.08362 },
title = {{Generalizing random real forcing for inaccessible cardinals}},
volume = {submitted},
abstract = {The two parallel concepts of ``small'' sets of the real line
are meagre sets and null sets. Those are equivalent to Cohen forcing
and Random real forcing for $\aleph^{\aleph_0}_0$; in spite of this
similarity, the Cohen forcing and Random Real forcing have very
different shapes. One of these differences is in the fact that the
Cohen forcing has an easy natural generalization for
$\lambda^{\lambda}$ while $\lambda > \aleph_0$, corresponding to an
extension for the meagre sets, while the Random real forcing didn't
see to have a natural generalization, as Lebesgue measure doesn't
have a generalization for space $\lambda^\lambda$ while $\lambda >
\aleph_0$. In work [1], Shelah found a forcing resembling the
properties of Random Real Forcing for $\lambda^\lambda$ while
$\lambda$ is a weakly compact cardinal. Here we describe, with
additional assumptions, such a forcing for $\lambda^\lambda$ while
$\lambda$ is an inaccessible cardinal; this forcing preserves cardinals
and cofinalities, however unlike Cohen forcing, does not add
dominating reals.},
},
@article{KoShSw:1086,
author = {Koszmider, Piotr and Shelah, Saharon and Swietek, Michal},
trueauthor = {Koszmider, Piotr and Shelah, Saharon and
\'Swi\c{e}tek, Micha{\l}},
fromwhere = {P,IL,P},
journal = {Advances in Mathematics},
title = {{There is no bound on sizes of indecomposable Banach Spaces}},
volume = {submitted},
abstract = {Assuming the generalized continuum hypothesis we construct
arbitrarily big indecomposable Banach spaces. i.e., such that
whenever they are decomposed as $X\oplus Y$, then one of the closed
subspaces $X$ or $Y$ must be finite dimensional. It requires
alternative techniques compared to those which were initiated by
Gowers and Maurey or Argyros with the coauthors. This is because
hereditarily indecompo- sable Banach spaces always embed into
$\ell_\infty$ and so their density and cardinality is bounded by
the continuum and because dual Banach spaces of densities bigger than
continuum are decomposable by a result due to Heinrich and
Mankiewicz. The obtained Banach spaces are of the form $C(K)$ for some
compact connected Hausdorff space and have few operators in the sense
that every linear bounded operator $T$ on $C(K)$ for every $f\in C(K)$
satisfies $T(f)=gf+S(f)$ where $g\in C(K)$ and $S$ is weakly compact
or equivalently strictly singular. In particular, the spaces carry
the structure of a Banach algebra and in the complex case even the
structure of a $C^*$-algebra.},
},
@article{GsSh:1087,
author = {Golshani, Mohammad and Shelah, Saharon},
trueauthor = {Golshani, Mohammad and Shelah, Saharon},
fromwhere = {IR,IL},
journal = {preprint},
note = { arxiv:math.LO/1604.06044 },
title = {{On cuts in ultraproducts on linear orders II}},
abstract = {We continue our study of the class $\mathcal{C}(D)$, where
$D$ is a uniform ultrafilter on a cardinal $\kappa$ and
$\mathcal{C}(D)$ is the class of all pairs $(\theta_1, \theta_2)$,
where $(\theta_1, \theta_2)$ is the cofinality of a cut in
$J^\kappa/D$ and $J$ is some $(\theta_1 + \theta_2)^+$-saturated dense
linear order. We show that if $(\theta_1, \theta_2) \in \mathcal{C}
(D)$ and $D$ is $\aleph_1$-complete or $\theta_1 + \theta_2 >
2^\kappa$, then $\theta_1 = \theta_2$.},
},
@article{ShSr:1088,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
fromwhere = {IL,3},
journal = {Annals of Pure and Applied Logic},
title = {{Universal graphs and functions on $\omega_1$}},
volume = {submitted},
abstract = {It is shown to be consistent with various values of
$mathfrak{b}$ and $\mathfrak{d}$ that there is a universal graph on
$\omega_1$. Moreover, it is also shown that it is consistent that
there is a ' universal graph on $\omega_1$ - in other words, a
universal symmetric function from $\omega^2_1$ to $2$ -- but no such
function from $\omega^2_1$ to $\omega$. The method used relies on
iterating well know reals, such as Miller and Laver reals, and
alternating this with the PID forcing which adds no new reals.},
},
@article{HwSh:1089,
author = {Horowitz, Haim and Shelah, Saharon},
trueauthor = {Horowitz, Haim and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {preprint},
title = {{A Borel maximal eventually different family}},
},
@article{HwSh:1090,
author = {Horowitz, Haim and Shelah, Saharon},
trueauthor = {Horowitz, Haim and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {preprint},
title = {{Can you take Tornquist's inaccessible away?}},
},
@article{DgHeSh:1091,
author = {Dugas, Manfred and Herden, Daniel and Shelah, Saharon},
trueauthor = {Dugas, Manfred and Herden, Daniel and Shelah, Saharon},
fromwhere = {1,1,IL},
journal = {''Groups and Model Theory''},
title = {{An extension of M.C.R. Butler's theorem on endomorphism
rings}},
},
@article{BlSh:1092,
author = {Baldwin, John T. and Shelah, Saharon},
trueauthor = {Baldwin, John T. and Shelah, Saharon},
fromwhere = {1, IL},
journal = {Archive for Mathematical Logic},
title = {{Hanf numbers for extendibility and related phenomena}},
volume = {submitted},
abstract = {In this paper we discuss two theorems whose proofs depend on
extensions of the Fraiss\'e method. We prove the Hanf number for the
existence of an extendible model (has a proper extension in the class.
Here, this means an $\infty,\omega$-elementary extension) of a
(complete) sentence of $L_{\omega_1, \omega}$ is (modulo some mild set
theoretic hypotheses that we expect to remove in a later paper) the
first measurable cardinal. And we outline the description on an
explicit $L_{\omega_1, \omega}$-sentence $\phi_n$ characterizing
$\aleph_n$ for each $n$. We provide some context for these
developments as outlined in the lectures at IPM.},
},
@article{HwSh:1093,
author = {Horowitz, Haim and Shelah, Saharon},
trueauthor = {Horowitz, Haim and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {preprint},
title = {{Maximal independent sets in Borel graphs and large
cardinals}},
abstract = {We construct a Borel graph $G$ such that $ZF + DC + $
``There are no maximal independent sets in $G$'' is equiconsistent
with $ZFC + $ ``There exists an inaccessible cardinal''},
},
@article{HwSh:1094,
author = {Horowitz, Haim and Shelah, Saharon},
trueauthor = {Horowitz, Haim and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {preprint},
title = {{Solovay's inaccessible over a weak set theory without
choice}},
abstract = {We study the consistency strength of Lebesgue measurability
for $\Sigma^1_3$ sets over a weak set theory in a completely
choiceless context. We establish a result analogous to the
Solovay-Shelah theorem},
},
@article{HwSh:1095,
author = {Horowitz, Haim and Shelah, Saharon},
trueauthor = {Horowitz, Haim and Shelah, Saharon},
fromwhere = {IL, IL},
journal = {preprint},
title = {{A Borel maximal cofinitary group}},
abstract = {We construct a Borel maximal cofinitary group},
},
@article{Sh:1096,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
title = {{Strong failure of 0-1 law for LFP and the path logics}},
abstract = {we continue [1062], in two issues. First we introduce a
logic, called path logic which seem natural there and try to prove for
it the 0-1 law Second, for the random graph $ G_{n, n^ \alpha} $ in
[1062] we get only a weak failure of the 01 law, the truth value of
the sentence is sometimes close to 1 and sometimes close to 0, but it
change slowly. Here we intend to prove that tehere is an interpretaion
which for a random enough graph give number theory on $ n $ , so we
can ``define n being odd/even''},
},
@article{HwSh:1097,
author = {Horowitz, Haim and Shelah, Saharon},
trueauthor = {Horowitz, Haim and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {preprint},
title = {{On the classification of definable ccc forcing notions}},
abstract = {We show that for a Suslin ccc forcing notion $\mathbb
Q$ adding a Hechler real, $ZF+DC_{\omega_1}+''$All sets of reals
are $I_{\mathbb Q,\aleph_0}$-measurable$''$ implies the existence of an
inner model with a measurable cardinal. We also further investigate the
forcing notions from {[}HwSh:1067{]}, showing that some of them add
Hechler reals (so the above result applies to them) while others don't
add dominating reals.},
},
@article{Sh:1098,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
title = {{LF groups, aec amalgamation, few automorphisms}},
abstract = {In \S1 we deal with amalgamation bases, e.g. we define when
an a.e.c. $\mathfrak k$ has $(\lambda,\kappa)$-amalgamation which means
``many'' $M \in K^{\mathfrak k}_\lambda$ are amalgamation bases. We
then consider what occurs for the class of lf groups. In \S2 we deal
with weak definability of $a \in N \backslash M$ over $M$, for $\bold
K_{\rm{exlf}}$. In \S3 we deal with indecomposable members of $K_{\rm
exlf}$ and with the existence of a universal members of $K^{\mathfrak
k}_\mu$, for $\mu$ strong limit of cofinality $\aleph_0$. Most note
worthy: if $\bold K_{\rm lf}$ has a universal model in $\mu$ then
it has a canonical one similar to the special models, (the parallel
to saturated ones in their cardinality). In \S4 we prove ``every $G
\in \bold K^{\rm lf}_{\le \mu}$ can be extended to a complete
$(\lambda,\theta)$-full $G$'' for many cardinals; we may consider
fixing the outer auto group. In \S(0C) we solve a problem of
Heckin on the automorphism group of the Hall group.},
},
@article{LwSh:1099,
author = {Laskowski, Michael C. and Shelah, Saharon},
trueauthor = {Laskowski, Michael C. and Shelah, Saharon},
fromwhere = {1,IL},
journal = {preprint},
title = {{A strong failure of $\aleph_0$ stability for atomic classes}},
},
@article{Sh:1100,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {preprint},
title = {{Creature iteration for inaccessibles}},
abstract = {Our starting point is \cite{Sh:1004}. As there we
concentrate on forcing for inaccessibles and our definition is by
induction when we like to get a nice forcing. (A) Mainly we deal with
iterations for $\lambda$ inaccessible of creature forcing (so getting
appropriate forcing axioms). We concentrate on the case the forcing is
strategically $(< \lambda)$-complete $\lambda^+$-c.c. (even
$\lambda$-centered) and mainly (i.e. in (A)) on cases leading to
$\lambda$-bounding forcing. In this case we can start with $2^\lambda
> \lambda^+$ and the forcing preserves various statements. We allow
$\bold U_{\mathfrak x}$, e.g. $= \lambda^+$ to deal, e.g. with the big
at universal graphs in $\lambda^+ < 2^\lambda$, while ${\mathfrak
d\/}_\lambda = \lambda^+$. The decision of ``weakly compact''
\underline{or} demand? is done via the choice of $\bold j$. (B) A
different case is in the same framework but naturally assuming
$2^\lambda = \lambda^+$. The forcing satisfies only
the $\lambda^{++}$-c.c. and is $\kappa$-proper so do not collapse
$\lambda^+$. We may make $2^\lambda$ arbitrarily large \underline{or}
weaken the demands on forcing and get $2^\lambda = \lambda^{++}$. '
We only later do something concerning this. (C) Changing the
frame somewhat, we allow adding unbounded $\lambda$-reals (i.e. $\eta
\in {}^\lambda \lambda$) without adding $\lambda$-Cohens. For this we
need to assume $\lambda$ is measurable and use a fix normal ultrafilter
$\mathbb{E}$ on it. \endgraf (D) For some purposes we need stronger
changes in the framework: allowing $H$'s in the $\bold i$'s. This
includes $(f,g)$-bounding.},
},
@article{Sh:1101,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
journal = {Israel Journal of Mathematics},
title = {{Isomorphic limit ultrapowers for infinitary logic}},
volume = {submitted},
abstract = {161218 This was section 3 of [1019]},
},
@article{KmSh:1102,
author = {Kumar, Ashutosh and Shelah, Saharon},
trueauthor = {Kumar, Ashutosh and Shelah, Saharon},
fromwhere = {IN,IL},
journal = {Journal of Mathematical Logic},
title = {{On possible restrictions of the null ideal}},
volume = {preprint},
abstract = {We prove that the null ideal restricted to a non null set of
reals could be isomorphic to a variety of sigma ideals. Using this, we
show that the following are consistent: (1) There is a non null subset
of plane each of whose non null subsets contains three collinear
points. (2) There is a partition of a non null set of reals into null
sets, each of size $\aleph_1$, such that every transversal of this
partition is null.},
},
@article{HwSh:1103,
author = {Horowitz, Haim and Shelah, Saharon},
trueauthor = {Horowitz, Haim and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {preprint},
title = {{Mad families and non-meager filters}},
abstract = {We prove the consistency of $ZF + DC + $ ``there are no mad
families'' + ``there exists a non-meager filter on $\omega$'' relative
to $ZFC$, answering a question of Neeman and Norwood. We also
introduce a weaker version of madness, and we strengthen the result
from [HwSh:1090] by showing that no such families exist in our
model.},
},
@article{KmSh:1104,
author = {Kumar, Ashutosh and Shelah, Saharon},
trueauthor = {Kumar, Ashutosh and Shelah, Saharon},
fromwhere = {IN,IL},
journal = {Transactions of the AMS},
title = {{Saturated null and meager ideal}},
volume = {submitted},
abstract = {We prove that the meager ideal and the null ideal could both
be somewhere $\aleph_1$-saturated},
},
@article{LrSh:1105,
author = {Larson, Paul and Shelah, Saharon},
trueauthor = {Larson, Paul and Shelah, Saharon},
fromwhere = {1, IL},
journal = {Archive for Mathematical Logic},
title = {{A model of $\mathsf{ZFA}$ with no outer model of
$\mathsf{ZFAC}$ with the same pure part}},
volume = {submitted},
abstract = {We produce a model of $\mathsf{ZFA} + \mathsf{PAC}$ such
that no outer model of $\mathsf{ZFAC}$ has the same pure sets,
answering a question asked privately by Eric Hall.},
},
@article{PaSh:1106,
author = {Paolini, Gianluca and Shelah, Saharon},
trueauthor = {Paolini, Gianluca and Shelah, Saharon},
fromwhere = {IT, IL},
journal = {Proceedings of the American Mathematical Society},
title = {{The automorphism group of Hall's Universal group}},
volume = {accepted},
abstract = {We study the automorphism group of Hall's universal
locally finite group $H$. We show that in $Aut(H)$ every subgroup of
index $< 2^\omega$ lies between the pointwise and the setwise
stabilizer of a unique finite subgroup $A$ of $H$, and use this to
prove that $Aut(H)$ is complete. We further show that $Inn(H)$ is the
largest locally finite normal subgroup of $Aut(H)$. Finally, we
observe that from the work of [312] it follows that for every countable
locally finite $G$ there exists $G \cong G' \leq H$ such that every $f
\in Aut(G')$ extends to an $\hat{f} \in Aut(H)$ in such a way that $f
\mapsto \hat{f}$ embeds $Aut(G')$ into $Aut(H)$. In particular,
we solve the three open questions of Hickin on $Aut(H)$ from [3] and
give a partial answer to Question VI.5 of Kegel and Wehrfritz from
[6].},
},
@article{PaSh:1108,
author = {Paolini, Gianluca and Shelah, Saharon},
trueauthor = {Paolini, Gianluca and Shelah, Saharon},
fromwhere = {IT, IL},
journal = {Discrete Mathematics},
title = {{The strong small index property for free homogeneous
structures}},
volume = {submitted},
abstract = { We show that in countable homogeneous structures with
canonical amalgamation and locally finite algebraicity the small index
property implies the strong small index property. We use this and the
main result of [12] to deduce that countable free homogeneous
structures in a locally finite irreflexive relational language
have the strong small index property. As an application, we exhibit
new continuum sized classes of $\aleph_0$-categorical structures with
the strong small index property whose automorphism groups are pairwise
non-isomorphic.},
},
@article{PaSh:1109,
author = {Paolini, Gianluca and Shelah, Saharon},
trueauthor = {Paolini, Gianluca and Shelah, Saharon},
fromwhere = {IT, IL},
journal = {Fundamenta Mathematicae},
title = {{Reconstructing structures with the strong small index property
up to bi-definability}},
volume = {submitted},
abstract = {Let $\mathbf{K}$ be the class of countable structures $M$
with the strong small index property and locally finite algebraicity,
and $\mathbf{K}_*$ the class of $M \in \mathbf{K}$ such that $acl_M(\{
a \}) = \{ a \}$ for every $a \in M$. For homogeneous $M \in
\mathbf{K}$, we introduce what we call the expanded group of
automorphisms of $M$, and show that it is second-order definable in
$Aut(M)$. We use this to prove that for $M, N \in \mathbf{K}_*$,
$Aut(M)$ and $Aut(N)$ are isomorphic as abstract groups if and only
if $(Aut(M), M)$ and $(Aut(N), N)$ are isomorphic as permutation
groups. In particular, we deduce that for $\aleph_0$-categorical
structures the combination of strong small index property and no
algebraicity implies reconstruction up to bi-definability, in
analogy with Rubin's well-known $\forall \exists$-interpretation
technique of [7]. Finally, we show that every finite group can be
realized as the outer automorphism group of $Aut(M)$ for some
countable $\aleph_0$-categorical homogeneous structure $M$ with the
strong small index property and no algebraicity.},
},
@article{ShSi:1110,
author = {Shelah, Saharon and Spinas, Otmar},
trueauthor = {Shelah, Saharon and Spinas, Otmar},
fromwhere = {IL, CH},
journal = {Annals of Pure and Applied Logic},
title = {{Different cofinalities of tree ideals}},
volume = {submitted},
abstract = {We introduce a general framework of generalized tree
forcings, GTF for short, that includes the classical tree forcings
like Sacks, Silver, Laver or Miller forcing. Using this concept we
study the cofinality of the ideal $\mathcal{I}(\mathbf {Q})$
associated with a GTF $\mathbf {Q}$. We show that if for two GTF's
$\mathbf{Q_0}$ and $\mathbf{Q_1}$ the consistency of
$add(\mathcal{I}(\mathbf{Q_0})) < add(\mathcal{I}(\mathbf{Q_1}))$
holds, then we can obtain the consistency of
$cof(\mathcal{I}(\mathbf{Q_1})) < cof(\mathcal{I} (\mathbf{Q_0}))$. We
also show that $cof(\mathcal{I}(\mathbf{Q}))$ can consistently be any
cardinal of cofinality larger than the continuum.},
},
@article{GaSh:1111,
author = {Garti, Shimon and Shelah, Saharon},
trueauthor = {Garti, Shimon and Shelah, Saharon},
fromwhere = {IL,IL},
journal = {preprint},
title = {{Double weakness}},
},
@article{PaSh:1112,
author = {Paolini, Gianluca and Shelah, Saharon},
trueauthor = {Paolini, Gianluca and Shelah, Saharon},
fromwhere = {IT,IL},
journal = {Axioms Topical Collection ``Topological Groups''},
title = {{No uncountable Polish group can be a right-angled Artin
group}},
volume = {accepted},
abstract = {We prove that no uncountable Polish group can admit a system
of generators whose associated length function satisfies the
following conditions: (i) if $0 < k < \omega$, then $lg(x) \leq
lg(x^k)$; (ii) if $lg(y) < k < \omega$ and $x^k = y$, then $x = e$. In
particular, the automorphism group of a countable structure cannot be
an uncountable right-angled Artin group. This generalizes results from
[Sh:744] and ``Polish Group Topologies'' by S. Solecki, where this is
proved for free and free abelian uncountable groups.},
},
@article{HwSh:1113,
author = {Horowitz, Haim and Shelah, Saharon},
trueauthor = {Horowitz, Haim and Shelah, Saharon},
fromwhere = {IL, IL},
journal = {preprint},
title = {{Madness and regularity properties}},
abstract = {Starting from an inaccessible cardinal, we construct a model
of $ZF + DC$ where there exists a mad family and all sets of reals are
$\mathbb{Q}$-measurable for $\omega^\omega$-bounding sufficiently
absolute forcing notions $\mathbb{Q}$.},
},
@article{ShSr:1114,
author = {Shelah, Saharon and Steprans, Juris},
trueauthor = {Shelah, Saharon and Stepr\={a}ns, Juris},
fromwhere = {IL, 3},
journal = {Fundamenta Mathematicae},
title = {{Trivial and non-trivial automorphisms of $\mathcal{P}
(\omega_1)/[\omega_1]^{<\aleph_0}$}},
volume = {submitted},
abstract = {The following statement is shown to be independent of set
theory with the Continuum Hypothesis: There is an automorphism of
$\mathcal{P}(\omega_1)/[\omega_1]^{<\aleph_0}$ whose restriction to
$\mathcal{P} (\alpha) / [\alpha]^{<\aleph_0}$ is induced by a
bijection for every $\alpha \in \omega_1$, but the automorphism
itself is not induced by any bijection on $\omega_1$.},
},
@article{PaSh:1115,
author = {Paolini, Gianluca and Shelah, Saharon},
trueauthor = {Paolini, Gianluca and Shelah, Saharon},
fromwhere = {IT,IL},
journal = {Israel Journal of Mathematics},
title = {{Polish topologies for graph products of cyclic groups}},
volume = {submitted},
abstract = {We give a complete characterization of the graph products
of cyclic groups admitting a Polish group topology, and show that
they are all realizable as the group of automorphisms of a
countable structure. In particular, we characterize the right-angled
Coxeter groups (resp. Artin groups) admitting a Polish group topology.
This generalizes results from [5], [7], and [4]. },
},
@article{CaSh:1116,
author = {Casanovas, Enrique and Shelah, Saharon},
trueauthor = {Casanovas, Enrique and Shelah, Saharon},
fromwhere = {IL},
journal = {Fundamenta Mathematicae},
title = {{Universal theories and compactly expandable models}},
abstract = {Our aim is to solve a quite old question on the difference
between expandability and compact expandability. Toward this, we
further investigate the logic of countable cofinality},
},
@article{PaSh:1117,
author = {Paolini, Gianluca and Shelah, Saharon},
trueauthor = {Paolini, Gianluca and Shelah, Saharon},
fromwhere = {IT, IL},
journal = {Topology and its Applications},
title = {{Group metrics for graph products of cyclic groups}},
volume = {submitted},
abstract = {We complement the characterization of the graph products of
cyclic groups $G(\Gamma, {\mathfrak p})$ admitting a Polish group
topology of [9] with the following result. Let $G = G(\Gamma,
{\mathfrak p})$, then the following are equivalent: [i] there is a
metric on $\Gamma$ which induces a separable topology in which
$E_{\Gamma}$ is closed; [ii] $G(\Gamma, {\mathfrak p})$ is embeddable
into a Polish group; [iii] $G(\Gamma, {\mathfrak p})$ is embeddable
into a non-Archimedean Polish group. We also construct left-invariant
separable group ultrametrics for $G = G(\Gamma, {\mathfrak p})$ and
$\Gamma$ a closed graph on the Baire space, which is of independent
interest.},
},
@article{KpRaSh:1118,
author = {Kaplan, Itay and Ramsey, Nicholas and Shelah, Saharon},
trueauthor = {Kaplan, Itay and Ramsey, Nicholas and Shelah, Saharon},
fromwhere = {IL,1,IL},
journal = {Proceedings of the AMS},
title = {{Local character of Kim-independence}},
volume = {submitted},
abstract = {We show that NSOP$_1$ theories are exactly the theories in
which Kim-independence satisfies a form of local character. In
particular, we show that if $T$ is NSOP$_1$, $M \models T$, and $p$ is
a type over $M$, then the collection of elementary submodels of size
$|T|$ over which $p$ does not Kim-fork is a club of $[M]^{|T|}$ and
that this characterizes NSOP$_1$.},
},
@article{ShVe:1119,
author = {Shelah, Saharon and Vasey, Sebastien},
trueauthor = {Shelah, Saharon and Vesey, Sebastien},
fromwhere = {IL,1},
journal = {Annals of Pure and Applied Logic},
note = { arxiv:math.LO/1702.08281 },
title = {{Abstract elementary classes stable in $\aleph_0$}},
abstract = {We study abstract elementary classes (AECs) that, in
$\aleph_0$, have amalgamation, joint embedding, no maximal models and
are stable (in terms of the number of orbital types). We prove that
such classes exhibit super stable-like behavior at $\aleph_0$. More
precisely, there is a superlimit model of cardinality $\aleph_0$ and
the class generated by this superlimit has a type-full good
$\aleph_0$-frame (a local notion of nonforking independence) and a
superlimit model of cardinality $\aleph_1$. This extends the first
author's earlier study of PC$_{\aleph_0}$-representable AECs and also
improves results of Hyttinen-Kesala and Baldwin-Kueker-VanDieren.},
},
@article{GsSh:1120,
author = {Golshani, Mohammad and Shelah, Saharon},
trueauthor = {Golshani, Mohammad and Shelah, Saharon},
fromwhere = {IR, IL},
journal = {preprint},
note = { arxiv:math.LO/1708.02719 },
title = {{Specializing trees and answer to a question of Williams}},
abstract = {We show that if $cf(2^{\aleph_0}) = \aleph_1$, then any
non-trivial $\aleph_1$-closed forcing notion of size $\le
2^{\aleph_0}$ is forcing equivalent to $Add(\aleph_1, 1)$, the Cohen
forcing for adding a new Cohen subset of $\omega_1$. We also produce,
relative the existence of some large cardinals, a model of $ZFC$ in
which $2^{\aleph_0} = \aleph_2$ and all $\aleph_1$-closed forcing
notion of size $\le 2^{\aleph_0}$ collapse $\aleph_2$, and hence are
forcing equivalent to $Add(\aleph_1, 1)$. Our results answer a
question of Scott Williams from 1978. We also extend a result of
Todorcevic and Foreman-Magidor-Shelah by showing that it is
consistent that every partial order which adds a new subset of
$\aleph_2$, collapes $\aleph_2$ or $\aleph_3$.},
},
@article{PaSh:1121,
author = {Paolini, Gianluca and Shelah, Saharon},
trueauthor = {Paolini, Gianluca and Shelah, Saharon},
fromwhere = {IT,IL},
journal = {Transactions of the AMS},
title = {{Polish topologies for graph products of groups}},
volume = {submitted},
abstract = { We give strong necessary conditions on the admissibility of
a Polish group topology for an arbitrary graph product of groups
$G(\Gamma, G_a)$, and use them to give a characterization modulo a
finite set of nodes. As a corollary, we give a complete
characterization in case all the factor groups $G_a$ are countable.},
},
@article{Sh:1122,
author = {Goldstern, Martin and Kellner, Jakob and Shelah, Saharon},
trueauthor = {Goldstern, Martin and Kellner, Jakob and Shelah, Saharon},
fromwhere = {A, IL, A},
note = { arxiv:1708.03691 },
title = {{Compact cardinals and all the values of Cicho\'n's Diagram}},
},
@article{Sh:1123,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
title = {{}},
},
@article{MiSh:1124,
author = {Malliaris, Maryanthe and Shelah, Saharon},
trueauthor = {Malliaris, Maryanthe and Shelah, Saharon},
fromwhere = {1, IL},
journal = {preprint},
title = {{A new look at interpretability and saturation}},
abstract = {We investigate the interpretability ordering
$\trianglelefteq^*$ using generalized Ehrenfeucht-Mostowski models.
This gives a new approach to proving inequalities and investigating
the structure of types.},
},
@article{Sh:1125,
author = {Shelah, Saharon},
trueauthor = {Shelah, Saharon},
fromwhere = {IL},
title = {{}},
},
*