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Saharon Shelah [412]S. Shelah [66]
  1. ▵13-Sets of Reals.Haim Judah & Saharon Shelah - 1993 - Journal of Symbolic Logic 58 (1):72 - 80.
    We build models where all $\underset{\sim}{\triangle}^1_3$ -sets of reals are measurable and (or) have the property of Baire and (or) are Ramsey. We will show that there is no implication between any of these properties for $\underset{\sim}{\triangle}^1_3$ -sets of reals.
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  2. Universal Graphs at the Successor of a Singular Cardinal.Mirna Džamonja & Saharon Shelah - 2003 - Journal of Symbolic Logic 68 (2):366-388.
    The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality ℵ0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are $\mu^{++}$ graphs on μ+ that taken jointly are universal for the graphs on μ+, while $2^{\mu^+} \gg \mu^{++}$ . The paper also addresses the general problem of obtaining a framework for consistency results at the (...)
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  3. Simple Unstable Theories.Saharon Shelah - 1980 - Annals of Mathematical Logic 19 (3):177-203.
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  4. Nonexistence of Universal Orders in Many Cardinals.Menachem Kojman & Saharon Shelah - 1992 - Journal of Symbolic Logic 57 (3):875-891.
    Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1 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 regular $\aleph_1 < (...)
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  5. Toward Classifying Unstable Theories.Saharon Shelah - 1996 - Annals of Pure and Applied Logic 80 (3):229-255.
  6.  6
    On Certain Indestructibility of Strong Cardinals and a Question of Hajnal.Moti Gitik & Saharon Shelah - 1989 - Archive for Mathematical Logic 28 (1):35-42.
    A model in which strongness ofκ is indestructible under κ+ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of {λ δ |2 δ <λ}.
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  7. Constructing Strongly Equivalent Nonisomorphic Models for Unsuperstable Theories, Part A.Tapani Hyttinen & Saharon Shelah - 1994 - Journal of Symbolic Logic 59 (3):984-996.
    In this paper we prove a strong nonstructure theorem for κ(T)-saturated models of a stable theory T with dop. This paper continues the work started in [1].
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  8.  7
    Non-Forking Frames in Abstract Elementary Classes.Adi Jarden & Saharon Shelah - 2013 - Annals of Pure and Applied Logic 164 (3):135-191.
    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-forking relations. Later, Shelah [17, II] introduced the good non-forking frame, an axiomatization of the non-forking notion.We improve results of Shelah on good non-forking frames, mainly by weakening the stability hypothesis in several important theorems, replacing it by the almost λ-stability hypothesis: The number of types over a model of cardinality (...)
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  9. The Cofinality of Cardinal Invariants Related to Measure and Category.Tomek Bartoszynski, Jaime I. Ihoda & Saharon Shelah - 1989 - Journal of Symbolic Logic 54 (3):719-726.
    We prove that the following are consistent with ZFC. 1. 2 ω = ℵ ω 1 + K C = ℵ ω 1 + K B = K U = ω 2 (for measure and category simultaneously). 2. 2 ω = ℵ ω 1 = K C (L) + K C (M) = ω 2 . This concludes the discussion about the cofinality of K C.
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  10.  8
    Reflecting Stationary Sets and Successors of Singular Cardinals.Saharon Shelah - 1991 - Archive for Mathematical Logic 31 (1):25-53.
    REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a κ which is κ+n -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the “bad” stationary set. It is shown that supercompactness (and even the failure (...)
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  11.  3
    Categoricity for Abstract Classes with Amalgamation.Saharon Shelah - 1999 - Annals of Pure and Applied Logic 98 (1-3):261-294.
    Let be an abstract elementary class with amalgamation, and Lowenheim Skolem number LS. We prove that for a suitable Hanf number gc0 if χ0 < λ0 λ1, and is categorical inλ1+ then it is categorical in λ0.
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  12.  6
    There May Be Simple Pℵ1 and Pℵ2-Points and the Rudin-Keisler Ordering May Be Downward Directed.Andreas Blass & Saharon Shelah - 1987 - Annals of Pure and Applied Logic 33 (3):213-243.
  13. Hanf Number of Omitting Type for Simple First-Order Theories.Saharon Shelah - 1979 - Journal of Symbolic Logic 44 (3):319-324.
    Let T be a complete countable first-order theory such that every ultrapower of a model of T is saturated. If T has a model omitting a type p in every cardinality $ then T has a model omitting p in every cardinality. There is also a related theorem, and an example showing the $\beth_\omega$ cannot be improved.
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  14. The Covering Numbers of Mycielski Ideals Are All Equal.Saharon Shelah & Juris Steprāns - 2001 - Journal of Symbolic Logic 66 (2):707-718.
    The Mycielski ideal M k is defined to consist of all sets $A \subseteq ^{\mathbb{N}}k$ such that $\{f \upharpoonright X: f \in A\} \neq ^Xk$ for all X ∈ [N] ℵ 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.
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  15. On the Structure of $\Operatorname{Ext}(a, \Mathbf{Z})$ in ZFC+.G. Sageev & S. Shelah - 1985 - Journal of Symbolic Logic 50 (2):302 - 315.
  16.  2
    On ◁∗-Maximality.Mirna Džamonja & Saharon Shelah - 2004 - Annals of Pure and Applied Logic 125 (1-3):119-158.
    This paper investigates a connection between the semantic notion provided by the ordering * among theories in model theory and the syntactic SOPn hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP2 and SOP1. It is shown here that SOP3 implies SOP2 implies SOP1. In Shelah's article 229) it was shown that SOP3 implies *-maximality and we prove here that *-maximality in a model of GCH implies a property called SOP2″. It has been (...)
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  17. The Strength of the Isomorphism Property.Renling Jin & Saharon Shelah - 1994 - Journal of Symbolic Logic 59 (1):292-301.
    In § 1 of this paper, we characterize the isomorphism property of nonstandard universes in terms of the realization of some second-order types in model theory. In § 2, several applications are given. One of the applications answers a question of D. Ross in [this Journal, vol. 55 (1990), pp. 1233-1242] about infinite Loeb measure spaces.
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  18.  35
    A Definable Nonstandard Model of the Reals.Vladimir Kanovei & Saharon Shelah - 2004 - Journal of Symbolic Logic 69 (1):159-164.
    We prove, in ZFC,the existence of a definable, countably saturated elementary extension of the reals.
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  19.  1
    More on SOP 1 and SOP 2.Saharon Shelah & Alexander Usvyatsov - 2008 - Annals of Pure and Applied Logic 155 (1):16-31.
    This paper continues the work in [S. Shelah, Towards classifying unstable theories, Annals of Pure and Applied Logic 80 229–255] and [M. Džamonja, S. Shelah, On left triangle, open*-maximality, Annals of Pure and Applied Logic 125 119–158]. We present a rank function for NSOP1 theories and give an example of a theory which is NSOP1 but not simple. We also investigate the connection between maximality in the ordering left triangle, open* among complete first order theories and the SOP2 property. We (...)
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  20.  1
    Some Exact Equiconsistency Results in Set Theory.Leo Harrington & Saharon Shelah - 1985 - Notre Dame Journal of Formal Logic 26 (2):178-188.
  21.  1
    Stability, the F.C.P., and Superstability; Model Theoretic Properties of Formulas in First Order Theory.Saharon Shelah - 1971 - Annals of Mathematical Logic 3 (3):271-362.
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  22. Applications of PCF Theory.Saharon Shelah - 2000 - Journal of Symbolic Logic 65 (4):1624-1674.
    We deal with several pcf problems: we characterize another version of exponentiation: maximal number of κ-branches in a tree with λ nodes, deal with existence of independent sets in stable theories, possible cardinalities of ultraproducts and the depth of ultraproducts of Boolean Algebras. Also we give cardinal invariants for each λ with a pcf restriction and investigate further T D (f). The sections can be read independently, although there are some minor dependencies.
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  23.  2
    Finite Diagrams Stable in Power.Saharon Shelah - 1970 - Annals of Mathematical Logic 2 (1):69-118.
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  24.  1
    Strong Splitting in Stable Homogeneous Models.Tapani Hyttinen & Saharon Shelah - 2000 - Annals of Pure and Applied Logic 103 (1-3):201-228.
    In this paper we study elementary submodels of a stable homogeneous structure. We improve the independence relation defined in Hyttinen 167–182). We apply this to prove a structure theorem. We also show that dop and sdop are essentially equivalent, where the negation of dop is the property we use in our structure theorem and sdop implies nonstructure, see Hyttinen.
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  25.  2
    Δ-Logics and Generalized Quantifiers.J. A. Makowsky, Saharon Shelah & Jonathan Stavi - 1976 - Annals of Mathematical Logic 10 (2):155-192.
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  26.  5
    Fixed-Point Extensions of First-Order Logic.Yuri Gurevich & Saharon Shelah - 1986 - Annals of Pure and Applied Logic 32 (3):265-280.
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  27.  9
    The Tree Property at Successors of Singular Cardinals.Menachem Magidor & Saharon Shelah - 1996 - Archive for Mathematical Logic 35 (5-6):385-404.
    Assuming some large cardinals, a model of ZFC is obtained in which $\aleph_{\omega+1}$ carries no Aronszajn trees. It is also shown that if $\lambda$ is a singular limit of strongly compact cardinals, then $\lambda^+$ carries no Aronszajn trees.
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  28. Diamonds, Uniformization.Saharon Shelah - 1984 - Journal of Symbolic Logic 49 (4):1022-1033.
    Assume G.C.H. We prove that for singular λ, □ λ implies the diamonds hold for many $S \subseteq \lambda^+$ (including $S \subseteq \{\delta:\delta \in \lambda^+, \mathrm{cf}\delta = \mathrm{cf}\delta = \mathrm{cf}\lambda\}$ . We also have complementary consistency results.
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  29.  1
    Superstable Fields and Groups.G. Cherlin & S. Shelah - 1980 - Annals of Mathematical Logic 18 (3):227-270.
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  30.  9
    A Δ22 Well-Order of the Reals and Incompactness of L.Uri Abraham & Saharon Shelah - 1993 - Annals of Pure and Applied Logic 59 (1):1-32.
    A forcing poset of size 221 which adds no new reals is described and shown to provide a Δ22 definable well-order of the reals . 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 noncompact.
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  31.  5
    Toward Categoricity for Classes with No Maximal Models.Saharon Shelah & Andrés Villaveces - 1999 - Annals of Pure and Applied Logic 97 (1-3):1-25.
    We provide here the first steps toward a Classification Theory ofElementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some λ greater than its Löwenheim-Skolem number. We study the degree to which amalgamation may be recovered, the behaviour of non μ-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 o Conjecture for these classes. Further (...)
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  32. Martin's Axioms, Measurability and Equiconsistency Results.Jaime I. Ihoda & Saharon Shelah - 1989 - Journal of Symbolic Logic 54 (1):78-94.
    We deal with the consistency strength of ZFC + variants of MA + suitable sets of reals are measurable (and/or Baire, and/or Ramsey). We improve the theorem of Harrington and Shelah [2] repairing the asymmetry between measure and category, obtaining also the same result for Ramsey. We then prove parallel theorems with weaker versions of Martin's axiom (MA(σ-centered), (MA(σ-linked)), MA(Γ + ℵ 0 ), MA(K)), getting Mahlo, inaccessible and weakly compact cardinals respectively. We prove that if there exists r ∈ (...)
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  33. On Polynomial Time Computation Over Unordered Structures.Andreas Blass, Yuri Gurevich & Saharon Shelah - 2002 - Journal of Symbolic Logic 67 (3):1093-1125.
    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 (...)
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  34. Δ12-Sets of Reals.Jaime I. Ihoda & Saharon Shelah - 1989 - Annals of Pure and Applied Logic 42 (3):207-223.
  35. Uniformization Principles.Alan H. Mekler & Saharon Shelah - 1989 - Journal of Symbolic Logic 54 (2):441-459.
    It is consistent that for many cardinals λ there is a family of at least λ + unbounded subsets of λ which have uniformization properties. In particular if it is consistent that a supercompact cardinal exists, then it is consistent that ℵ ω has such a family. We have applications to point set topology, Whitehead groups and reconstructing separable abelian p-groups from their socles.
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  36.  11
    The Bounded Proper Forcing Axiom.Martin Goldstern & Saharon Shelah - 1995 - Journal of Symbolic Logic 60 (1):58-73.
    The bounded proper forcing axiom BPFA is the statement that for any family of ℵ 1 many maximal antichains of a proper forcing notion, each of size ℵ 1 , there is a directed set meeting all these antichains. A regular cardinal κ is called Σ 1 -reflecting, if for any regular cardinal χ, for all formulas $\varphi, "H(\chi) \models`\varphi'"$ implies " $\exists\delta . We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the bounded (...)
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  37.  6
    Interpreting Groups and Fields in Some Nonelementary Classes.Tapani Hyttinen, Olivier Lessmann & Saharon Shelah - 2005 - Journal of Mathematical Logic 5 (01):1-47.
  38.  4
    Remarks on Superatomic Boolean Algebras.James E. Baumgartner & Saharon Shelah - 1987 - Annals of Pure and Applied Logic 33 (2):109-129.
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  39.  24
    Successors of Singular Cardinals and Coloring Theorems I.Todd Eisworth & Saharon Shelah - 2005 - Archive for Mathematical Logic 44 (5):597-618.
  40. On Closed P-Sets with in the Ω.Rvszard Frankiewicz, Saharon Shelah & Paweł Zbierski - forthcoming - Journal of Symbolic Logic.
  41.  7
    Categoricity of Theories in Lϰω, with Κ a Compact Cardinal.Saharon Shelah & Michael Makkai - 1990 - Annals of Pure and Applied Logic 47 (1):41-97.
  42.  4
    On the Consistency of Some Partition Theorems for Continuous Colorings, and the Structure of ℵ< Sub> 1-Dense Real Order Types.Uri Abraham, Matatyahu Rubin & Saharon Shelah - 1985 - Annals of Pure and Applied Logic 29 (2):123-206.
    We present some techniques in c.c.c. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of ℵ 1 -dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two ℵ 1 -dense subsets of R are isomorphic, is consistent. We e.g. prove Con). Let K H , be the set of order types of ℵ 1 -dense homogeneous subsets of (...)
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  43. On Finite Rigid Structures.Yuri Gurevich & Saharon Shelah - 1996 - Journal of Symbolic Logic 61 (2):549-562.
    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 ω ∞,ω formula with counting quantifiers defines a linear order.
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  44. The Cofinality Spectrum of the Infinite Symmetric Group.Saharon Shelah & Simon Thomas - 1997 - Journal of Symbolic Logic 62 (3):902-916.
    Let S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals λ such that S can be expressed as the union of a chain of λ proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the cofinality spectrum of S. The following theorem is the main result of this paper. Theorem. Suppose that $V \models GCH$ . Let C be (...)
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  45.  9
    The Kunen-Miller Chart (Lebesgue Measure, the Baire Property, Laver Reals and Preservation Theorems for Forcing).Haim Judah & Saharon Shelah - 1990 - Journal of Symbolic Logic 55 (3):909-927.
    In this work we give a complete answer as to the possible implications between some natural properties of Lebesgue measure and the Baire property. For this we prove general preservation theorems for forcing notions. Thus we answer a decade-old problem of J. Baumgartner and answer the last three open questions of the Kunen-Miller chart about measure and category. Explicitly, in \S1: (i) We prove that if we add a Laver real, then the old reals have outer measure one. (ii) We (...)
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  46.  28
    Forcing Closed Unbounded Sets.Uri Abraham & Saharon Shelah - 1983 - Journal of Symbolic Logic 48 (3):643-657.
    We discuss the problem of finding forcing posets which introduce closed unbounded subsets to a given stationary set.
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  47.  6
    Choiceless Polynomial Time.Andreas Blass, Yuri Gurevich & Saharon Shelah - 1999 - Annals of Pure and Applied Logic 100 (1-3):141-187.
    Turing machines define polynomial time 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 ? Earlier, one of us conjectured a negative answer. The problem motivated a quest for stronger and stronger (...)
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  48.  8
    Reasonable Ultrafilters, Again.Andrzej Rosłanowski & Saharon Shelah - 2010 - Notre Dame Journal of Formal Logic 52 (2):113-147.
    We continue investigations of reasonable ultrafilters on uncountable cardinals defined in previous work by Shelah. We introduce stronger properties of ultrafilters and we show that those properties may be handled in λ-support iterations of reasonably bounding forcing notions. We use this to show that consistently there are reasonable ultrafilters on an inaccessible cardinal λ with generating systems of size less than $2^\lambda$ . We also show how ultrafilters generated by small systems can be killed by forcing notions which have enough (...)
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  49.  3
    Combinatorial Properties of Hechler Forcing.Jörg Brendle, Haim Judah & Saharon Shelah - 1992 - Annals of Pure and Applied Logic 58 (3):185-199.
    Brendle, J., H. Judah and S. Shelah, Combinatorial properties of Hechler forcing, Annals of Pure and Applied Logic 59 185–199. Using a notion of rank for Hechler forcing we show: assuming ωV1 = ωL1, there is no real in V[d] which is eventually different from the reals in L[ d], where d is Hechler over V; adding one Hechler real makes the invariants on the left-hand side of Cichoń's diagram equal ω1 and those on the right-hand side equal 2ω and (...)
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  50.  2
    A Dichotomy Theorem for Regular Types.Ehud Hrushovski & Saharon Shelah - 1989 - Annals of Pure and Applied Logic 45 (2):157-169.
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