347 found
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  1.  12
    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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  2.  29
    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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  3.  21
    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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  4.  20
    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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  5.  9
    Creature Forcing and Five Cardinal Characteristics in Cichoń’s Diagram.Arthur Fischer, Martin Goldstern, Jakob Kellner & Saharon Shelah - 2017 - Archive for Mathematical Logic 56 (7-8):1045-1103.
    We use a creature construction to show that consistently $$\begin{aligned} \mathfrak d=\aleph _1= {{\mathrm{cov}}}< {{\mathrm{non}}}< {{\mathrm{non}}}< {{\mathrm{cof}}} < 2^{\aleph _0}. \end{aligned}$$The same method shows the consistency of $$\begin{aligned} \mathfrak d=\aleph _1= {{\mathrm{cov}}}< {{\mathrm{non}}}< {{\mathrm{non}}}< {{\mathrm{cof}}} < 2^{\aleph _0}. \end{aligned}$$.
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  6.  18
    On the Consistency of Some Partition Theorems for Continuous Colorings, and the Structure of ℵ 1 -Dense Real Order Types.J. Steprans, Uri Abraham, Matatyahu Rubin & Saharon Shelah - 2002 - Bulletin of Symbolic Logic 8 (2):303.
    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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  7.  12
    Toward Classifying Unstable Theories.Saharon Shelah - 1996 - Annals of Pure and Applied Logic 80 (3):229-255.
  8.  41
    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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  9.  17
    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.
  10.  14
    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.  15
    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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  12.  46
    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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  13.  71
    The Monadic Theory of Ω2.Yuri Gurevich, Menachem Magidor & Saharon Shelah - 1983 - Journal of Symbolic Logic 48 (2):387-398.
    Assume ZFC + "There is a weakly compact cardinal" is consistent. Then: (i) For every $S \subseteq \omega, \mathrm{ZFC} +$ "S and the monadic theory of ω 2 are recursive each in the other" is consistent; and (ii) ZFC + "The full second-order theory of ω 2 is interpretable in the monadic theory of ω 2 " is consistent.
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  14.  20
    Some Exact Equiconsistency Results in Set Theory.Leo Harrington & Saharon Shelah - 1985 - Notre Dame Journal of Formal Logic 26 (2):178-188.
  15.  5
    Δ12-Sets of Reals.Jaime I. Ihoda & Saharon Shelah - 1989 - Annals of Pure and Applied Logic 42 (3):207-223.
  16.  21
    Interpreting Groups and Fields in Some Nonelementary Classes.Tapani Hyttinen, Olivier Lessmann & Saharon Shelah - 2005 - Journal of Mathematical Logic 5 (1):1-47.
    This paper is concerned with extensions of geometric stability theory to some nonelementary classes. We prove the following theorem: Theorem. Let [Formula: see text] be a large homogeneous model of a stable diagram D. Let p, q ∈ SD, where p is quasiminimal and q unbounded. Let [Formula: see text] and [Formula: see text]. Suppose that there exists an integer n < ω such that [Formula: see text] for any independent a1, …, an ∈ P and finite subset C ⊆ (...)
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  17.  7
    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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  18.  22
    On Weak and Strong Interpolation in Algebraic Logics.Gábor Sági & Saharon Shelah - 2006 - Journal of Symbolic Logic 71 (1):104 - 118.
    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 [12].
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  19. 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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  20.  13
    The Spectrum of Independence.Vera Fischer & Saharon Shelah - 2019 - Archive for Mathematical Logic 58 (7-8):877-884.
    We study the set of possible sizes of maximal independent families to which we refer as spectrum of independence and denote \\). Here mif abbreviates maximal independent family. We show that:1.whenever \ are finitely many regular uncountable cardinals, it is consistent that \\); 2.whenever \ has uncountable cofinality, it is consistent that \=\{\aleph _1,\kappa =\mathfrak {c}\}\). Assuming large cardinals, in addition to above, we can provide that $$\begin{aligned} \cap \hbox {Spec}=\emptyset \end{aligned}$$for each i, \.
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  21.  39
    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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  22. Constructing Strongly Equivalent Nonisomorphic Models for Unsuperstable Theories, Part A.Tapani Hyttinen & Saharon Shelah - 1994 - Journal of Symbolic Logic 59 (3):984-996.
    We study how equivalent nonisomorphic models an unsuperstable theory can have. We measure the equivalence by Ehrenfeucht-Fraisse games. This paper continues the work started in $[HT]$.
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  23. 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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  24.  22
    Examples of Non-Locality.John T. Baldwin & Saharon Shelah - 2008 - Journal of Symbolic Logic 73 (3):765-782.
    We use κ-free but not Whitehead Abelian groups to constructElementary Classes (AEC) which satisfy the amalgamation property but fail various conditions on the locality of Galois-types. We introduce the notion that an AEC admits intersections. We conclude that for AEC which admit intersections, the amalgamation property can have no positive effect on locality: there is a transformation of AEC's which preserves non-locality but takes any AEC which admits intersections to one with amalgamation. More specifically we have: Theorem 5.3. There is (...)
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  25.  16
    Reasonable Ultrafilters, Again.Andrzej Rosłanowski & Saharon Shelah - 2011 - 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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  26. On the Number of Nonisomorphic Models of an Infinitary Theory Which has the Infinitary Order Property. Part A.Rami Grossberg & Saharon Shelah - 1986 - Journal of Symbolic Logic 51 (2):302-322.
    Let κ and λ be infinite cardinals such that κ ≤ λ (we have new information for the case when $\kappa ). Let T be a theory in L κ +, ω of cardinality at most κ, let φ(x̄, ȳ) ∈ L λ +, ω . Now define $\mu^\ast_\varphi (\lambda, T) = \operatorname{Min} \{\mu^\ast:$ If T satisfies $(\forall\mu \kappa)(\exists M_\chi \models T)(\exists \{a_i: i Our main concept in this paper is $\mu^\ast_\varphi (\lambda, \kappa) = \operatorname{Sup}\{\mu^\ast(\lambda, T): T$ is a theory (...)
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  27.  15
    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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  28.  31
    Sacks Forcing, Laver Forcing, and Martin's Axiom.Haim Judah, Arnold W. Miller & Saharon Shelah - 1992 - Archive for Mathematical Logic 31 (3):145-161.
    In this paper we study the question assuming MA+⌝CH does Sacks forcing or Laver forcing collapse cardinals? We show that this question is equivalent to the question of what is the additivity of Marczewski's ideals 0. We give a proof that it is consistent that Sacks forcing collapses cardinals. On the other hand we show that Laver forcing does not collapse cardinals.
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  29. Semiproper Forcing Axiom Implies Martin Maximum but Not PFA+.Saharon Shelah - 1987 - Journal of Symbolic Logic 52 (2):360-367.
    We prove that MM (Martin maximum) is equivalent (in ZFC) to the older axiom SPFA (semiproper forcing axiom). We also prove that SPFA does not imply SPFA + or even PFA + (using the consistency of a large cardinal).
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  30.  11
    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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  31.  35
    On the Existence of Large Subsets of [Λ] Κ Which Contain No Unbounded Non-Stationary Subsets.Saharon Shelah - 2002 - Archive for Mathematical Logic 41 (3):207-213.
    Here we deal with some problems posed by Matet. The first section deals with the existence of stationary subsets of [λ]<κ with no unbounded subsets which are not stationary, where, of course, κ is regular uncountable ≤λ. In the second section we deal with the existence of such clubs. The proofs are easy but the result seems to be very surprising. Theorem 1.2 was proved some time ago by Baumgartner (see Theorem 2.3 of [Jo88]) and is presented here for the (...)
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  32.  45
    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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  33.  49
    The Stability Spectrum for Classes of Atomic Models.John T. Baldwin & Saharon Shelah - 2012 - Journal of Mathematical Logic 12 (1):1250001-.
    We prove two results on the stability spectrum for Lω1,ω. Here [Formula: see text] denotes an appropriate notion of Stone space of m-types over M. Theorem for unstable case: Suppose that for some positive integer m and for every α μ, K is not i-stable in μ. 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 (...)
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  34.  9
    Decisive Creatures and Large Continuum.Jakob Kellner & Saharon Shelah - 2009 - Journal of Symbolic Logic 74 (1):73-104.
    For f, g $ \in \omega ^\omega $ let $c_{f,g}^\forall $ be the minimal number of uniform g-splitting trees (or: Slaloms) to cover the uniform f-splitting tree, i.e., for every branch v of the f-tree, one of the g-trees contains v. $c_{f,g}^\exists $ is the dual notion: For every branch v, one of the g-trees guesses v(m) infinitely often. It is consistent that $c_{f \in ,g \in }^\exists = c_{f \in ,g \in }^\forall = k_ \in $ for N₁ many (...)
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  35.  27
    Creature Forcing and Large Continuum: The Joy of Halving.Jakob Kellner & Saharon Shelah - 2012 - Archive for Mathematical Logic 51 (1-2):49-70.
    For ${f,g\in\omega^\omega}$ let ${c^\forall_{f,g}}$ be the minimal number of uniform g-splitting trees needed to cover the uniform f-splitting tree, i.e., for every branch ν of the f-tree, one of the g-trees contains ν. Let ${c^\exists_{f,g}}$ be the dual notion: For every branch ν, one of the g-trees guesses ν(m) infinitely often. We show that it is consistent that ${c^\exists_{f_\epsilon,g_\epsilon}{=}c^\forall_{f_\epsilon,g_\epsilon}{=}\kappa_\epsilon}$ for continuum many pairwise different cardinals ${\kappa_\epsilon}$ and suitable pairs ${(f_\epsilon,g_\epsilon)}$ . For the proof we introduce a new mixed-limit creature forcing (...)
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  36. ▵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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  37.  11
    Abstract Elementary Classes Stable in ℵ 0.Saharon Shelah & Sebastien Vasey - 2018 - Annals of Pure and Applied Logic 169 (7):565-587.
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  38.  21
    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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  39.  14
    Canonical Models for ℵ1-Combinatorics.Saharon Shelah & Jindr̆ich Zapletal - 1999 - Annals of Pure and Applied Logic 98 (1-3):217-259.
    We define the property of Π2-compactness of a statement Φ of set theory, meaning roughly that the hard core of the impact of Φ on combinatorics of 1 can be isolated in a canonical model for the statement Φ. We show that the following statements are Π2-compact: “dominating NUMBER = 1,” “cofinality of the meager IDEAL = 1”, “cofinality of the null IDEAL = 1”, “bounding NUMBER = 1”, existence of various types of Souslin trees and variations on uniformity of (...)
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  40. 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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  41. 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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  42. 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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  43. 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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  44.  10
    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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  45.  51
    Superdestructibility: A Dual to Laver's Indestructibility.Joel David Hamkins & Saharon Shelah - 1998 - Journal of Symbolic Logic 63 (2):549-554.
    After small forcing, any $ -closed forcing will destroy the supercompactness and even the strong compactness of κ.
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  46.  22
    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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  47.  25
    Many Simple Cardinal Invariants.Martin Goldstern & Saharon Shelah - 1993 - Archive for Mathematical Logic 32 (3):203-221.
  48. 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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  49. 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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  50. 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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