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Saharon Shelah [355]S. Shelah [47]
  1. Rvszard Frankiewicz, Saharon Shelah & Paweł Zbierski (forthcoming). On Closed P-Sets with in the Ω. Journal of Symbolic Logic.
     
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  2. Saharon Shelah (2015). A.E.C. With Not Too Many Models. In Andrés Villaveces, Roman Kossak, Juha Kontinen & Åsa Hirvonen (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics. De Gruyter. 367-402.
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  3. Saharon Shelah & Maryanthe Malliaris (2015). Saturating the Random Graph with an Independent Family of Small Range. [REVIEW] In Andrés Villaveces, Roman Kossak, Juha Kontinen & Åsa Hirvonen (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics. De Gruyter. 319-338.
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  4. John T. Baldwin & Saharon Shelah (2014). A Hanf Number for Saturation and Omission: The Superstable Case. Mathematical Logic Quarterly 60 (6):437-443.
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  5. Noam Greenberg & Saharon Shelah (2014). Models of Cohen Measurability. Annals of Pure and Applied Logic 165 (10):1557-1576.
    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.
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  6. Heike Mildenberger & Saharon Shelah (2014). Many Countable Support Iterations of Proper Forcings Preserve Souslin Trees. Annals of Pure and Applied Logic 165 (2):573-608.
    We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support iterations in the so-called Case A that does not need a division into forcings that add reals and those who do not.
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  7. Moti Gitik & Saharon Shelah (2013). Applications of Pcf for Mild Large Cardinals to Elementary Embeddings. Annals of Pure and Applied Logic 164 (9):855-865.
    The following pcf results are proved:1. Assume thatκ>ℵ0κ>ℵ0is a weakly compact cardinal. Letμ>2κμ>2κbe a singular cardinal of cofinality κ. Then for every regularView the MathML sourceλ sup{suppcfσ⁎-complete|a⊆Reg∩and|a|<μ}.Turn MathJax onAs an application we show that:if κ is a measurable cardinal andj:V→Mj:V→Mis the elementary embedding by a κ-complete ultrafilter over κ, then for every τ the following holds:1. ifjjis a cardinal thenj=τj=τ;2. |j|=|j)||j|=|j)|;3. for any κ-complete ultrafilter W on κ, |j|=|jW||j|=|jW|.The first two items provide affirmative answers to questions from Gitik and Shelah (...)
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  8. Victor Harnik, Terrence S. Millar, Michael L. Wage, Saharon Shelah, Helmut Schwichtenberg, Daniel Lascar, Bruno Poizat, Warren D. Goldfarb, On Carnap & Hugues Leblanc (2013). The Journal of Symbolic Logic Publishes Original Scholarly Work in Symbolic Logic. Founded in 1936, It has Become the Leading Research Journal in the Field. The Journal Aims to Represent Logic Broadly, Including its Connections with Mathematics and Philosophy as Well as Newer Aspects Related to Computer Science and Linguistics. [REVIEW] Journal of Symbolic Logic 309 (318).
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  9. Adi Jarden & Saharon Shelah (2013). Non-Forking Frames in Abstract Elementary Classes. 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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  10. Itay Kaplan & Saharon Shelah (2013). Chain Conditions in Dependent Groups. Annals of Pure and Applied Logic 164 (12):1322-1337.
    In this note we prove and disprove some chain conditions in type definable and definable groups in dependent, strongly dependent and strongly2 dependent theories.
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  11. Andrzej Rosłanowski & Saharon Shelah (2013). More About Λ-Support Iterations of (<Λ)-Complete Forcing Notions. Archive for Mathematical Logic 52 (5-6):603-629.
    This article continues Rosłanowski and Shelah (Int J Math Math Sci 28:63–82, 2001; Quaderni di Matematica 17:195–239, 2006; Israel J Math 159:109–174, 2007; 2011; Notre Dame J Formal Logic 52:113–147, 2011) and we introduce here a new property of (<λ)-strategically complete forcing notions which implies that their λ-support iterations do not collapse λ + (for a strongly inaccessible cardinal λ).
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  12. John T. Baldwin & Saharon Shelah (2012). The Stability Spectrum for Classes of Atomic Models. Journal of Mathematical Logic 12 (01):1250001-.
  13. Shimon Garti & Saharon Shelah (2012). A Strong Polarized Relation. Journal of Symbolic Logic 77 (3):766-776.
    We prove that the strong polarized relation $\left( {\mu _\mu ^ + } \right) \to \left( {\mu _\mu ^ + } \right)_2^{1.1}$ is consistent with ZFC, for a singular ì which is a limit of measurable cardinals.
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  14. Jakob Kellner & Saharon Shelah (2012). Creature Forcing and Large Continuum: The Joy of Halving. 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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  15. Philipp Lücke & Saharon Shelah (2012). External Automorphisms of Ultraproducts of Finite Models. Archive for Mathematical Logic 51 (3-4):433-441.
    Let ${\fancyscript{L}}$ be a finite first-order language and ${\langle{\fancyscript{M}_n} \,|\, {n < \omega}\rangle}$ be a sequence of finite ${\fancyscript{L}}$ -models containing models of arbitrarily large finite cardinality. If the intersection of less than continuum-many dense open subsets of Cantor Space ω 2 is non-empty, then there is a non-principal ultrafilter ${\fancyscript{U}}$ over ω such that the corresponding ultraproduct ${\prod_\fancyscript{U}\fancyscript{M}_n}$ has an automorphism that is not induced by an element of ${\prod_{n<\omega}{\rm Aut}(\fancyscript{M}_n)}$.
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  16. Saharon Shelah & Pierre Simon (2012). Adding Linear Orders. Journal of Symbolic Logic 77 (2):717-725.
    We address the following question: Can we expand an NIP theory by adding a linear order such that the expansion is still NIP? Easily, if acl(A)= A for all A, then this is true. Otherwise, we give counterexamples. More precisely, there is a totally categorical theory for which every expansion by a linear order has IP. There is also an ω-stable NDOP theory for which every expansion by a linear order interprets pseudofinite arithmetic.
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  17. J. Kellner & S. Shelah (2011). 1 Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Strasse 25, 1090 Wien 2 Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, Hebrew University of Jerusalem, Jerusalem, 91904 3 Department of Mathematics, Rutgers University, New Brunswick, NJ 08854. [REVIEW] Journal of Symbolic Logic 76 (4):1153-1183.
     
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  18. Jakob Kellner & Saharon Shelah (2011). More on the Pressing Down Game. Archive for Mathematical Logic 50 (3-4):477-501.
    We investigate the pressing down game and its relation to the Banach Mazur game. In particular we show: consistently, there is a nowhere precipitous normal ideal I on ${\aleph_2}$ such that player nonempty wins the pressing down game of length ${\aleph_1}$ on I even if player empty starts.
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  19. Jakob Kellner & Saharon Shelah (2011). Saccharinity. Journal of Symbolic Logic 76 (4):1153-1183.
    We present a method to iterate finitely splitting lim-sup tree forcings along non-wellfounded linear orders. As an application, we introduce a new method to force (weak) measurability of all definable sets with respect to a certain (non-ccc) ideal.
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  20. Heike Mildenberger & Saharon Shelah (2011). The Minimal Cofinality of an Ultrapower of Ω and the Cofinality of the Symmetric Groupcan Be Larger Than. Journal of Symbolic Logic 76 (4):1322-1340.
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  21. Saharon Shelah (2011). Models of Expansions of Documentclass{Article}Usepackage{Amssymb}Begin{Document}Pagestyle{Empty}${Mathbb N}$End{Document} with No End Extensions. Mathematical Logic Quarterly 57 (4):341-365.
    We deal with models of Peano arithmetic . The methods are from creature forcing. We find an expansion of equation image such that its theory has models with no end extensions. In fact there is a Borel uncountable set of subsets of equation image such that expanding equation image by any uncountably many of them suffice. Also we find arithmetically closed equation image with no ultrafilter on it with suitable definability demand . © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, (...)
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  22. Saharon Shelah (2011). Models of Expansions of with No End Extensions. Mathematical Logic Quarterly 57 (4):341-365.
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  23. Tomek Bartoszynski & Saharon Shelah (2010). Dual Borel Conjecture and Cohen Reals. Journal of Symbolic Logic 75 (4):1293-1310.
    We construct a model of ZFC satisfying the Dual Borel Conjecture in which there is a set of size ℵ₁ that does not have measure zero.
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  24. Ilijas Farah & Saharon Shelah (2010). A Dichotomy for the Number of Ultrapowers. Journal of Mathematical Logic 10 (01n02):45-81.
  25. Shimon Garti & Saharon Shelah (2010). Depth of Boolean Algebras. Notre Dame Journal of Formal Logic 52 (3):307-314.
    Suppose $D$ is an ultrafilter on $\kappa$ and $\lambda^\kappa = \lambda$. We prove that if ${\bf B}_i$ is a Boolean algebra for every $i.
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  26. Jakob Kellner & Saharon Shelah (2010). A Sacks Real Out of Nowhere. Journal of Symbolic Logic 75 (1):51-76.
    There is a proper countable support iteration of length ω adding no new reals at finite stages and adding a Sacks real in the limit.
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  27. Andrzej Rosłanowski & Saharon Shelah (2010). Reasonable Ultrafilters, Again. 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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  28. Saharon Shelah (2010). On Long Increasing Chains Modulo Flat Ideals. Mathematical Logic Quarterly 56 (4):397-399.
    We prove that, e.g., in there is no sequence of length W4 increasing modulo the ideal of countable sets.
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  29. Saharon Shelah & Lutz Strüngmann (2010). Filtration-Equivalent ℵ1-Separable Abelian Groups of Cardinality ℵ1. Annals of Pure and Applied Logic 161 (7):935-943.
    We show that it is consistent with ordinary set theory ZFC and the generalized continuum hypothesis that there exist two 1-separable abelian groups of cardinality 1 which are filtration-equivalent and one is a Whitehead group but the other is not. This solves one of the open problems from Eklof and Mekler [2].
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  30. John T. Baldwin, Alexei Kolesnikov & Saharon Shelah (2009). The Amalgamation Spectrum. Journal of Symbolic Logic 74 (3):914-928.
    We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals. Theorem A For every natural number k, there is a class $K_k $ defined by a sentence in $L_{\omega 1.\omega } $ that has no models of cardinality greater than $ \supset _{k - 1} $ , but $K_k $ has the disjoint amalgamation property on models of cardinality less than or equal to $\mathfrak{N}_{k - 3} $ and has models of cardinality $\mathfrak{N}_{k (...)
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  31. Todd Eisworth & Saharon Shelah (2009). Successors of Singular Cardinals and Coloring Theorems II. Journal of Symbolic Logic 74 (4):1287 - 1309.
    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.
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  32. Itay Kaplan & Saharon Shelah (2009). The Automorphism Tower of a Centerless Group Without Choice. Archive for Mathematical Logic 48 (8):799-815.
    For a centerless group G, we can define its automorphism tower. We define G α : G 0 = G, G α+1 = Aut(G α ) and for limit ordinals ${G^{\delta}=\bigcup_{\alpha<\delta}G^{\alpha}}$ . Let τ 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 (using Fodor’s lemma), we have here a more direct proof with little set theory. However, set theoretically we get a parallel theorem without the (...)
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  33. Jakob Kellner & Saharon Shelah (2009). Decisive Creatures and Large Continuum. 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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  34. Paul Larson & Saharon Shelah (2009). Splitting Stationary Sets From Weak Forms of Choice. Mathematical Logic Quarterly 55 (3):299-306.
    Working in the context of restricted forms of the Axiom of Choice, we consider the problem of splitting the ordinals below λ of cofinality θ into λ many stationary sets, where θ < λ are regular cardinals. This is a continuation of [4].
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  35. John T. Baldwin & Saharon Shelah (2008). Examples of Non-Locality. 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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  36. Stefan Geschke & Saharon Shelah (2008). The Number of Openly Generated Boolean Algebras. Journal of Symbolic Logic 73 (1):151-164.
    This article is devoted to two different generalizations of projective Boolean algebras: openly generated Boolean algebras and tightly ϭ-filtered Boolean algebras. We show that for every uncountable regular cardinal κ there are 2κ pairwise non-isomorphic openly generated Boolean algebras of size κ > N1 provided there is an almost free non-free abelian group of size κ. The openly generated Boolean algebras constructed here are almost free. Moreover, for every infinite regular cardinal κ we construct 2κ pairwise non-isomorphic Boolean algebras of (...)
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  37. Juliette Kennedy, Saharon Shelah & Jouko Väänänen (2008). Regular Ultrafilters and Finite Square Principles. Journal of Symbolic Logic 73 (3):817-823.
    We show that many singular cardinals λ above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle $\square _{\lambda ,D}^{\mathit{fin}}$ introduced in [3]. For such ultrafilters D and cardinals λ there are models of size λ for which Mλ / D is not λ⁺⁺-universal and elementarily equivalent models M and N of size λ for which Mλ / D and Nλ / D are non-isomorphic. The question of the existence of such ultrafilters and models was (...)
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  38. Paul B. Larson & Saharon Shelah (2008). The Stationary Set Splitting Game. Mathematical Logic Quarterly 54 (2):187-193.
    The stationary set splitting game is a game of perfect information of length ω1 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 Σ22 maximality with a predicate for the nonstationary ideal on ω1, and an example of a consistently (...)
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  39. Andrzej Roslanowski & Saharon Shelah (2008). Generating Ultrafilters in a Reasonable Way. Mathematical Logic Quarterly 54 (2):202-220.
    We continue investigations of reasonable ultrafilters on uncountable cardinals defined in Shelah [8]. We introduce a general scheme of generating a filter on λ from filters on smaller sets and we investigate the combinatorics of objects obtained this way.
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  40. Saharon Shelah (2008). Groupwise Density Cannot Be Much Bigger Than the Unbounded Number. Mathematical Logic Quarterly 54 (4):340-344.
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  41. Saharon Shelah & Alexander Usvyatsov (2008). More on SOP 1 and SOP 2. 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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  42. Shimon Garti & Saharon Shelah (2007). Two Cardinal Models for Singular Μ. Mathematical Logic Quarterly 53 (6):636-641.
    We deal here with colorings of the pair , when μ is a strong limit and singular cardinal. We show that there exists a coloring c with no refinement. It follows that the properties of colorings of when μ is singular differ in an essential way from the case of regular μ.
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  43. Chanoch Havlin & Saharon Shelah (2007). Existence of EF-Equivalent Non-Isomorphic Models. Mathematical Logic Quarterly 53 (2):111-127.
    We prove the existence of pairs of models of the same cardinality which are very equivalent according to EF games, but not isomorphic. We continue the paper [4], but we do not rely on it. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).
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  44. Jakob Kellner, Matti Pauna & Saharon Shelah (2007). Winning the Pressing Down Game but Not Banach-Mazur. Journal of Symbolic Logic 72 (4):1323 - 1335.
    Let S be the set of those α ∈ ω₂ that have cofinality ω₁. It is consistent relative to a measurable that the nonempty player wins the pressing down game of length ω₁, but not the Banach-Mazur game of length ω + 1 (both games starting with S).
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  45. Heike Mildenberger & Saharon Shelah (2007). Increasing the Groupwise Density Number by Ccc Forcing. Annals of Pure and Applied Logic 149 (1):7-13.
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  46. Andrzej Rosłanowski & Saharon Shelah (2007). Universal Forcing Notions and Ideals. Archive for Mathematical Logic 46 (3-4):179-196.
    Our main result states that 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 σ-ideals determined by those universality parameters.
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  47. Saharon Shelah (2007). Power Set Modulo Small, the Singular of Uncountable Cofinality. Journal of Symbolic Logic 72 (1):226 - 242.
    Let μ be singular of uncountable cofinality. If μ > 2cf(μ), we prove that in P = ([μ]μ, ⊇) as a forcing notion we have a natural complete embedding of Levy (‮א‬₀, μ⁺) (so P collapses μ⁺ to ‮א‬₀) and even Levy ($(\aleph _{0},U_{J_{\kappa}^{{\rm bd}}}(\mu))$). The "natural" means that the forcing ({p ∈ [μ]μ: p closed}, ⊇) is naturally embedded and is equivalent to the Levy algebra. Also if P fails the χ-c.c. then it collapses χ to ‮א‬₀ (and the (...)
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  48. Saharon Shelah & Mor Doron (2007). Relational Structures Constructible by Quantifier Free Definable Operations. Journal of Symbolic Logic 72 (4):1283 - 1298.
    We consider the notion of bounded m-ary patch-width defined in [9], and its very close relative m-constructibility defined below. We show that the notions of m-constructibility all coincide for m ≥ 3, while 1-constructibility is a weaker notion. The same holds for bounded m-ary patch-width. The case m = 2 is left open.
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  49. Mirna Džamonja & Saharon Shelah (2006). On Properties of Theories Which Preclude the Existence of Universal Models. Annals of Pure and Applied Logic 139 (1):280-302.
    We introduce the oak property of first order theories, which is a syntactical condition that we show to be sufficient for a theory not to have universal models in cardinality λ when certain cardinal arithmetic assumptions about λ implying the failure of GCH hold. We give two examples of theories that have the oak property and show that none of these examples satisfy SOP4, not even SOP3. This is related to the question of the connection of the property SOP4 to (...)
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