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Shelah Saharon - One of the best experts on this subject based on the ideXlab platform.
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Categoricity and multidimensional diagrams
2018Co-Authors: Shelah Saharon, Vasey SebastienAbstract:We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high-enough cardinality will have a single model in any high-enough cardinal. Assuming a weak version of the Generalized Continuum Hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation.Comment: 63 page
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There is no bound on sizes of indecomposable Banach spaces
2016Co-Authors: Koszmider Piotr, Shelah Saharon, Świȩtek MichałAbstract: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 indecomposable 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
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Filtration-equivalent ℵ1-separable abelian groups of cardinality ℵ1
Elsevier B.V., 2010Co-Authors: Shelah Saharon, Strüngmann LutzAbstract:AbstractWe show that it is consistent with ordinary set theory ZFC and the Generalized Continuum Hypothesis that there exist two ℵ1-separable abelian groups of cardinality ℵ1 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 (2002) [2]
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Filtration equivalent aleph_1-separable abelian groups of cardinality aleph_1
2006Co-Authors: Shelah Saharon, Strüngmann LutzAbstract:We show that it is consistent with ordinary set theory ZFC and the Generalized Continuum Hypothesis that there exist two 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
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Localizations of Groups
2000Co-Authors: Göbel Rüdiger, Shelah SaharonAbstract:A group homomorphism eta:A-> H is called a localization of A if every homomorphism phi:A-> H can be `extended uniquely' to a homomorphism Phi:H-> H in the sense that Phi eta = phi. 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-> 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-> SO_{n-1}(R) for a natural embedding of the alternating group A_n is a localization if n even and n >= 10 . Answering an immediate question by Dror Farjoun and assuming the Generalized Continuum Hypothesis GCH we recently showed in math.LO/9912191 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
Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.
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Bext 2 (G, T) CAN BE NONTRIVIAL, EVEN ASSUMING GCH
2011Co-Authors: Menachem Magidor, Saharon ShelahAbstract: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 ℵω+1 and for some torsion group T Bext 2 (G, T) ̸ = 0. Hence G.C.H. is not sufficient for getting the results of [10]. 514 revision:1994-05-23 modified:1994-05-23 1
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Filtration-equivalent ℵ1-separable abelian groups of cardinality ℵ1
2011Co-Authors: Saharon ShelahAbstract:and Lutz Strüngmann Abstract. We show that it is consistent with ordinary set theory ZFC and the Generalized Continuum Hypothesis that there exist two ℵ1-separable abelian groups of cardinality ℵ1 which are filtration-equivalent and one is a Whitehead group but the other is not. This solves one of the open problems from [EkMe]
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CONSTRUCTING SIMPLE GROUPS FOR LOCALIZATIONS
2001Co-Authors: Rüdiger Göbel, Saharon ShelahAbstract:A group homomorphism η: A → H is called a localization of A if every homomorphism ϕ: A → H can be ‘extended uniquely ’ to a homomorphism Φ: H → H in the sense that Φη = ϕ. This categorical concept, 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, see the introduction. For localizations η: A → 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 739 revision:2001-02-20 modified:2001-02-23 showed that An → SOn−1(R) for a natural embedding of the alternating group An is a localization if n is even and n ≥ 10. Answering an immediate question by Dror Farjoun and assuming the Generalized Continuum Hypothesis GCH we recently showed in [12] 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. The new techniques exploit abelian centralizers of free (non-abelian) subgroups of H which constitute a rigid system of cotorsion-free abelian groups. A known strong theorem on the existence of such abelian groups turns out to be very helpful, see [5]. Like [12], this shows (now in ZFC) that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg–Mac Lane space K(A, 1) for many groups A. The Main Theorem 1.3 is also used to answer a question by Philip Hall in [13]
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LOCALIZATIONS OF GROUPS
2000Co-Authors: Rüdiger Göbel, Saharon ShelahAbstract:A group homomorphism η: A → H is called a localization of A if every homomorphism ϕ: A → H can be ‘extended uniquely ’ to a homomorphism Φ: H → H in the sense that Φη = ϕ. 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, see the introduction. For localizations η: A → 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 An → SOn−1(R) for a natural embedding of the alternating group An is a localization if n even and n ≥ 10. Answering an immediate question by Dror Farjoun and assuming the Generalized Continuum Hypothesis GCH we recently showed in [12] 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. The new techniques exploit abelian centralizers of free (non-abelian) subgroups of H which constitute a rigid system of cotorsion-free abelian groups. A known strong theorem on the existence of such abelian groups turns out to be very helpful, see [5]. Like [12], this shows (now in ZFC) that there is a proper class of distinct homotopy types which are localizations of a given Eilenberg–Mac Lane space K(A, 1) for many groups A. The Main Theorem 1.3 is also used to answer a question by Philip Hall in [13]
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bext 2 g t can be nontrivial even assuming gch
arXiv: Logic, 1994Co-Authors: Menachem Magidor, Saharon ShelahAbstract: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 ℵ!+1 and for some torsion group T Bext 2 (G,T) 6 0. Hence G.C.H. is not sufficient for getting the results of (10).
Vasey Sebastien - One of the best experts on this subject based on the ideXlab platform.
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The categoricity spectrum of large abstract elementary classes
2019Co-Authors: Vasey SebastienAbstract:The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the Generalized Continuum Hypothesis, GCH), we give a complete list of the possible categoricity spectrums of an abstract elementary class with amalgamation and arbitrarily large models. Specifically, the categoricity spectrum is either empty, an end segment starting below the Hanf number, or a closed interval consisting of finite successors of the L\"owenheim-Skolem-Tarski number (there are examples of each type). We also prove (assuming a strengthening of the GCH) that the categoricity spectrum of an abstract elementary class with no maximal models is either bounded or contains an end segment. This answers several longstanding questions around Shelah's categoricity conjecture.Comment: 50 page
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Categoricity and multidimensional diagrams
2018Co-Authors: Shelah Saharon, Vasey SebastienAbstract:We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high-enough cardinality will have a single model in any high-enough cardinal. Assuming a weak version of the Generalized Continuum Hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation.Comment: 63 page
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Downward categoricity from a successor inside a good frame
'Elsevier BV', 2016Co-Authors: Vasey SebastienAbstract:We use orthogonality calculus to prove a downward transfer from categoricity in a successor in abstract elementary classes (AECs) that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals: $\mathbf{Theorem}$ Let $K$ be an AEC and let $\text{LS} (K) \le \lambda < \theta$ be cardinals. If $K$ has a type-full good $[\lambda, \theta]$-frame and $K$ is categorical in both $\lambda$ and $\theta^+$, then $K$ is categorical in all $\lambda' \in [\lambda, \theta]$. We deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from $\beth_{\beth_{\left(2^{\text{LS} (K)}\right)^+}}$ to $\beth_{\left(2^{\text{LS} (K)}\right)^+}$ assuming that the AEC is $\text{LS} (K)$-tame. The successor Hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form $M \cup \{a\}$ or (using an unpublished claim of Shelah) that the weak Generalized Continuum Hypothesis holds.Comment: 63 pages. Was previously named "A downward categoricity transfer for tame abstract elementary classes
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Building independence relations in abstract elementary classes
'Elsevier BV', 2016Co-Authors: Vasey SebastienAbstract:We study general methods to build forking-like notions in the framework of tame abstract elementary classes (AECs) with amalgamation. We show that whenever such classes are categorical in a high-enough cardinal, they admit a good frame: a forking-like notion for types of singleton elements. $\mathbf{Theorem}$ (Superstability from categoricity) Let $K$ be a $( \text{LS} (K)$ and $K$ is categorical in a $\lambda > \kappa$, then: * $K$ is stable in all cardinals $\ge \kappa$. * $K$ is categorical in $\kappa$. * There is a type-full good $\lambda$-frame with underlying class $K_\lambda$. Under more locality conditions, we prove that the frame extends to a global independence notion (for types of arbitrary length). $\mathbf{Theorem}$ (A global independence notion from categoricity) Let $K$ be a densely type-local, fully tame and type short AEC with amalgamation. If $K$ is categorical in unboundedly many cardinals, then there exists $\lambda \ge \text{LS} (K)$ such that $K_{\ge \lambda}$ admits a global independence relation with the properties of forking in a superstable first-order theory. As an application, we deduce (modulo an unproven claim of Shelah) that Shelah's eventual categoricity conjecture for AECs (without assuming categoricity in a successor cardinal) follows from the weak Generalized Continuum Hypothesis and a large cardinal axiom. $\textbf{Corollary}$ Assume $2^{\lambda} < 2^{\lambda^+}$ for all cardinals $\lambda$, as well as an unpublished claim of Shelah. If there exists a proper class of strongly compact cardinals, then any AEC categorical in some high-enough cardinal is categorical in all high-enough cardinals.Comment: 96 pages. Was initially part of Infinitary stability theory (arXiv:1412.3313). Early versions were called "Independence in abstract elementary classes
Nambiar Kannan - One of the best experts on this subject based on the ideXlab platform.
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Generalized Continuum Hypothesis and the Axiom of Combinatorial Sets
2002Co-Authors: Nambiar KannanAbstract:Axiom of Combinatorial Sets is defined and used to derive Generalized Continuum Hypothesis
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Definition of Intuitive Set Theory
2001Co-Authors: Nambiar KannanAbstract:The two axioms which define intuitive set theory, Axiom of Combinatorial Sets and Axiom of Infinitesimals, are stated. Generalized Continuum Hypothesis is derived from the first axiom, and the infinitesimal is visualized using the latter
Tomita, Artur Hideyuki - One of the best experts on this subject based on the ideXlab platform.
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Abelian torsion groups with a countably compact group topology
Elsevier B.V., 2010Co-Authors: Castro-pereira Irene, Tomita, Artur HideyukiAbstract:AbstractComfort and Remus [W.W. Comfort, D. Remus, Abelian torsion groups with a pseudocompact group topology, Forum Math. 6 (3) (1994) 323–337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the Ulm–Kaplansky invariants.We show, under a condition weaker than the Generalized Continuum Hypothesis, that an Abelian torsion group (of any cardinality) admits a pseudocompact group topology if and only if it admits a countably compact group topology. Dikranjan and Tkachenko [D. Dikranjan, M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811–837], and Dikranjan and Shakhmatov [D. Dikranjan, D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on Abelian groups, Topology Appl. 151 (1–3) (2005) 2–54] showed this equivalence for groups of cardinality not greater than 2c.We also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality κω, for any infinite cardinal κ. In particular, it is consistent that for every cardinal κ there are countably compact groups without non-trivial convergent sequences whose weight λ has countable cofinality and λ>κ
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Abelian torsion groups with a countably compact group topology
ELSEVIER SCIENCE BV, 2010Co-Authors: Castro-pereira Irene, Tomita, Artur HideyukiAbstract:Comfort and Remus [W.W. Comfort, D. Remus, Abelian torsion groups with a pseudo-compact group topology, Forum Math. 6 (3) (1994) 323-337] characterized algebraically the Abelian torsion groups that admit a pseudocompact group topology using the Ulm-Kaplansky invariants. We show, under a condition weaker than the Generalized Continuum Hypothesis, that an Abelian torsion group (of any cardinality) admits a pseudocompact group topology if and only if it admits a countably compact group topology. Dikranjan and Tkachenko [D. Dikranjan. M. Tkachenko, Algebraic structure of small countably compact Abelian groups, Forum Math. 15 (6) (2003) 811-837], and Dikranjan and Shakhmatov [D. Dikranjan. D. Shakhmatov, Forcing hereditarily separable compact-like group topologies on Abelian groups, Topology Appl. 151 (1-3) (2005) 2-54] showed this equivalence for groups of cardinality not greater than 2(c). We also show, from the existence of a selective ultrafilter, that there are countably compact groups without non-trivial convergent sequences of cardinality kappa(omega), for any infinite cardinal kappa. In particular, it is consistent that for every cardinal kappa there are countably compact groups without non-trivial convergent sequences whose weight lambda has countable cofinality and lambda > kappa. (C) 2009 Elsevier B.V. All rights reserved