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  1. Limit Models in Metric Abstract Elementary Classes: The Categorical Case.Andrés Villaveces & Pedro Zambrano - 2016 - Mathematical Logic Quarterly 62 (4-5):319-334.
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  • Categoricity From One Successor Cardinal in Tame Abstract Elementary Classes.Rami Grossberg & Monica Vandieren - 2006 - Journal of Mathematical Logic 6 (2):181-201.
  • Infinitary Stability Theory.Sebastien Vasey - 2016 - Archive for Mathematical Logic 55 (3-4):567-592.
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  • Categoricity in Homogeneous Complete Metric Spaces.Åsa Hirvonen & Tapani Hyttinen - 2009 - Archive for Mathematical Logic 48 (3-4):269-322.
    We introduce a new approach to the model theory of metric structures by defining the notion of a metric abstract elementary class (MAEC) closely resembling the notion of an abstract elementary class. Further we define the framework of a homogeneous MAEC were we additionally assume the existence of arbitrarily large models, joint embedding, amalgamation, homogeneity and a property which we call the perturbation property. We also assume that the Löwenheim-Skolem number, which in this setting refers to the density character of (...)
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  • Galois-Stability for Tame Abstract Elementary Classes.Rami Grossberg & Monica Vandieren - 2006 - Journal of Mathematical Logic 6 (01):25-48.
  • Shelah's Categoricity Conjecture From a Successor for Tame Abstract Elementary Classes.Rami Grossberg & Monica Vandieren - 2006 - Journal of Symbolic Logic 71 (2):553 - 568.
    We prove a categoricity transfer theorem for tame abstract elementary classes. Theorem 0.1. Suppose that K is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ.LS(K)⁺}. If K is categorical in λ and λ⁺, then K is categorical in λ⁺⁺. Combining this theorem with some results from [37], we derive a form of Shelah's Categoricity Conjecture for tame abstract elementary classes: Corollary 0.2. Suppose K is a χ-tame (...)
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  • Compactness and Independence in Non First Order Frameworks.Itay Ben-Yaacov - 2005 - Bulletin of Symbolic Logic 11 (1):28-50.
    This communication deals with positive model theory, a non first order model theoretic setting which preserves compactness at the cost of giving up negation. Positive model theory deals transparently with hyperimaginaries, and accommodates various analytic structures which defy direct first order treatment. We describe the development of simplicity theory in this setting, and an application to the lovely pairs of models of simple theories without the weak non finite cover property.
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  • Toward a Stability Theory of Tame Abstract Elementary Classes.Sebastien Vasey - 2018 - Journal of Mathematical Logic 18 (2):1850009.
    We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness, and are stable in some cardinal. Assuming the singular cardinal hypothesis, we prove a full characterization of the stability cardinals, and connect the stability spectrum with the behavior of saturated models. We deduce that if a class is stable on a tail of cardinals, then it has no long splitting chains. This indicates that there is a clear notion of superstability in this framework. We also (...)
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  • Categoricity Transfer in Simple Finitary Abstract Elementary Classes.Tapani Hyttinen & Meeri Kesälä - 2011 - Journal of Symbolic Logic 76 (3):759 - 806.
    We continue our study of finitary abstract elementary classes, defined in [7]. In this paper, we prove a categoricity transfer theorem for a case of simple finitary AECs. We introduce the concepts of weak κ-categoricity and f-primary models to the framework of א₀-stable simple finitary AECs with the extension property, whereby we gain the following theorem: Let (, ≼ ) be a simple finitary AEC, weakly categorical in some uncountable κ. Then (, ≼ ) is weakly categorical in (...)
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  • 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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  • Ranks and Pregeometries in Finite Diagrams.Olivier Lessmann - 2000 - Annals of Pure and Applied Logic 106 (1-3):49-83.
    The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D. In this work, we introduce a natural dependence relation on the subsets of the models for the 0-stable case which share many of the formal properties of forking. This (...)
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  • The Hanf Number for Amalgamation of Coloring Classes.Alexei Kolesnikov & Chris Lambie-Hanson - 2016 - Journal of Symbolic Logic 81 (2):570-583.
  • Forking and Superstability in Tame Aecs.Sebastien Vasey - 2016 - Journal of Symbolic Logic 81 (1):357-383.
  • Equivalent Definitions of Superstability in Tame Abstract Elementary Classes.Rami Grossberg & Sebastien Vasey - 2017 - Journal of Symbolic Logic 82 (4):1387-1408.
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  • Tameness and Frames Revisited.Will Boney & Sebastien Vasey - 2017 - Journal of Symbolic Logic 82 (3):995-1021.
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  • 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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  • Categoricity in Abstract Elementary Classes with No Maximal Models.Monica VanDieren - 2006 - Annals of Pure and Applied Logic 141 (1):108-147.
    The results in this paper are in a context of abstract elementary classes identified by Shelah and Villaveces in which the amalgamation property is not assumed. The long-term goal is to solve Shelah’s Categoricity Conjecture in this context. Here we tackle a problem of Shelah and Villaveces by proving that in their context, the uniqueness of limit models follows from categoricity under the assumption that the subclass of amalgamation bases is closed under unions of bounded, -increasing chains.
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  • A Topology for Galois Types in Abstract Elementary Classes.Michael Lieberman - 2011 - Mathematical Logic Quarterly 57 (2):204-216.
    We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
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  • Around Independence and Domination in Metric Abstract Elementary Classes: Assuming Uniqueness of Limit Models.Andrés Villaveces & Pedro Zambrano - 2014 - Mathematical Logic Quarterly 60 (3):211-227.
  • A Presentation Theorem for Continuous Logic and Metric Abstract Elementary Classes.Will Boney - 2017 - Mathematical Logic Quarterly 63 (5):397-414.
    In recent years, model theory has widened its scope to include metric structures by considering real-valued models whose underlying set is a complete metric space. We show that it is possible to carry out this work by giving presentation theorems that translate the two main frameworks into discrete settings. We also translate various notions of classification theory.
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  • Building Independence Relations in Abstract Elementary Classes.Sebastien Vasey - 2016 - Annals of Pure and Applied Logic 167 (11):1029-1092.
  • Saturation and Solvability in Abstract Elementary Classes with Amalgamation.Sebastien Vasey - 2017 - Archive for Mathematical Logic 56 (5-6):671-690.
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  • As an Abstract Elementary Class.John T. Baldwin, Paul C. Eklof & Jan Trlifaj - 2007 - Annals of Pure and Applied Logic 149 (1):25-39.
    In this paper we study abstract elementary classes of modules. We give several characterizations of when the class of modules A with is abstract elementary class with respect to the notion that M1 is a strong submodel M2 if the quotient remains in the given class.
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  • On the Uniqueness Property of Forking in Abstract Elementary Classes.Sebastien Vasey - 2017 - Mathematical Logic Quarterly 63 (6):598-604.
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  • Independence in Finitary Abstract Elementary Classes.Tapani Hyttinen & Meeri Kesälä - 2006 - Annals of Pure and Applied Logic 143 (1):103-138.
    In this paper we study a specific subclass of abstract elementary classes. We construct a notion of independence for these AEC’s and show that under simplicity the notion has all the usual properties of first order non-forking over complete types. Our approach generalizes the context of 0-stable homogeneous classes and excellent classes. Our set of assumptions follow from disjoint amalgamation, existence of a prime model over 0/, Löwenheim–Skolem number being ω, -tameness and a property we call finite character. We also (...)
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  • Forking in Short and Tame Abstract Elementary Classes.Will Boney & Rami Grossberg - 2017 - Annals of Pure and Applied Logic 168 (8):1517-1551.
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  • Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes.M. M. VanDieren - 2016 - Annals of Pure and Applied Logic 167 (4):395-407.
  • Uncountable Categoricity of Local Abstract Elementary Classes with Amalgamation.John T. Baldwin & Olivier Lessmann - 2006 - Annals of Pure and Applied Logic 143 (1):29-42.
    We give a complete and elementary proof of the following upward categoricity theorem: let be a local abstract elementary class with amalgamation and joint embedding, arbitrarily large models, and countable Löwenheim–Skolem number. If is categorical in 1 then is categorical in every uncountable cardinal. In particular, this provides a new proof of the upward part of Morley’s theorem in first order logic without any use of prime models or heavy stability theoretic machinery.
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  • The Amalgamation Spectrum.John T. Baldwin, Alexei Kolesnikov & Saharon Shelah - 2009 - 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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  • Superstability From Categoricity in Abstract Elementary Classes.Will Boney, Rami Grossberg, Monica M. VanDieren & Sebastien Vasey - 2017 - Annals of Pure and Applied Logic 168 (7):1383-1395.
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  • Good Frames in the Hart–Shelah Example.Will Boney & Sebastien Vasey - 2018 - Archive for Mathematical Logic 57 (5-6):687-712.
    For a fixed natural number \, the Hart–Shelah example is an abstract elementary class with amalgamation that is categorical exactly in the infinite cardinals less than or equal to \. We investigate recently-isolated properties of AECs in the setting of this example. We isolate the exact amount of type-shortness holding in the example and show that it has a type-full good \-frame which fails the existence property for uniqueness triples. This gives the first example of such a frame. Along the (...)
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  • Categoricity and Universal Classes.Tapani Hyttinen & Kaisa Kangas - 2018 - Mathematical Logic Quarterly 64 (6):464-477.
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  • Abstract Elementary Classes and Infinitary Logics.David W. Kueker - 2008 - Annals of Pure and Applied Logic 156 (2):274-286.
    In this paper we study abstract elementary classes using infinitary logics and prove a number of results relating them. For example, if is an a.e.c. with Löwenheim–Skolem number κ then is closed under L∞,κ+-elementary equivalence. If κ=ω and has finite character then is closed under L∞,ω-elementary equivalence. Analogous results are established for . Galois types, saturation, and categoricity are also studied. We prove, for example, that if is finitary and λ-categorical for some infinite λ then there is some σLω1,ω such (...)
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  • Downward Categoricity From a Successor Inside a Good Frame.Sebastien Vasey - 2017 - Annals of Pure and Applied Logic 168 (3):651-692.
  • Shelah's Eventual Categoricity Conjecture in Universal Classes: Part I.Sebastien Vasey - 2017 - Annals of Pure and Applied Logic 168 (9):1609-1642.
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  • Superstability and Symmetry.Monica M. VanDieren - 2016 - Annals of Pure and Applied Logic 167 (12):1171-1183.
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  • Canonical Forking in AECs.Will Boney, Rami Grossberg, Alexei Kolesnikov & Sebastien Vasey - 2016 - Annals of Pure and Applied Logic 167 (7):590-613.
  • Shelah's Eventual Categoricity Conjecture in Tame Abstract Elementary Classes with Primes.Sebastien Vasey - 2018 - Mathematical Logic Quarterly 64 (1-2):25-36.
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  • Uniqueness of Limit Models in Classes with Amalgamation.Rami Grossberg, Monica VanDieren & Andrés Villaveces - 2016 - Mathematical Logic Quarterly 62 (4-5):367-382.
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  • Tameness and Extending Frames.Will Boney - 2014 - Journal of Mathematical Logic 14 (2):1450007.