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  1. Categoricity and U-Rank in Excellent Classes.Olivier Lessmann - 2003 - Journal of Symbolic Logic 68 (4):1317-1336.
    Let K be the class of atomic models of a countable first order theory. We prove that if K is excellent and categorical in some uncountable cardinal, then each model is prime and minimal over the basis of a definable pregeometry given by a quasiminimal set. This implies that K is categorical in all uncountable cardinals. We also introduce a U-rank to measure the complexity of complete types over models. We prove that the U-rank has the usual additivity properties, that (...)
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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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  • A Rank for the Class of Elementary Submodels of a Superstable Homogeneous Model.Tapani Hyttinen & Olivier Lessmann - 2002 - Journal of Symbolic Logic 67 (4):1469-1482.
    We study the class of elementary submodels of a large superstable homogeneous model. We introduce a rank which is bounded in the superstable case, and use it to define a dependence relation which shares many (but not all) of the properties of forking in the first order case. The main difference is that we do not have extension over all sets. We also present an example of Shelah showing that extension over all sets may not hold for any dependence relation (...)
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  • Interpreting Groups and Fields in Some Nonelementary Classes.Tapani Hyttinen, Olivier Lessmann & Saharon Shelah - 2005 - Journal of Mathematical Logic 5 (1):1-47.
  • 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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  • 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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  • Potential Isomorphism of Elementary Substructures of a Strictly Stable Homogeneous Model.Sy-David Friedman, Tapani Hyttinen & Agatha C. Walczak-Typke - 2011 - Journal of Symbolic Logic 76 (3):987 - 1004.
    The results herein form part of a larger project to characterize the classification properties of the class of submodels of a homogeneous stable diagram in terms of the solvability (in the sense of [1]) of the potential isomorphism problem for this class of submodels. We restrict ourselves to locally saturated submodels of the monster model m of some power π. We assume that in Gödel's constructible universe , π is a regular cardinal at least the successor of the first cardinal (...)
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  • Main Gap for Locally Saturated Elementary Submodels of a Homogeneous Structure.Tapani Hyttinen & Saharon Shelah - 2001 - Journal of Symbolic Logic 66 (3):1286-1302.
    We prove a main gap theorem for locally saturated submodels of a homogeneous structure. We also study the number of locally saturated models, which are not elementarily embeddable into each other.
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  • Measuring Dependence in Metric Abstract Elementary Classes with Perturbations.Åsa Hirvonen & Tapani Hyttinen - 2017 - Journal of Symbolic Logic 82 (4):1199-1228.
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  • Simplicity and Uncountable Categoricity in Excellent Classes.Tapani Hyttinen & Olivier Lessmann - 2006 - Annals of Pure and Applied Logic 139 (1):110-137.
    We introduce Lascar strong types in excellent classes and prove that they coincide with the orbits of the group generated by automorphisms fixing a model. We define a new independence relation using Lascar strong types and show that it is well-behaved over models, as well as over finite sets. We then develop simplicity and show that, under simplicity, the independence relation satisfies all the properties of nonforking in a stable first order theory. Further, simplicity for an excellent class, as well (...)
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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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  • Notes on Quasiminimality and Excellence.John T. Baldwin - 2004 - Bulletin of Symbolic Logic 10 (3):334-366.
    This paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for L ω 1 ,ω (Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) (...)
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  • Exponentially closed fields and the conjecture on intersections with tori.Jonathan Kirby & Boris Zilber - 2014 - Annals of Pure and Applied Logic 165 (11):1680-1706.
    We give an axiomatization of the class ECF of exponentially closed fields, which includes the pseudo-exponential fields previously introduced by the second author, and show that it is superstable over its interpretation of arithmetic. Furthermore, ECF is exactly the elementary class of the pseudo-exponential fields if and only if the Diophantine conjecture CIT on atypical intersections of tori with subvarieties is true.
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  • Categoricity and Universal Classes.Tapani Hyttinen & Kaisa Kangas - 2018 - Mathematical Logic Quarterly 64 (6):464-477.
    No categories
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  • 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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