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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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  • 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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  • Galois-Stability for Tame Abstract Elementary Classes.Rami Grossberg & Monica Vandieren - 2006 - Journal of Mathematical Logic 6 (01):25-48.
  • 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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  • Infinitary Stability Theory.Sebastien Vasey - 2016 - Archive for Mathematical Logic 55 (3-4):567-592.
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  • The Spectrum of Resplendency.John T. Baldwin - 1990 - Journal of Symbolic Logic 55 (2):626-636.
    Let T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2 λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least $\min(2^\lambda,\beth_2)$ resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, (...)
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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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  • Upward Categoricity From a Successor Cardinal for Tame Abstract Classes with Amalgamation.Olivier Lessmann - 2005 - Journal of Symbolic Logic 70 (2):639 - 660.
    This paper is devoted to the proof of the following upward categoricity theorem: Let K be a tame abstract elementary class with amalgamation, arbitrarily large models, and countable Löwenheim-Skolem number. If K is categorical in ‮א‬₁ then K is categorical in every uncountable cardinal. More generally, we prove that if K is categorical in a successor cardinal λ⁺ then K is categorical everywhere above λ⁺.
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  • Upward Categoricity From a Successor Cardinal for Tame Abstract Classes with Amalgamation.Olivier Lessmann - 2005 - Journal of Symbolic Logic 70 (2):639-660.
    This paper is devoted to the proof of the following upward categoricity theorem: Let.
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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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  • 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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  • 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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  • 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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  • 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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  • 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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  • 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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  • 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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  • The Primal Framework II: Smoothness.J. T. Baldwin & S. Shelah - 1991 - Annals of Pure and Applied Logic 55 (1):1-34.
    Let be a class of models with a notion of ‘strong’ submodel and of canonically prime model over an increasing chain. We show under appropriate set-theoretic hypotheses that if K is not smooth , then K has many models in certain cardinalities. On the other hand, if K is smooth, we show that in reasonable cardinalities K has a unique homogeneous-universal model. In this situation we introduce the notion of type and prove the equivalence of saturated with homogeneous-universal.
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  • Fundamentals of Forking.Victor Harnik & Leo Harrington - 1984 - Annals of Pure and Applied Logic 26 (3):245-286.
  • 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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  • Building Prime Models in Fully Good Abstract Elementary Classes.Sebastien Vasey - 2017 - Mathematical Logic Quarterly 63 (3-4):193-201.
    We show how to build prime models in classes of saturated models of abstract elementary classes having a well-behaved independence relation: Let math formula be an almost fully good AEC that is categorical in math formula and has the math formula-existence property for domination triples. For any math formula, the class of Galois saturated models of math formula of size λ has prime models over every set of the form math formula. This generalizes an argument of Shelah, who proved the (...)
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  • 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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  • 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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  • A New Proof of a Theorem of Shelah.John W. Rosenthal - 1972 - Journal of Symbolic Logic 37 (1):133-134.
  • Categoricity and Universal Classes.Tapani Hyttinen & Kaisa Kangas - 2018 - Mathematical Logic Quarterly 64 (6):464-477.
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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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  • 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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  • Model Theory Without Choice? Categoricity.Saharon Shelan - 2009 - Journal of Symbolic Logic 74 (2):361-401.
    We prove Łos conjecture = Morley theorem in ZF, with the same characterization, i.e., of first order countable theories categorical in $N_\alpha $ for some (equivalently for every ordinal) α > 0. Another central result here in this context is: the number of models of a countable first order T of cardinality $N_\alpha $ is either ≥ |α| for every α or it has a small upper bound (independent of α close to Ð₂).
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