Shelah's Categoricity Conjecture from a Successor for Tame Abstract Elementary Classes

Journal of Symbolic Logic 71 (2):553 - 568 (2006)

Abstract
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 abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ₀:= Hanf(K). If χ ≤ ‮ב‬(2μ0)+ and K is categorical in some λ⁺ > ‮ב‬(2μ0)+, then K is categorical in μ for all μ > ‮ב‬(2μ0)+
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DOI 10.2178/jsl/1146620158
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References found in this work BETA

Categoricity for Abstract Classes with Amalgamation.Saharon Shelah - 1999 - Annals of Pure and Applied Logic 98 (1-3):261-294.
Positive Model Theory and Compact Abstract Theories.Itay Ben-Yaacov - 2003 - Journal of Mathematical Logic 3 (01):85-118.
Toward Categoricity for Classes with No Maximal Models.Saharon Shelah & Andrés Villaveces - 1999 - Annals of Pure and Applied Logic 97 (1-3):1-25.
Finite Diagrams Stable in Power.Saharon Shelah - 1970 - Annals of Pure and Applied Logic 2 (1):69.
Strong Splitting in Stable Homogeneous Models.Tapani Hyttinen & Saharon Shelah - 2000 - Annals of Pure and Applied Logic 103 (1-3):201-228.

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Citations of this work BETA

Building Independence Relations in Abstract Elementary Classes.Sebastien Vasey - 2016 - Annals of Pure and Applied Logic 167 (11):1029-1092.
Shelah's Eventual Categoricity Conjecture in Universal Classes: Part I.Sebastien Vasey - 2017 - Annals of Pure and Applied Logic 168 (9):1609-1642.
Superstability and Symmetry.Monica M. VanDieren - 2016 - Annals of Pure and Applied Logic 167 (12):1171-1183.
Downward Categoricity From a Successor Inside a Good Frame.Sebastien Vasey - 2017 - Annals of Pure and Applied Logic 168 (3):651-692.

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