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

Rami Grossberg and Monica Vandieren
The Journal of Symbolic Logic
Vol. 71, No. 2 (Jun., 2006), pp. 553-568
Stable URL: http://www.jstor.org/stable/27588465
Page Count: 16
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Shelah's Categoricity Conjecture from a Successor for Tame Abstract Elementary Classes
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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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