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June 2006 Shelah’s categoricity conjecture from a successor for tame abstract elementary classes
Rami Grossberg, Monica VanDieren
J. Symbolic Logic 71(2): 553-568 (June 2006). DOI: 10.2178/jsl/1146620158

Abstract

We prove a categoricity transfer theorem for tame abstract elementary classes.

Theorem. 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. 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)⁺.

Citation

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Rami Grossberg. Monica VanDieren. "Shelah’s categoricity conjecture from a successor for tame abstract elementary classes." J. Symbolic Logic 71 (2) 553 - 568, June 2006. https://doi.org/10.2178/jsl/1146620158

Information

Published: June 2006
First available in Project Euclid: 2 May 2006

zbMATH: 1100.03023
MathSciNet: MR2225893
Digital Object Identifier: 10.2178/jsl/1146620158

Subjects:
Primary: 03C45, 03C52, 03C75
Secondary: 03C05, 03C55, 03C95

Rights: Copyright © 2006 Association for Symbolic Logic

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Vol.71 • No. 2 • June 2006
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