## Journal of Symbolic Logic

### Shelah’s categoricity conjecture from a successor for tame abstract elementary classes

#### 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)⁺.

#### Article information

Source
J. Symbolic Logic, Volume 71, Issue 2 (2006), 553-568.

Dates
First available in Project Euclid: 2 May 2006

https://projecteuclid.org/euclid.jsl/1146620158

Digital Object Identifier
doi:10.2178/jsl/1146620158

Mathematical Reviews number (MathSciNet)
MR2225893

Zentralblatt MATH identifier
1100.03023

#### Citation

Grossberg, Rami; VanDieren, Monica. Shelah’s categoricity conjecture from a successor for tame abstract elementary classes. J. Symbolic Logic 71 (2006), no. 2, 553--568. doi:10.2178/jsl/1146620158. https://projecteuclid.org/euclid.jsl/1146620158

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