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Shelah’s categoricity conjecture from a successor for tame abstract elementary classes. (English) Zbl 1100.03023
Summary: We prove a categoricity transfer theorem for tame abstract elementary classes.
Theorem 0.1. Suppose that \({\mathcal K}\) is a \(\chi\)-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let \(\lambda \geq\text{Max}\{\chi,\text{LS} ({\mathcal K})^+\}\). If \({\mathcal K}\) is categorical in \(\lambda\) and \(\lambda^+\), then \({\mathcal K}\) is categorical in \(\lambda^{++}\).
Combining this theorem with some results from [S. Shelah, Ann. Pure Appl. Logic 98, 261–294 (1999; Zbl 0945.03049)], we derive a form of Shelah’s categoricity conjecture for tame abstract elementary classes:
Corollary 0.2. Suppose \({\mathcal K}\) is a \(\chi\)-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let \(\mu_0:=\text{Hanf}({\mathcal K})\). If \(\chi\leq \beth_{(2^{\mu_0})^+}\) and \({\mathcal K}\) is categorical in some \(\lambda^+> \beth_{(2^{\mu_0})^+}\), then \({\mathcal K}\) is categorical in \(\mu\) for all \(\mu> \beth_{(2^{\mu_0})^+}\).

MSC:
03C45 Classification theory, stability, and related concepts in model theory
03C35 Categoricity and completeness of theories
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