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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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