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

Abstract

Abstract We prove a categoricity transfer theorem for tame abstract elementary classes. 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: Suppose K is χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ 0 ≔ Hanf( K ). If and K is categorical in some then K is categorical in μ for all μ .