On categoricity in successive cardinals

Abstract

We investigate, in ZFC, the behavior of abstract elementary classes (AECs) categorical in many successive small cardinals. We prove for example that a universal ${\mathbb{L}}_{{\omega }_1,\omega }$ sentence categorical on an end segment of cardinals below ${\beth }_{\omega }$ must be categorical also everywhere above ${\beth }_{\omega }$. This is done without any additional model-theoretic hypotheses (such as amalgamation or arbitrarily large models) and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.

19 pages