On categoricity in successive cardinals

TL;DR

This paper explores the behavior of abstract elementary classes (AECs) in ZFC that are categorical across many successive small cardinals, establishing broad categoricity transfer results without extra assumptions.

Contribution

It proves that a universal L_{ω_1, ω} sentence categorical on an end segment below beth_ω is also categorical above beth_ω, generalizing to tame AECs with weak amalgamation.

Findings

Categoricity in many small cardinals implies categoricity above beth_ω.

Results hold without additional model-theoretic hypotheses.

Generalizes to tame AECs with weak amalgamation.

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.