On Stability and Existence of Models in Abstract Elementary Classes

TL;DR

This paper establishes a deep connection between stability and the existence of models in larger cardinals within abstract elementary classes, under mild assumptions, and proves a categoricity theorem without requiring amalgamation or large models.

Contribution

It proves that stability in a cardinal is equivalent to the existence of a larger model under certain conditions and introduces a categoricity theorem without assuming amalgamation or large models.

Findings

Stability in  implies existence of models in b2.

Categoricity in two successive cardinals leads to stability and model existence.

New categoricity theorem for classes with weak amalgamation and tameness.

Abstract

For an abstract elementary class $\mathbf{K}$ and a cardinal $\lambda \geq LS(\mathbf{K})$, we prove under mild cardinal arithmetic assumptions, categoricity in two succesive cardinals, almost stability for $\lambda^+$-minimal types and continuity of splitting in $\lambda$, that stability in $\lambda$ is equivalent to the existence of a model in $\lambda^{++}$. The forward direction holds without any cardinal or categoricity assumptions, this result improves both [Vas18b, 12.1] and [MaYa24, 3.14]. Moreover, we prove a categoricity theorem for abstract elementary classes with weak amalgamation and tameness under mild structural assumptions in $\lambda$. A key feature of this result is that we do not assume amalgamation or arbitrarily large models.