Shelah-Villaveces revisited

TL;DR

This paper explores the conditions under which limit models are unique in abstract elementary classes, and establishes categoricity transfer results assuming certain set-theoretic principles and tameness.

Contribution

It revisits Shelah and Villaveces's work, proving new categoricity transfer theorems under assumptions like diamonds and tameness in AECs.

Findings

Categoricity in a specific cardinal implies uniqueness of limit models in subsequent cardinals.

Tameness plus categoricity in a proper class of cardinals leads to categoricity on a tail of cardinals.

First such categoricity transfer theorem in this context.

Abstract

We study uniqueness of limit models in abstract elementary classes (AECs) with no maximal models. We prove (assuming instances of diamonds) that categoricity in a cardinal of the form $\mu^{+(n + 1)}$ implies the uniqueness of limit models of cardinality $\mu^{+}, \mu^{++}, \ldots, \mu^{+n}$. This sheds light on a paper of Shelah and Villaveces, who were the first to consider uniqueness of limit models in this context. We also prove that (again assuming instances of diamonds) in an AEC with no maximal models, tameness (a locality property for types) together with categoricity in a proper class of cardinals imply categoricity on a tail of cardinals. This is the first categoricity transfer theorem in that setup and answers a question of Baldwin.