Shelah’s eventual categoricity conjecture in universal classes: part II

Abstract

We prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the categoricity cardinal to be a successor, do not assume amalgamation, and do not use large cardinals. Moreover we give an explicit bound on the “high-enough” threshold:

Theorem 0.1 Let $\psi$ be a universal $\mathbb {L}_{\omega _1, \omega }$ sentence (in a countable vocabulary). If $\psi$ is categorical in some $\lambda \ge \beth _{\beth _{\omega _1}}$, then $\psi$ is categorical in all $\lambda ' \ge \beth _{\beth _{\omega _1}}$.

As a byproduct of the proof, we show that a conjecture of Grossberg holds in universal classes:

Corollary 0.2 Let $\psi$ be a universal $\mathbb {L}_{\omega _1, \omega }$ sentence (in a countable vocabulary) that is categorical in some $\lambda \ge \beth _{\beth _{\omega _1}}$, then the class of models of $\psi$ has the amalgamation property for models of size at least $\beth _{\beth _{\omega _1}}$.

We also establish generalizations of these two results to uncountable languages. As part of the argument, we develop machinery to transfer model-theoretic properties between two different classes satisfying a compatibility condition (agreeing on any sufficiently large cardinals in which either is categorical). This is used as a bridge between Shelah’s milestone study of universal classes (which we use extensively) and a categoricity transfer theorem of the author for abstract elementary classes that have amalgamation, are tame, and have primes over sets of the form $M \cup \{a\}$.