Shelah's eventual categoricity conjecture in universal classes: part I

TL;DR

This paper proves that universal classes categorical in arbitrarily high cofinalities are eventually categorical on a tail of cardinals, using ZFC without successor categoricity assumptions, and extends results to certain AECs with explicit bounds.

Contribution

It establishes a tail categoricity transfer for universal classes and extends to AECs with amalgamation, providing explicit Hanf number bounds without relying on successor categoricity.

Findings

Categoricity in high cofinality implies tail categoricity for universal classes.

The proof uses ZFC and does not assume categoricity in a successor cardinal.

Explicit Hanf number bounds are given for AECs with amalgamation.

Abstract

We prove: $\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in cardinals of arbitrarily high cofinality, then $K$ is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on the Hanf number and get rid of the cofinality restrictions: $\mathbf{Theorem}$ Let $K$ be an AEC with amalgamation. Assume that $K$ is fully $\operatorname{LS} (K)$-tame and short and has primes over sets of the form $M \cup \{a\}$. Write $H_2 := \beth_{\left(2^{\beth_{\left(2^{\operatorname{LS} (K)}\right)^+}}\right)^+}$. If $K$ is categorical in a $\lambda > H_2$, then $K$ is categorical in all $\lambda' \ge H_2$.