Categoricity and Universal Classes

TL;DR

This paper proves categoricity transfer results for universal classes under certain conditions and shows that models in the class are essentially vector spaces or trivial, advancing understanding in model theory.

Contribution

It establishes categoricity in all larger cardinals for a class under specific assumptions and characterizes the models as vector spaces or trivial, extending prior categoricity transfer results.

Findings

Categoricity in all cardinals above b5+b5^+

Models are essentially vector spaces or trivial

Includes models of arbitrarily large size

Abstract

Let $(\mathcal{K} ,\subseteq )$ be a universal class with $LS(\mathcal{K})=\lambda$ categorical in regular $\kappa >\lambda^+$ with arbitrarily large models, and let $\mathcal{K}^*$ be the class of all $\mathcal{A}\in\mathcal{K}_{>\lambda}$ for which there is $\mathcal{B} \in \mathcal{K}_{\ge\kappa}$ such that $\mathcal{A}\subseteq\mathcal{B}$. We prove that $\mathcal{K}^*$ is categorical in every $\xi >\lambda^+$, $\mathcal{K}_{\ge\beth_{(2^{\lambda^+})^+}} \subseteq \mathcal{K}^{*}$, and the models of $\mathcal{K}^*_{>\lambda^+}$ are essentially vector spaces (or trivial i.e. disintegrated).