Universal classes near $\aleph_1$

TL;DR

This paper establishes conditions under which universal $L_{1,omega}$-sentences have large models and categoricity transfer properties, focusing on the roles of $eth_1$ and $eth_2$ and their set-theoretic assumptions.

Contribution

It provides new sufficient conditions for categoricity transfer and the existence of arbitrarily large models for universal $L_{1,omega}$-sentences, extending Shelah's results.

Findings

Categoricity in $eth_0$ and $eth_1$ implies arbitrarily large models.

Categoricity in $eth_1$ implies categoricity in all uncountable cardinals.

Assumes $2^{eth_0} < 2^{eth_1}$ for the results.

Abstract

Shelah has provided sufficient conditions for an $L_{\omega_1, \omega}$-sentence $\psi$ to have arbitrarily large models and for a Morley-like theorem to hold of $\psi$. These conditions involve structural and set-theoretic assumptions on all the $\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\aleph_0$ and $\aleph_1$ which suffice when $\psi$ is restricted to be universal: $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $\psi$ be a universal $L_{\omega_{1}, \omega}$-sentence. - If $\psi$ is categorical in $\aleph_{0}$ and $1 \leq I(\psi, \aleph_{1}) < 2 ^{\aleph_{1}}$, then $\psi$ has arbitrarily large models and categoricity of $\psi$ in some uncountable cardinal implies categoricity of $\psi$ in all uncountable cardinals. - If $\psi$ is categorical in $\aleph_1$, then $\psi$ is categorical in all uncountable cardinals. The theorem generalizes to the framework of $L_{\omega_1, \omega}$-definable tame abstract elementary classes with primes.