Universal classes near $\aleph_1$

Abstract

Shelah has provided sufficient conditions for an $L_{ω_1, ω}$-sentence $ψ$ to have arbitrarily large models and for a Morley-like theorem to hold of $ψ$. 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 $ψ$ is restricted to be universal: $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $ψ$ be a universal $L_{ω_{1}, ω}$-sentence. - If $ψ$ is categorical in $\aleph_{0}$ and $1 \leq I(ψ, \aleph_{1}) < 2 ^{\aleph_{1}}$, then $ψ$ has arbitrarily large models and categoricity of $ψ$ in some uncountable cardinal implies categoricity of $ψ$ in all uncountable cardinals. - If $ψ$ is categorical in $\aleph_1$, then $ψ$ is categorical in all uncountable cardinals. The theorem generalizes to the framework of $L_{ω_1, ω}$-definable tame abstract elementary classes with primes.