Universal classes near ${\aleph }_1$

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 $1\le I(\psi ,{\aleph }_1)<2^{{\aleph }_1}$, 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.

12 pages; Corrected typos; Rewrote part of the introduction

References in corpus (5)

• Shelah's eventual categoricity conjecture in universal classes: part I
• Shelah's eventual categoricity conjecture in universal classes. Part II
• Downward categoricity from a successor inside a good frame
• Abstract elementary classes stable in ${\aleph }_0$
• Shelah's eventual categoricity conjecture in tame AECs with primes

Cited by in corpus (2)

• Building models in small cardinals in local abstract elementary classes
• On categoricity in successive cardinals