Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
157 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Universal classes near $\aleph_1$ (1712.02880v5)

Published 7 Dec 2017 in math.LO

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.

Summary

We haven't generated a summary for this paper yet.