John Baldwin : Uncountable models still count

Posted by Matthias Aschenbrenner , part of the Logic Seminar.

At

Feb. 14, 2006, 4 p.m.

In

SEO 427

Abstract

I will discuss approaches to Vaught's Conjecture by counting the number of models with cardinality $ \aleph_1$. The main actual result will be a theorem of Gao: An infinite structure $M$ is totally categorical if and only if the automorphism group of $M$ admits a complete left-invariant metric. We will observe that that fact prevents naive attempts to construct sentences of $\mathcal L_{\omega_1,\omega}$ with a prescribed number of models of cardinality $\aleph_1$.