John Baldwin : The Hanf number for Extendability

Posted by Sherwood Hachtman , part of the Logic Seminar.

At

Oct. 25, 2016, 4 p.m.

In

SEO 427

Abstract

We construct a complete $L_{\omega_1,\omega}$-sentence $\phi$ such that $(\textbf{R},\subseteq)$ is an abstract elementary class with a proper class of models. <b>Theorem.</b> There is a maximal model $M \in \textbf{R}$ of cardinality $\lambda$ if there is no measurable cardinal $\rho$ with $\rho \leq \lambda$, $\lambda = \lambda^{< \lambda}$, and there is an $S \subseteq S^{\lambda}_{\aleph_0}$, that is stationary non-reflecting, and $\diamond_S$ holds. Thus in the absence of a measurable, $\phi$ has arbitrarily large maximal models. But in the presence of measurables there are maximal models cofinally in the first measurable and never again. I hope to say something about the removal of the set-theoretic hypotheses.