## Logic Seminar

John Baldwin
UIC
The Hanf number for Extendability
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.
Theorem. 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.
Tuesday October 25, 2016 at 4:00 PM in SEO 427
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