On September 7, 2018 I gave a talk in the CUNY Set Theory Seminar in New York.

How to obtain Woodin cardinals from the determinacy of long games

Abstract: We will study infinite two player games and the large cardinal strength corresponding to their determinacy. For games of length $\omega$ this is well understood and there is a tight connection between the determinacy of projective games and the existence of canonical inner models with Woodin cardinals. For games of arbitrary countable length, Itay Neeman proved the determinacy of analytic games of length $\omega \cdot \theta$ for countable $\theta > \omega$ from a sharp for $\theta$ Woodin cardinals. We aim for a converse at successor ordinals and sketch how to obtain $\omega+n$ Woodin cardinals from the determinacy of $\boldsymbol\Pi^1_{n+1}$ games of length $\omega^2$. Moreover, we outline how to generalize this to construct a model with $\omega+\omega$ Woodin cardinals from the determinacy games of length $\omega^2$ with $\Game^{\mathbb{R}}\boldsymbol\Pi^1_1$ payoff.

This is joint work with Juan P. Aguilera.