# 2015 seminar talk: Mice with finitely many Woodin cardinals from optimal determinacy hypotheses

Talk held by Sandra Uhlenbrock (Universität Münster, Germany) at the KGRC seminar on 2015-03-05.

### Abstract

Projective determinacy is the statement that for certain infinite games, where the winning condition is projective, there is always a winning strategy for one of the two players. It has many nice consequences which are not decided by ZFC alone, e.g. that every projective set of reals is Lebesgue measurable. An old so far unpublished result by W. Hugh Woodin is that one can derive specific countable iterable models with Woodin cardinals, $M^\#_n$, from this assumption. Work by Itay Neeman shows the converse direction, i.e. projective determinacy is in fact equivalent to the existence of such models. These results connect the areas of inner model theory and descriptive set theory. This talk will be an overview of the relevant topics in both fields and sketch a proof of the result that boldface $\Pi^1_{n+1}$ determinacy implies the existence of $M^\#_n(x)$ for all reals $x$.

This is joint work with Ralf Schindler and W. Hugh Woodin.