# Producing $M_n^{\#}$ from Boldface Level-wise Projective Determinacy

## Sandra Uhlenbrock

### University of Münster

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 An old so far unpublished result by W. Hugh Woodin is that one can derive specific countable iterable models with Woodin cardinals, $M_n^{\#}$, from this assumption.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. We will give an overview of the relevant topics in both fields and, if time allows, sketch a proof of the result that for the odd levels of the projective hierarchy boldface $\Pi^1_{2n+1}$-determinacy implies the existence of $M_{2n}^{\#}(x)$ for all reals $x$.

Sandra Uhlenbrock is a PhD student at the University of Münster studying under the supervision of Ralf Schindler. She received her Masters Degree in Mathematics from the University of Münster. Her interests lie in the area of descriptive set theory and inner model theory.