Projective Games on the Reals

Abstract

Let $M_n^{♯}\left(\mathbb{R}\right)$ denote the minimal active iterable extender model which has $n$ Woodin cardinals and contains all reals, if it exists, in which case we denote by $M_n\left(\mathbb{R}\right)$ the class-sized model obtained by iterating the topmost measure of $M_n\left(\mathbb{R}\right)$ class-many times. We characterize the sets of reals which are ${\Sigma }_1$-definable from $\mathbb{R}$ over $M_n\left(\mathbb{R}\right)$, under the assumption that projective games on reals are determined:

1. for even $n$, ${\Sigma }_1^{M_n\left(\mathbb{R}\right)}={⅁}^{\mathbb{R}}{\Pi }_{n+1}^1$;

2. for odd $n$, ${\Sigma }_1^{M_n\left(\mathbb{R}\right)}={⅁}^{\mathbb{R}}{\Sigma }_{n+1}^1$.

This generalizes a theorem of Martin and Steel for $L\left(\mathbb{R}\right)$, that is, the case $n=0$. As consequences of the proof, we see that determinacy of all projective games with moves in $\mathbb{R}$ is equivalent to the statement that $M_n^{♯}\left(\mathbb{R}\right)$ exists for all $n\in \mathbb{N}$, and that determinacy of all projective games of length ${\omega }^2$ with moves in $\mathbb{N}$ is equivalent to the statement that $M_n^{♯}\left(\mathbb{R}\right)$ exists and satisfies $\mathsf{AD}$ for all $n\in \mathbb{N}$.