Projective Games on the Reals

Abstract

Let $M^\sharp_n(\mathbb{R})$ 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 $M_n(\mathbb{R})$ the class-sized model obtained by iterating the topmost measure of $M_n(\mathbb{R})$ class-many times. We characterize the sets of reals which are $\Sigma_1$-definable from $\mathbb{R}$ over $M_n(\mathbb{R})$, under the assumption that projective games on reals are determined: (1) for even $n$, $\Sigma_1^{M_n(\mathbb{R})} = \Game^\mathbb{R}\Pi^1_{n+1}$; (2) for odd $n$, $\Sigma_1^{M_n(\mathbb{R})} = \Game^\mathbb{R}\Sigma^1_{n+1}$. This generalizes a theorem of Martin and Steel for $L(\mathbb{R})$, i.e., 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 $M^\sharp_n(\mathbb{R})$ 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 $M^\sharp_n(\mathbb{R})$ exists and satisfies $\mathsf{AD}$ for all $n\in\mathbb{N}$.