Projective Games on the Reals

TL;DR

This paper characterizes the sets of reals definable over certain minimal extender models with Woodin cardinals, linking their definability to projective game determinacy assumptions.

Contribution

It generalizes Martin and Steel's theorem for $L(R)$ to models with multiple Woodin cardinals, connecting determinacy and the existence of specific extender models.

Findings

For even n, $Sigma_1^{M_n(R)} = ext{Game}^R Pi^1_{n+1}$

For odd n, $Sigma_1^{M_n(R)} = ext{Game}^R Sigma^1_{n+1}$

Determinacy of all projective games with moves in $R$ is equivalent to the existence of $M^lat_n(R)$ for all n.

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 $n\in\mathbb{N}$.