Projective Games on the Reals

Abstract

Let Mn♯(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 Mn(R) the class-sized model obtained by iterating the topmost measure of Mn(R) class-many times. We characterize the sets of reals which are Σ1-definable from R over Mn(R), under the assumption that projective games on reals are determined: 1. for even n, Σ1Mn(R)=⅁RΠn+11; 2. for odd n, Σ1Mn(R)=⅁RΣn+11. This generalizes a theorem of Martin and Steel for L(R), that is, the case n=0. As consequences of the proof, we see that determinacy of all projective games with moves in R is equivalent to the statement that Mn♯(R) exists for all n∈N, and that determinacy of all projective games of length ω2 with moves in N is equivalent to the statement that Mn♯(R) exists and satisfies AD for all n∈N.