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J. Aguilera Ozuna, S. Müller:
"Projective Games on the Reals";
Notre Dame Journal of Formal Logic, Advance publication (2020), 17 pages.

Let M♯n(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
, ΣMn(R)1=⅁RΠ1n+1

;

2. for odd n
, ΣMn(R)1=⅁RΣ1n+1

.

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 M♯n(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 M♯n(R) exists and satisfies AD for all n∈N.

http://dx.doi.org/10.1215/00294527-2020-0027