Games of Length Ω1

We prove determinacy for open length ω1 games. Going further we introduce, and prove determinacy for, a stronger class of games of length ω1, with payoff conditions involving the entire run, the club filter on ω1, and a sequence of ω1 disjoint stationary subsets of ω1. The determinacy proofs use an iterable model with a class of indiscernible Woodin cardinals, and we show that the games precisely capture the theory of the minimal model for this assumption. The purpose of this paper is to bring determinacy to the level of games of length ω1. For a set A ⊂ ω1 define Gopen−ω1(A) to be the following game: Players I and II alternate playing natural numbers as in Diagram 1 to create r ∈ ω1 . Player I wins if there exists some α < ω1 so that r↾α belongs to A, and otherwise II wins. Such games are called open length ω1 games, as victory by player I, if achieved, is secured at a strict initial segment of the run. By definable open length ω1 games we mean games Gopen−ω1(A) with A which is Π 1 1 in the codes. (We could relax to projective in the codes, or to lightface definable over L(R), instead of Π1. This would not affect the strength of the resulting class of games, since any number of extra real quantifiers in the payoff can be absorbed by moves in Gopen−ω1 .) These games trace back to Steel [7, 5] who proved various results assuming their determinacy, including propagation of scales and existence of definable winning strategies. I r(0) r(2) . . . r(2ξ) . . . . . . II r(1) . . . r(2ξ + 1) . . . . . . Diagram 1. Games of length ω1. We prove in this paper that these and even stronger games are determined, assuming the existence of an iterable model with a class of indiscernible Woodin cardinals. This precise large cardinal assumption had been expected, and indeed it was already known through work of Steel to be optimal, in the sense that no weaker large cardinal assumption proves the determinacy of definable open games of length ω1. Steel showed this by noting that the minimal iterable model with a class of indiscernible Woodin cardinals does not satisfy definable open length ω1 determinacy. It had also been known from work of Steel and This material is based upon work supported by the National Science Foundation under Grant No. DMS-0094174.