Determinacy

Summary

Determinacy is the property of an infinite two-player game of perfect information that one of the two players has a winning strategy. In the games of descriptive set theory, the players alternately choose natural numbers forever. The first player wins when the resulting sequence lies in a fixed payoff set. Gale and Stewart proved that games with an open or a closed payoff set are determined. The Borel determinacy theorem extends this to all Borel payoff sets. Stronger determinacy hypotheses, such as projective determinacy and the axiom of determinacy, imply strong regularity properties of sets of reals.