Projective determinacy

Summary

Projective determinacy, written PD, says that every game of natural numbers whose payoff set is a projective set is determined. Unlike the axiom of determinacy, it is consistent with the axiom of choice, and it is often added to ZFC. Donald A. Martin and John R. Steel proved that PD follows from the existence of infinitely many Woodin cardinals. Under PD, all projective sets of reals are Lebesgue measurable, have the Baire property and have the perfect set property.