Projective determinacy

Projective determinacy (PD) is the axiom that every projective set of real numbers is determined — that is, for every projective subset A of the Baire space, the infinite game with payoff set A has a winning strategy for one of the two players. PD is strictly stronger than Borel determinacy (which is a theorem of ZFC) and strictly weaker than the full Axiom of Determinacy, which applies to all sets of reals.

The projective hierarchy classifies sets of reals by how many alternating quantifiers over the reals are needed to define them, starting from the Borel sets. In ZFC, the projective hierarchy rapidly escapes control: ZFC cannot decide whether projective sets are Lebesgue measurable, have the Baire property, or contain perfect subsets. PD settles all of these questions positively.

The landmark result of Martin, Steel, and Woodin established that PD follows from the existence of sufficiently large cardinals — specifically, from the existence of infinitely many Woodin cardinals with a measurable cardinal above them. This connection between large cardinal axioms and determinacy is one of the deepest structural results in modern set theory: it shows that the regularity of definable sets of reals is not an isolated combinatorial principle but a consequence of the same axioms that calibrate consistency strength across mathematics.