Axiom of projective determinacy

In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets.

We may define the set of "first projective" subsets of Rⁿ to be the set of all subsets which are projections of Borel subsets of Rⁿ⁺¹; then we may define "second projective" sets as projections of first projective sets or complements thereof, and so on. A set is said to be projective if it belongs to some level of this hierarchy.

The axiom of projective determinacy states that for any Banach-Mazur game on the real numbers, if the victory set (for either player, since the projective sets are closed under complementation) is projective, then the game has a winning strategy.

The axiom is undecidable in ZFC, unlike the full axiom, which contradicts the Axiom of Choice; it follows from certain large cardinal axioms, such as the existence of infinitely many Woodin cardinals.