Determinacy

Determinacy is a property of two-player, perfect-information games in which players take turns making moves, and the outcome is determined by whether the infinite sequence of moves belongs to a specified winning set. A game is called determined if one of the players has a winning strategy that guarantees a win no matter how the opponent plays.

In finite two-player games of perfect information, determinacy is guaranteed by Zermelo’s theorem: one player has a winning strategy. For infinite games of length omega, determinacy depends on the underlying set-theoretic axioms. The Gale–Stewart theorem shows that every infinite game with perfect information and an open or closed winning set is determined. Martin’s theorem extends this to all Borel winning sets. More broadly, projective determinacy (PD) asserts that all projective sets of reals are determined, a stance with strong consequences in descriptive set theory and typically derived from large-cardinal hypotheses.

The Axiom of Determinacy (AD) states that all such infinite games are determined, but AD conflicts with the Axiom of Choice, so it is not embraced within ZFC. AD implies powerful regularity properties for sets of reals, including Lebesgue measurability, the Baire property, and the perfect set property. PD, while weaker than AD, is still incompatible with full AC but is studied within frameworks that preserve weaker forms of choice, aided by large cardinals.

Determinacy thus organizes the landscape of definable sets and their regularity properties under different foundational assumptions. It also informs questions about the existence and complexity of winning strategies in logic, computability, and related fields.