Borel Determinacy Theorem

In descriptive set theory, the Borel determinacy theorem states that any Gale-Stewart game whose winning set is a Borel set is determined, meaning that one of the two players will have a winning strategy for the game. It was proved by Donald A. Martin in 1975. The theorem is applied in descriptive set theory to show that Borel sets in Polish spaces have regularity properties such as the perfect set property and the property of Baire.

The theorem is also known for its metamathematical properties. In 1971, before the theorem was proved, Harvey Friedman showed that any proof of the theorem in Zermelo-Fraenkel set theory must make repeated use of the axiom of replacement. Later results showed that stronger determinacy theorems cannot be proven in Zermelo-Fraenkel set theory, although they are relatively consistent with it.