In descriptive set theory, the Borel determinacy theorem states that any Gale–Stewart game whose payoff set is a Borel set is determined, meaning that one of the two players will have a winning strategy for the game. A Gale–Stewart game is a possibly infinite two-player game, where both players have perfect information and no randomness is involved.
The theorem is a far reaching generalization of Zermelo's theorem about the determinacy of finite games. It was proved by Donald A. Martin in 1975, and is applied in descriptive set theory to show that Borel sets in Polish spaces have regularity properties such as the perfect set property.
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, if certain large cardinals are consistent.
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In descriptive set theory, the Boreldeterminacytheorem states that any Gale–Stewart game whose payoff set is a Borel set is determined, meaning that...
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\Gamma } has the property of Baire. Therefore, it follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective...
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