Borel determinacy theorem

In <a href="/facts/Descriptive_set_theory/ek0tvUXb">descriptive set theory</a>, the Borel determinacy theorem states that any Gale–Stewart game whose payoff set is a <a href="/facts/Borel_set/jvSsfpkP">Borel set</a> is <a href="/facts/Determinacy/5GMp9B6z">determined</a>, meaning that one of the two players will have a <a href="/facts/Winning_strategy/5GMp9B6z">winning strategy</a> 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 <a href="/facts/Zermelo%2527s_theorem_(game_theory)/rev7L9zS">Zermelo's theorem</a> about the determinacy of finite games. It was proved by <a href="/facts/Donald_A._Martin/KsNsJnDP">Donald A. Martin</a> in 1975, and is applied in descriptive <a href="/facts/Set_theory/1ptfsvUP">set theory</a> to show that Borel sets in <a href="/facts/Polish_space/CkUFcGap">Polish spaces</a> have regularity properties such as the <a href="/facts/Perfect_set_property/S8A7wxcN">perfect set property</a>.
The theorem is also known for its <a href="/facts/Metamathematics/yeIm8vqC">metamathematical</a> properties. In 1971, before the theorem was proved, <a href="/facts/Harvey_Friedman_(mathematician)/Yq10dMrt">Harvey Friedman</a> showed that any proof of the theorem in <a href="/facts/Zermelo%25E2%2580%2593Fraenkel_set_theory/UYIsPbXw">Zermelo–Fraenkel set theory</a> must make repeated use of instances of the <a href="/facts/Axiom_schema_of_replacement/dmJn0BaS">axiom schema of replacement</a>. Later results showed that stronger determinacy theorems cannot be proven in Zermelo–Fraenkel set theory, although they are relatively <a href="/facts/Consistent/IHkdnA9M">consistent</a> with it, if certain <a href="/facts/Large_cardinals/jkSeV92D">large cardinals</a> are consistent.

Borel determinacy theorem open-in-new

Borel determinacy theorem