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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.

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rdf:type	yago:WikicatTheoremsInTheFoundationsOfMathematics yago:Abstraction100002137 yago:Communication100033020 yago:Message106598915 yago:Proposition106750804 yago:Statement106722453 yago:Theorem106752293
rdfs:label	Borel determinacy theorem (en)
rdfs:comment	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. (en)
dct:subject	Theorems in the foundations of mathematics Determinacy
Wikipage page ID	15483838 (xsd:integer)
Wikipage revision ID	1115668735 (xsd:integer)
Link from a Wikipage to another Wikipage	Power set Product topology Metamathematics Determinacy Annals of Mathematics Cumulative hierarchy Descriptive set theory Donald A. Martin Consistent Countably infinite Analytic set Measurable cardinal Measure (mathematics) Stanford Encyclopedia of Philosophy Strongly inaccessible cardinal Closed set Zermelo–Fraenkel set theory Strategy Mathematical Reviews Topological product Topology Pruned tree Game tree Harvey Friedman Large cardinal Theorems in the foundations of mathematics Beth numbers Graduate Texts in Mathematics Baire space Determinacy Large cardinal axiom Discrete topology Axiom of choice Axiom of dependent choice Axiom of determinacy Axiom of projective determinacy Borel hierarchy Borel set Polish space Open set Cantor space Set theory Perfect set property Zermelo's theorem (game theory) Property of Baire Subset Large cardinals Axiom of replacement Basis (topology) Projective set
Link from a Wikipage to an external page	http://plato.stanford.edu/entries/large-cardinals-determinacy/ http://www.cas.unt.edu/~rdb0003/thesis/thesis.pdf https://archive.org/details/classicaldescrip0000kech
sameAs	Borel determinacy theorem Borel determinacy theorem Borel determinacy theorem Borel determinacy theorem
dbp:wikiPageUsesTemplate	dbt:Center dbt:Citation_needed dbt:Cite_book dbt:Cite_journal dbt:Main dbt:Reflist dbt:MR
has abstract	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 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, if certain large cardinals are consistent. (en)
gold:hypernym	Borel
prov:wasDerivedFrom	wikipedia-en:Borel_determinacy_theorem?oldid=1115668735&ns=0
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foaf:isPrimaryTopicOf	wikipedia-en:Borel_determinacy_theorem
is Link from a Wikipage to another Wikipage of	Determinacy Émile Borel Axiom of determinacy Axiom schema of replacement List of theorems The Higher Infinite Borel determinacy
is Wikipage redirect of	Borel determinacy
is foaf:primaryTopic of	wikipedia-en:Borel_determinacy_theorem

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