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Mathematical logic and foundations Definition

Axiom of determinacy

Builds on DeterminacyBaire space of sequencesZermelo–Fraenkel set theoryAxiom of choice

Definition

The axiom of determinacy (AD) is the following statement in the language of set theory. For every set X⊆ℕℕX \subseteq \N^{\N} of sequences of natural numbers, the game G(ℕ,X)G(\N, X) is determined. Here, in G(ℕ,X)G(\N, X), players I and II alternately choose natural numbers a0,a1,a2,…a_0, a_1, a_2, \dots, and player I wins if (a0,a1,a2,… )∈X(a_0, a_1, a_2, \dots) \in X.

Remarks

Example: for each Borel set XX of the Baire space ℕℕ\N^{\N}, the instance of AD for XX holds. It is even a theorem of ZFC, by the Borel determinacy theorem.

Non-example: AD is false in ZFC. The axiom of choice gives a well-ordering of all strategies. A diagonal argument along it builds a set XX that defeats every strategy of each player.

AD is therefore studied over ZF without choice. It still implies the axiom of countable choice for sets of reals.

Reading path

To understand this page from the ground up, read these 35 pages in order. The foundations come first.

Show the path as a graph
Baire space, concept, no evidence levelBaire spaceconceptComplete metric space, concept, no evidence levelComplete metric spaceconceptDeterminacy, concept, no evidence levelDeterminacyconceptElement, definition, no evidence levelElementdefinitionIrrational number, concept, no evidence levelIrrational numberconceptNatural numbers, object, no evidence levelNatural numbersobjectProduct topology, concept, no evidence levelProduct topologyconceptProposition, concept, no evidence levelPropositionconceptSeparable space, concept, no evidence levelSeparable spaceconceptSet, definition, no evidence levelSetdefinitionBaire space of sequences, object, no evidence levelBaire space of sequen…objectConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUnion, definition, no evidence levelUniondefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionAxiom of extensionality, statement, no evidence levelAxiom of extensionali…statementSubset, definition, no evidence levelSubsetdefinitionPropositional logic, concept, no evidence levelPropositional logicconceptUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionPredicate logic, concept, no evidence levelPredicate logicconceptZermelo–Fraenkel set theory, concept, no evidence levelZermelo–Fraenkel set …conceptFunction, definition, no evidence levelFunctiondefinitionChoice function, definition, no evidence levelChoice functiondefinitionAxiom of choice, statement, no evidence levelAxiom of choicestatementAxiom of determinacy, definition, no evidence levelAxiom of determinacydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Baire space concept
  2. Complete metric space concept
  3. Determinacy concept
  4. Element definition
  5. Irrational number concept
  6. Natural numbers object
  7. Product topology concept
  8. Proposition concept
  9. Separable space concept
  10. Set definition
  11. Baire space of sequences object
  12. Conjunction definition
  13. Disjunction definition
  14. Implication definition
  15. Negation definition
  16. Predicate definition
  17. Family of sets definition
  18. Ordered pair definition
  19. Union definition
  20. Biconditional definition
  21. Existential quantifier definition
  22. Universal quantifier definition
  23. Cartesian product definition
  24. Union of a family of sets definition
  25. Axiom of extensionality statement
  26. Subset definition
  27. Propositional logic concept
  28. Uniqueness quantifier definition
  29. Binary relation definition
  30. Predicate logic concept
  31. Zermelo–Fraenkel set theory concept
  32. Function definition
  33. Choice function definition
  34. Axiom of choice statement
  35. Axiom of determinacy definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Axiom_of_determinacy
About this page
TagWC5JZVG82R
URIm2://WC5JZVG82R
Kinddefinition
WritingW0 stub
MSC03E60, 03E25
DefinesAxiom of determinacy
UsesDeterminacy, Baire space of sequences, Zermelo–Fraenkel set theory, Axiom of choice
Changed2 November 2026

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