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Mathematical logic and foundations Definition

Projective determinacy

Builds on DeterminacyProjective hierarchyBaire space of sequences

Definition

Projective determinacy (PD) is the following statement in the language of set theory. For every projective set X⊆ℕℕX \subseteq \N^{\N}, that is, every set in some level 𝚺n1\boldsymbol{\Sigma}^1_n of the projective hierarchy, the game G(ℕ,X)G(\N, X) is determined.

Remarks

Example: every Borel set XX of the Baire space is projective, and its game is determined in ZFC by the Borel determinacy theorem. So the Borel instances of PD are theorems of ZFC.

Non-example: the axiom of choice gives a set X⊆ℕℕX \subseteq \N^{\N} whose game is not determined. PD does not cover this set, and under PD such a set is not projective.

For each nn, the statement that all 𝚺n1\boldsymbol{\Sigma}^1_n games are determined is a separate hypothesis, and PD is the conjunction of all of them. The statement that all analytic games, the case n=1n = 1, are determined follows from the existence of a measurable cardinal.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Baire space, concept, no evidence levelBaire spaceconceptComplete metric space, concept, no evidence levelComplete metric spaceconceptDeterminacy, concept, no evidence levelDeterminacyconceptIrrational number, concept, no evidence levelIrrational numberconceptNatural numbers, object, no evidence levelNatural numbersobjectProduct topology, concept, no evidence levelProduct topologyconceptProjective hierarchy, concept, no evidence levelProjective hierarchyconceptSeparable space, concept, no evidence levelSeparable spaceconceptBaire space of sequences, object, no evidence levelBaire space of sequen…objectProjective determinacy, definition, no evidence levelProjective 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. Irrational number concept
  5. Natural numbers object
  6. Product topology concept
  7. Projective hierarchy concept
  8. Separable space concept
  9. Baire space of sequences object
  10. Projective determinacy definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Projective_determinacy
About this page
TagC5MA8PA2Q5
URIm2://C5MA8PA2Q5
Kinddefinition
WritingW0 stub
MSC03E60, 03E15, 03E55
DefinesProjective determinacy
UsesDeterminacy, Projective hierarchy, Baire space of sequences
Changed2 November 2026

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