---
tag: 2GY3131ZPS
kind: concept
name: Projective determinacy
msc: ["03E60", "03E15", "03E55"]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_determinacy
same_as:
  wikidata: Q4830557
  wikipedia: Projective determinacy
---

## Summary

Projective determinacy, written PD, says that every game of natural numbers whose payoff set is a projective set is [[determined|QXECDPAYTN]].
Unlike the [[axiom of determinacy|PEXTM3MJ5Z]], it is consistent with the axiom of choice, and it is often added to ZFC.
Donald A. Martin and John R. Steel proved that PD follows from the existence of infinitely many [[Woodin cardinals|CMC2C1D53W]].
Under PD, all projective sets of reals are Lebesgue measurable, have the [[Baire property|FZ7237VY65]] and have the [[perfect set property|H47ZGRR13V]].

## Explanations

### L1 human

Projective sets are the sets of reals one gets from Borel sets by taking continuous images and complements, again and again, finitely many times.
They include almost every set of reals that arises in ordinary analysis.
PD says that infinite games with such a payoff set always have a winning strategy for one player.
Many set theorists regard PD as the natural extension of ZFC for questions about projective sets, because it settles most of them.