Editing Projective determinacy

In [[mathematical logic]], '''projective determinacy''' is the special case of the [[axiom of determinacy]] applying only to [[projective set]]s.

The '''axiom of projective determinacy''', abbreviated '''PD''', states that for any two-player infinite game of [[perfect information]] of length [[Ω (ordinal number)|ω]] in which the players play [[natural number]]s, if the victory set (for either player, since the projective sets are closed under complementation) is projective, then one player or the other has a [[winning strategy]].

The axiom is not a theorem of [[ZFC]] (assuming ZFC is consistent), but unlike the full axiom of determinacy (AD), which contradicts the [[axiom of choice]], it is not known to be inconsistent with ZFC. PD follows from certain [[large cardinal]] axioms, such as the existence of infinitely many [[Woodin cardinal]]s.

==Consequences==
PD implies that all projective sets are [[Lebesgue measurable]] (in fact, [[universally measurable]]) and have the [[perfect set property]] and the [[property of Baire]].  It also implies that every projective [[binary relation]] may be [[Uniformization (set theory)|uniformized]] by a projective set.

PD implies that for all positive integers <math>n</math>, there is a largest countable <math>\Sigma^1_{2n}</math> set.<ref>Donald A. Martin, "The largest countable this, that, and the other". Cabal seminar 79–81, Proceedings, Caltech-UCLA Logic Seminar 1979–81, edited by A. S. Kechris, D. A. Martin, and Y. N. Moschovakis, Lecture notes in mathematics, vol. 1019, Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo, 1983, pp. 97–106.</ref>

==References==
{{refbegin}}
* {{cite journal|last1=Martin |first1=Donald A. |author1-link=Donald A. Martin |first2=John R. |last2=Steel |author2-link=John R. Steel |date=Jan 1989 |title=A Proof of Projective Determinacy |journal=[[Journal of the American Mathematical Society]] |volume=2 |issue=1 |pages=71–125 |doi=10.2307/1990913 |jstor=1990913 |doi-access=free }}
* {{cite book|last1=Moschovakis |first1=Yiannis N. |authorlink1=Yiannis N. Moschovakis |title=Descriptive set theory |date=2009 |publisher=American Mathematical Society |location=Providence, R.I. |isbn=978-0-8218-4813-5 |edition=2nd |url=http://www.math.ucla.edu/~ynm/lectures/dst2009/dst2009.pdf |url-status=bot: unknown |archiveurl=https://web.archive.org/web/20141112111558/http://www.math.ucla.edu/~ynm/lectures/dst2009/dst2009.pdf |archivedate=2014-11-12 }}
{{refend}}

===Citations===
{{reflist}}

[[Category:Axioms of set theory]]
[[Category:Descriptive set theory]]
[[Category:Determinacy]]
[[Category:Large cardinals]]


{{settheory-stub}}