Editing Projective hierarchy

{{Short description|Descriptive set theory concept}}
{{redirect|Projective set|the card game|Projective Set (game)}}

In the mathematical field of [[descriptive set theory]], a subset <math>A</math> of a [[Polish space]] <math>X</math> is '''projective''' if it is <math>\boldsymbol{\Sigma}^1_n</math> for some positive integer <math>n</math>.  Here <math>A</math> is
* <math>\boldsymbol{\Sigma}^1_1</math> if <math>A</math> is [[analytic set|analytic]]
* <math>\boldsymbol{\Pi}^1_n</math> if the [[Complement (set theory)|complement]] of <math>A</math>, <math>X\setminus A</math>, is <math>\boldsymbol{\Sigma}^1_n</math>
* <math>\boldsymbol{\Sigma}^1_{n+1}</math> if there is a Polish space <math>Y</math> and a <math>\boldsymbol{\Pi}^1_n</math> subset <math>C\subseteq X\times Y</math> such that <math>A</math> is the [[projection (mathematics)|projection]] of <math>C</math> onto <math>X</math>; that is, <math>A=\{x\in X \mid \exists y\in Y : (x,y)\in C\}.</math>

The choice of the Polish space <math>Y</math> in the third clause above is not very important; it could be replaced in the definition by a fixed [[uncountable]] Polish space, say [[Baire space (set theory)|Baire space]] or [[Cantor space]] or the [[real line]].

== Relationship to the analytical hierarchy ==

There is a close relationship between the relativized [[analytical hierarchy]] on subsets of Baire space (denoted by lightface letters <math>\Sigma</math> and <math>\Pi</math>) and the projective hierarchy on subsets of Baire space (denoted by boldface letters <math>\boldsymbol{\Sigma}</math> and <math>\boldsymbol{\Pi}</math>).  Not every <math>\boldsymbol{\Sigma}^1_n</math> subset of Baire space is <math>\Sigma^1_n</math>.   It is true, however, that if a subset ''X'' of Baire space is <math>\boldsymbol{\Sigma}^1_n</math> then there is a set of [[natural number]]s ''A'' such that ''X'' is <math>\Sigma^{1,A}_n</math>.    A similar statement holds for <math>\boldsymbol{\Pi}^1_n</math> sets.  Thus the sets classified by the projective hierarchy are exactly the sets classified by the relativized version of the analytical hierarchy.    This relationship is important in [[effective descriptive set theory]]. Stated in terms of definability, a set of reals is projective iff it is definable in the language of [[second-order arithmetic]] from some real parameter.<ref>J. Steel, "[https://www.ams.org/notices/200709/tx070901146p.pdf What is... a Woodin cardinal?]". Notices of the American Mathematical Society vol. 54, no. 9 (2007), p.1147.</ref>

A similar relationship between the projective hierarchy and the relativized analytical hierarchy holds for subsets of Cantor space and, more generally, subsets of any [[effective Polish space]].

==Table==

{{pointclasses}}

==See also==
* [[Borel hierarchy]]

== References ==
{{Reflist}}
* {{Citation | last1=Kechris | first1=A. S. | author-link=Alexander Kechris | title=Classical Descriptive Set Theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94374-9 | year=1995 | url-access=registration | url=https://archive.org/details/classicaldescrip0000kech }}
* {{Citation | last1=Rogers | first1=Hartley |author-link= Hartley Rogers|title=The Theory of Recursive Functions and Effective Computability | orig-year=1967 | publisher=First MIT press paperback edition | isbn=978-0-262-68052-3 | year=1987}}

[[Category:Descriptive set theory]]
[[Category:Mathematical logic hierarchies]]