Editing Projective hierarchy (section)

== 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]].