Editing Descriptive set theory (section)

== Projective sets and Wadge degrees ==

Many questions in descriptive set theory ultimately depend upon [[set theory|set-theoretic]] considerations and the properties of [[ordinal number|ordinal]] and [[cardinal number]]s. This phenomenon is particularly apparent in the '''projective sets'''.   These are defined via the [[projective hierarchy]] on a Polish space ''X'':
* A set is declared to be <math>\mathbf{\Sigma}^1_1</math> if it is analytic.
* A set is <math>\mathbf{\Pi}^1_1</math> if it is coanalytic.
* A set ''A'' is <math>\mathbf{\Sigma}^1_{n+1}</math> if there is a <math>\mathbf{\Pi}^1_n</math> subset ''B'' of <math>X \times X</math> such that ''A'' is the projection of ''B'' to the first coordinate.
* A set ''A'' is <math>\mathbf{\Pi}^1_{n+1}</math> if there is a <math>\mathbf{\Sigma}^1_n</math> subset ''B'' of <math>X \times X</math> such that ''A'' is the projection of ''B'' to the first coordinate. 
* A set is <math>\mathbf{\Delta}^1_{n}</math> if it is both <math>\mathbf{\Pi}^1_n</math> and <math>\mathbf{\Sigma}^1_n</math> .

As with the Borel hierarchy, for each ''n'', any <math>\mathbf{\Delta}^1_n</math> set is both <math>\mathbf{\Sigma}^1_{n+1}</math> and <math>\mathbf{\Pi}^1_{n+1}</math>.

The properties of the projective sets are not completely determined by ZFC. Under the assumption [[axiom of constructibility|''V = L'']], not all projective sets have the perfect set property or the property of Baire.  However, under the assumption of [[projective determinacy]], all projective sets have both the perfect set property and the property of Baire. This is related to the fact that ZFC proves [[Borel determinacy]], but not projective determinacy.

There are also generic extensions of <math>L</math> for any natural number <math>n>2</math> in which <math>\mathcal P(\omega)\cap L</math> consists of all the lightface <math>\Delta^1_n</math> subsets of <math>\omega</math>.<ref>V. Kanovei, V. Lyubetsky, "[https://www.mdpi.com/2227-7390/8/9/1477 On the <math>\Delta^1_n</math> problem of Harvey Friedman]. In ''Mathematical Logic and its Applications'' (2020), DOI [https://doi.org/10.3390/math8091477 10.3380/math8091477].</ref>

More generally, the entire collection of sets of elements of a Polish space ''X'' can be grouped into equivalence classes, known as [[Wadge degree]]s, that generalize the projective hierarchy. These degrees are ordered in the [[Wadge hierarchy]]. The [[axiom of determinacy]] implies that the Wadge hierarchy on any Polish space is well-founded and of length [[Θ (set theory)|Θ]], with structure extending the projective hierarchy.