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* 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}
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.
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* {{cite book | authorlink=Alexander Kechris | author=Kechris, Alexander S. | title=Classical Descriptive Set Theory | url=https://archive.org/details/classicaldescrip0000kech | url-access=registration | publisher=Springer-Verlag | year=1994 | isbn=0-387-94374-9}}
* {{cite book | authorlink=Yiannis N. Moschovakis | author=Moschovakis, Yiannis N. | title=Descriptive Set Theory|url=https://www.math.ucla.edu/~ynm/books.htm | publisher=North Holland | year=1980|page=2 |isbn=0-444-70199-0}}
===Citations===
{{Reflist}}
== External links ==
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