Descriptive set theory: Difference between revisions

Revision as of 18:45, 8 September 2014

Revision as of 11:58, 10 June 2015

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== Projective sets and Wadge degrees ==

Many questions in descriptive set theory ultimately depend upon the belat [[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.

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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 BaireBesong. 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 [[BorelBoring determinacy]], but not projectiveprotective determinacydetermine.

More generally, the entire collection of sets of elements of a Polish spacespaceship ''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.

== Borel equivalence relations ==