Descriptive Set Theory - Projective Sets and Wadge Degrees

Projective Sets and Wadge Degrees

Many questions in descriptive set theory ultimately depend upon set-theoretic considerations and the properties of ordinal and cardinal numbers. 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 if it is analytic.
• A set is if it is coanalytic.
• A set A is if there is a subset B of such that A is the projection of B to the first coordinate.
• A set A is if there is a subset B of such that A is the projection of B to the first coordinate.
• A set is if it is both and .

As with the Borel hierarchy, for each n, any set is both and

The properties of the projective sets are not completely determined by ZFC. Under the assumption 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.

More generally, the entire collection of sets of elements of a Polish space X can be grouped into equivalence classes, known as Wadge degrees, 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, with structure extending the projective hierarchy.