Editing Set theory (section)

=== Descriptive set theory ===
{{Main|Descriptive set theory}}

''Descriptive set theory'' is the study of subsets of the [[real line]] and, more generally, subsets of [[Polish space]]s. It begins with the study of [[pointclass]]es in the [[Borel hierarchy]] and extends to the study of more complex hierarchies such as the [[projective hierarchy]] and the [[Wadge hierarchy]]. Many properties of [[Borel set]]s can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.

The field of [[effective descriptive set theory]] is between set theory and [[recursion theory]]. It includes the study of [[lightface pointclass]]es, and is closely related to [[hyperarithmetical theory]]. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable.

A recent area of research concerns [[Borel equivalence relation]]s and more complicated definable [[equivalence relation]]s. This has important applications to the study of [[invariant (mathematics)|invariants]] in many fields of mathematics.