Index sets for Π01 classes Original Research Article

A Π01 class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of the members of a class P. Given an effective enumeration {Pe:e < ω} of the Π01 classes, the index set I for a certain property (such as having positive measure) is the set of indices e such that Pe has the property. For example, the index set of binary Π01 classes of positive measure is Σ02 complete. Various notions of boundedness (including a new notion of “almost bounded” classes) are discussed and classified. For example, the index set of the recursively bounded classes is Σ03 complete and the index set of the recursively bounded classes which have infinitely many recursive members is Π04 complete. Consideration of the Cantor-Bendixson derivative leads to index sets in the transfinite levels of the hyperarithmetic hierarchy.