Mass Problems and Hyperarithmeticity

A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y . A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of weak degrees of mass problems associated with nonempty Π1 subsets of the Cantor space. The lattice Pw has been studied in previous publications. The purpose of this paper is to show that Pw partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in Pw which are indexed by the ordinal numbers less than ω 1 and which correspond to the hyperarithmetical hierarchy. Namely, for each α < ω 1 let hα be the weak degree of 0 , the αth Turing jump of 0. If p is the weak degree of any mass problem P , let p∗ be the weak degree of the mass problem P ∗ = {Y | ∃X (X ∈ P and BLR(X) ⊆ BLR(Y ))} where BLR(X) is the set of functions which are boundedly limit recursive in X. Let 1 be the top degree in Pw. We prove that all of the weak degrees inf(hα,1), α < ω CK 1 , are distinct and belong to Pw. In addition, we prove that certain index sets associated with Pw are Π1 complete.