## Mass problems and hyperarithmeticity (2006)

Citations: | 24 - 16 self |

### BibTeX

@TECHREPORT{Cole06massproblems,

author = {Joshua A. Cole and Stephen G. Simpson},

title = {Mass problems and hyperarithmeticity},

institution = {},

year = {2006}

}

### OpenURL

### Abstract

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 Π 0 1 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 CK 1 and which correspond to the hyperarithmetical hierarchy. Namely, for each α<ω CK 1 let hα be the weak degree of 0 (α),theαth Turing jump of 0. If p is the weak degree of any mass problem P,letp ∗ be the weak degree