Turing reducibility in the fine hierarchy

Abstract

© 2019 Elsevier B.V. In the late 1980s, Selivanov used typed Boolean combinations of arithmetical sets to extend the Ershov hierarchy beyond Δ20 sets. Similar to the Ershov hierarchy, Selivanov's fine hierarchy {Σγ}γ<ε0 proceeds through transfinite levels below ε0 to cover all arithmetical sets. In this paper we use a 0‴ construction to show that the Σ30 Turing degrees are properly contained in the Σωω+2 Turing degrees (to be defined); intuitively, the latter class consists of “non-uniformly Σ30 sets” in the sense that will be clarified in the introduction. The question whether the hierarchy was proper at this level with respect to Turing reducibility remained open for over 20 years.