Extending Cooper’s theorem to Δ30 Turing degrees

Abstract

© 2018 - IOS Press and the authors. All rights reserved. In 1971 B. Cooper proved that there exists a 2-c.e. Turing degree which doesn't contain a c.e. set. Thus, he showed that the second level of the Ershov hierarchy is proper. In this paper we investigate proper levels of some extensions of the Ershov hierarchy to higher levels of the arithmetical hierarchy. Thus we contribute to the theory of ' " 3 0 -degrees by extending Cooper's theorem to some levels of the fine hierarchy within ' " 3 0 -sets.