Definable relations in Turing degree structures

Abstract

In this paper we investigate questions about the definability of classes of n-computably enumerable (c. e.) sets and degrees in the Ershov difference hierarchy. It is proved that the class of all c. e. sets is definable under the set inclusion ⊆ in all finite levels of the difference hierarchy. It is also proved the definability of all m-c. e. degrees in each higher level of the hierarchy. Besides, for each level n, n ≥ 2, of the hierarchy a definable non-trivial subset of n-c. e. degrees is established.