On the degree spectrum of a $\Pi ^0_1$ class

Abstract

Toward establishing the decidability of the two-quantifier theory of the Δ02 Turing degrees with join, we study extensions of embeddings of upper-semi-lattices into the initial segments of Turing degrees determined by computably enumerable sets, in particular, the degree of the halting set 0'. We obtain a good deal of sufficient and necessary conditions.