Extensions of embeddings below computably enumerable degrees

Abstract

Toward establishing the decidability of the two-quantifier theory of the Δ 2 0 \Delta ^0_2 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 ′ \boldsymbol {0}’ . We obtain a good deal of sufficient and necessary conditions.