TURING DEFINABILITY IN THE ERSHOV HIERARCHY

Abstract

The first nontrivial DCE (2-computably enumerable) Turing approximation to the class of computably enumerable degrees is obtained. This depends on the following extension of the splitting theorem for the DCE degrees. For any DCE degree ${\bf a}$ and any computably enumerable degree ${\bf b}$ , if ${\bf b} < {\bf a}$ , then there are DCE degrees ${\bf x_0}, {\bf x_1}$ such that ${\bf b} < {\bf x_0}, {\bf x_1} < {\bf a}$ and ${\bf a} = {\bf x_0} \lor {\bf x_1}$ . The construction is unusual in that it is incompatible with upper cone avoidance.