Realizing Computably Enumerable Degrees in Separating Classes

Abstract

We investigate what collections of c.e.\ Turing degrees can be realised as the collection of elements of a separating ${\Pi }_1^0$ class of c.e.\ degree. We show that for every c.e.\ degree $\mathbf{c}$, the collection $\{\mathbf{c},0^{\prime }\}$ can be thus realized. We also rule out several attempts at constructing separating classes realizing a unique c.e.\ degree. For example, we show that there is no \emph{super-maximal} pair: disjoint c.e.\ sets $A$ and $B$ whose separating class is infinite, but every separator of c.e.\ degree is a finite variant of either $A$ or $\overline{B}$.