Weak Density and Nondensity among Transfinite Levels of the Ershov Hierarchy

Abstract

We show that for any ω-r.e. degree d and n-r.e. degree b with d<b, there is an (ω+1)-r.e. degree a strictly between d and b. We also show that there is a maximal incomplete (ω+1)-r.e. degree. As a corollary, Dω is not a Σ1-elementary substructure of Dω+1.