Weak Density and Nondensity among Transfinite Levels of the Ershov Hierarchy

Abstract

We show that for any $\omega$-r.e. degree $\mathit{d}$ and $n$-r.e. degree $\mathit{b}$ with $\mathit{d}<\mathit{b}$, there is an $(\omega +1)$-r.e. degree $\mathit{a}$ strictly between $\mathit{d}$ and $\mathit{b}$. We also show that there is a maximal incomplete $(\omega +1)$-r.e. degree. As a corollary, ${\mathcal{D}}_{\omega }$ is not a ${\Sigma }_1$-elementary substructure of ${\mathcal{D}}_{\omega +1}$.