Low level nondefinability results: Domination and recursive enumeration

Abstract

We study low level nondefinability in the Turing degrees. We prove a variety of results, including, for example, that being array nonrecursive is not definable by a ${\mathrm{\Sigma }}_1$ or ${\mathrm{\Pi }}_1$ formula in the language $(\le ,\text{REA})$ where $\text{REA}$ stands for the ``r.e.\ in and above'' predicate. In contrast, this property is definable by a ${\mathrm{\Pi }}_2$ formula in this language. We also show that the ${\mathrm{\Sigma }}_1$-theory of $(\mathcal{D},\le ,\text{REA})$ is decidable.