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 $\Sigma_{1}$ or $\Pi_{1}$ formula in the language $(\leq ,\REA)$ where $\REA$ stands for the ``r.e.\ in and above'' predicate. In contrast, this property is definable by a $\Pi_{2}$ formula in this language. We also show that the $\Sigma_{1}$-theory of $(\mathcal{D},\leq ,\REA)$ is decidable.