## Journal of Symbolic Logic

### 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.

#### Article information

Source
J. Symbolic Logic, Volume 78, Issue 3 (2013), 1005-1024.

Dates
First available in Project Euclid: 6 January 2014

https://projecteuclid.org/euclid.jsl/1389032288

Digital Object Identifier
doi:10.2178/jsl.7803180

Mathematical Reviews number (MathSciNet)
MR3135511

Zentralblatt MATH identifier
1325.03047

Subjects
Primary: 03D28: Other Turing degree structures

#### Citation

Cai, Mingzhong; Shore, Richard A. Low level nondefinability results: Domination and recursive enumeration. J. Symbolic Logic 78 (2013), no. 3, 1005--1024. doi:10.2178/jsl.7803180. https://projecteuclid.org/euclid.jsl/1389032288