## Notre Dame Journal of Formal Logic

### Implicit Definability in Arithmetic

Stephen G. Simpson

#### Abstract

We consider implicit definability over the natural number system $\mathbb{N},+,\times,=$. We present a new proof of two theorems of Leo Harrington. The first theorem says that there exist implicitly definable subsets of $\mathbb{N}$ which are not explicitly definable from each other. The second theorem says that there exists a subset of $\mathbb{N}$ which is not implicitly definable but belongs to a countable, explicitly definable set of subsets of $\mathbb{N}$. Previous proofs of these theorems have used finite- or infinite-injury priority constructions. Our new proof is easier in that it uses only a nonpriority oracle construction, adapted from the standard proof of the Friedberg jump theorem.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 57, Number 3 (2016), 329-339.

Dates
Accepted: 8 January 2014
First available in Project Euclid: 30 March 2016

https://projecteuclid.org/euclid.ndjfl/1459343729

Digital Object Identifier
doi:10.1215/00294527-3507386

Mathematical Reviews number (MathSciNet)
MR3521483

Zentralblatt MATH identifier
06621292

#### Citation

Simpson, Stephen G. Implicit Definability in Arithmetic. Notre Dame J. Formal Logic 57 (2016), no. 3, 329--339. doi:10.1215/00294527-3507386. https://projecteuclid.org/euclid.ndjfl/1459343729

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