## Notre Dame Journal of Formal Logic

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

### Implicit Definability in Arithmetic

#### 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**

Received: 3 September 2013

Accepted: 8 January 2014

First available in Project Euclid: 30 March 2016

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00294527-3507386

**Mathematical Reviews number (MathSciNet)**

MR3521483

**Zentralblatt MATH identifier**

06621292

**Subjects**

Primary: 03D55: Hierarchies

Secondary: 03D30: Other degrees and reducibilities 03D28: Other Turing degree structures 03C40: Interpolation, preservation, definability 03D80: Applications of computability and recursion theory

**Keywords**

arithmetical hierarchy arithmetical singletons implicit definability hyperarithmetical sets Turing jump

#### 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