Notre Dame Journal of Formal Logic

Implicit Definability in Arithmetic

Stephen G. Simpson

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We consider implicit definability over the natural number system N,+,×,=. We present a new proof of two theorems of Leo Harrington. The first theorem says that there exist implicitly definable subsets of N which are not explicitly definable from each other. The second theorem says that there exists a subset of N which is not implicitly definable but belongs to a countable, explicitly definable set of subsets of 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

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

Received: 3 September 2013
Accepted: 8 January 2014
First available in Project Euclid: 30 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

arithmetical hierarchy arithmetical singletons implicit definability hyperarithmetical sets Turing jump


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

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