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2016 Implicit Definability in Arithmetic
Stephen G. Simpson
Notre Dame J. Formal Logic 57(3): 329-339 (2016). DOI: 10.1215/00294527-3507386


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.


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Stephen G. Simpson. "Implicit Definability in Arithmetic." Notre Dame J. Formal Logic 57 (3) 329 - 339, 2016.


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

zbMATH: 06621292
MathSciNet: MR3521483
Digital Object Identifier: 10.1215/00294527-3507386

Primary: 03D55
Secondary: 03C40, 03D28, 03D30, 03D80

Rights: Copyright © 2016 University of Notre Dame


Vol.57 • No. 3 • 2016
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