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2005 A Simple Proof of Parsons' Theorem
Fernando Ferreira
Notre Dame J. Formal Logic 46(1): 83-91 (2005). DOI: 10.1305/ndjfl/1107220675

Abstract

Let $\mathsf{I\Sigma_1}$ be the fragment of elementary Peano arithmetic in which induction is restricted to $\Sigma_1$-formulas. More than three decades ago, Parsons showed that the provably total functions of $\mathsf{I\Sigma_1}$ are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the $\exists \forall \exists$-consequences of universal theories. We give a self-contained proof requiring only basic knowledge of mathematical logic.

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Fernando Ferreira. "A Simple Proof of Parsons' Theorem." Notre Dame J. Formal Logic 46 (1) 83 - 91, 2005. https://doi.org/10.1305/ndjfl/1107220675

Information

Published: 2005
First available in Project Euclid: 31 January 2005

zbMATH: 1095.03063
MathSciNet: MR2131548
Digital Object Identifier: 10.1305/ndjfl/1107220675

Subjects:
Primary: 03F30

Rights: Copyright © 2005 University of Notre Dame

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