Notre Dame Journal of Formal Logic

A Simple Proof of Parsons' Theorem

Fernando Ferreira

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.

Article information

Source
Notre Dame J. Formal Logic, Volume 46, Number 1 (2005), 83-91.

Dates
First available in Project Euclid: 31 January 2005

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1107220675

Digital Object Identifier
doi:10.1305/ndjfl/1107220675

Mathematical Reviews number (MathSciNet)
MR2131548

Zentralblatt MATH identifier
1095.03063

Subjects
Primary: 03F30: First-order arithmetic and fragments

Keywords
finitism Hilbert's program conservativeness

Citation

Ferreira, Fernando. A Simple Proof of Parsons' Theorem. Notre Dame J. Formal Logic 46 (2005), no. 1, 83--91. doi:10.1305/ndjfl/1107220675. https://projecteuclid.org/euclid.ndjfl/1107220675


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