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.
Citation
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
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