A Simple Proof of Parsons' Theorem

Let IΣ1 be the fragment of elementary Peano Arithmetic in which induction is restricted to Σ1-formulas. More than three decades ago, Charles Parsons showed that the provably total functions of IΣ1 are exactly the primitive recursive functions. In this paper, we observe that Parsons’ result is a consequence of Herbrand’s theorem concerning the ∃∀∃-consequences of universal theories. We give a self-contained proof requiring only basic knowledge of mathematical logic.