The Internal Consistency of Arithmetic with Infinite Descent

Abstract

A proof of the consistency of arithmetic without the induction postulate, but with infinite descent is given in the following. No use is made of transfinite induction and “internal” means that infinite descent will be shown to be self-consistent. I call this arithmetic with infinite descent Fermat arithmetic (FA) to contrast it with Peano arithmetic (PA) (see Gauthier, 1989). The main idea is to translate logic into arithmetic via a polynomial interpretation with Kronecker’s indeterminates and is thus an attempt at the arithmetization of logic in the line of what can be called “Kronecker’s programme”. The logic is constructive, that is it has all the intuitionistic features plus some constructive (local) characteristics to be described below. Fermat arithmetic is minimal in the sense that it is sufficient for (elementary or constructive) number theory and algebra up to (some important part of) algebraic or arithmetic geometry. Andre Weil has stressed the import of Fermat’s infinite descent and Kronecker’s arithmetical theory of algebraic quantities in the making of modern mathematics, but the constructive nature of such proof methods has not been generally recognized by logicians. Rather, logicians in general have tended to assimilate infinite descent and complete induction on the one side and favor Dedekind’s transcendental method over Kronecker’s algorithmic approach on the other1 (see Edwards, 1987).