The internal consistency of arithmetic with infinite descent

Abstract

The consistency of arithmetic is shown to obtain without the recourse of transfinite induction or the detour of an infinite set. Arithmetic without an induction postulate, but with infinite descent, is Fermat Arithmetic coupled with Kronecker's "general arithmetic" of indeterminates. Fermat arithmetic is self-consistent or self-contained. The main idea is to interpret a local (constructive) logic with a local "effinite" quantifier in a polynomial translation and show how logic is eliminated by infinite descent in the same way as the content is exhausted in the decomposition of polynomials (or forms) where the method of infinite descent is at work. The arithmetization of logic (and the topological interpretation) is effected through the (combinatorial) convolution product of polynomials and amounts to a parametrization of logic by polynomials with indeterminates. Although not always effective, infinite descent provides a finite constructivist setting for an arithmetic that encompasses most of number theory and a large part of algebraic or arithmetic geometry. The resulting arithmetical logic can be seen as a vindication of Kronecker's foundational outlook beyond Hilbert's programme.