Induction the Hard Way

Abstract

The instructor sees that the student has discovered a way of proving P(N) without induction in roughly 2VNK steps. Can we do any better? That is, under the given restrictions, how short can a proof of P(N) be? The purpose of this note is to explore a few aspects of that question. The more-or-less optimal answer, given in Theorem 2 below, is a common piece of folklore among experts in the branch of logic known as proof theory, but it does not seem to be widely known elsewhere, even among other logicians. This omission is both surprising and unfortunate, since the question is simple and natural, and one direction of the solution (proving the best answer known, although not proving that it is the best) does not even require any specialized knowledge of logic. The proof presented here seems to be far simpler than any that can currently be dug out of the literature. First, let us make the ground rules precise: working in first-order logic, whether formally or informally, we may use any fact whatsoever about the natural numbers themselves, but we may not apply the principle of induction to any formula