Goldbach's Conjecture — Towards the Inconsistency of Arithmetic

Abstract

This paper proves that ZFC and Peano arithmetic (PA) are inconsistent, the latter result being a corollary of the former. We introduce a strengthened form of the strong Goldbach conjecture and, assuming consistency, establish the following contradiction. We either prove that the conjecture leads to FALSE, or that its negation leads to FALSE, which is equivalent to saying that we either have a proof of the conjecture or a proof of its negation. However, we actually have no proof for either of them. This contradiction is triggered by the conjunction of two properties of an infinite set, which we use to reformulate the conjecture. We apply elementary number theory, where the constructive role of prime numbers within the natural numbers is a key point.