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, we establish the contradiction that, on the one hand, we have derived either a proof for the conjecture or for its negation, but on the other hand, neither of them is actually contained in the paper. The 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.