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. Based on two properties of an infinite set, which we use to reformulate the conjecture, we either prove that the conjecture leads to FALSE, or that its negation leads to FALSE. This is equivalent to saying that we either have a proof of the conjecture or a proof of its negation. However, this is false, as we actually have no proof for either statement. We apply elementary number theory, where the constructive role of prime numbers within the natural numbers is a key point.

In a further corollary, we show that even if ZFC and PA were sound (which they are not due to the inconsistency), the conjecture would be independent of ZFC and PA, i.e., ZFC (PA) could neither prove nor disprove the conjecture. We also show that this is true for the original strong Goldbach conjecture. The consequence of this is that both the original and the strengthened conjecture are true.