Goldbach's Conjecture is Independent of ZFC and PA

Abstract

This paper proves that both the strong Goldbach conjecture and a strengthened form of it are independent of ZFC and Peano arithmetic (PA), with the latter result being a corollary of the former, assuming these theories are sound. For each of the conjectures, we define an infinite set with which we reformulate the conjecture and show that this set always remains the same, regardless of whether the conjecture or its negation is assumed. Then, based on this, both the assumption that there is a proof of the conjecture and the assumption that there is a proof of its negation lead to a contradiction. We use 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 the fact that ZFC (PA) can neither prove nor disprove the conjectures implies that both must be true.