Goldbach's Conjecture is Independent of ZFC and PA

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 proo