The Strong Goldbach Conjecture and the Inconsistency of ZFC and Peano Arithmetic

Abstract

This paper investigates the strong Goldbach conjecture and a strengthened variant in relation to the soundness and consistency of ZFC and Peano arithmetic (PA). By reformulating the strengthened conjecture in terms of an infinite set with specific structural properties, we show that assuming soundness makes particular formally derived statements, starting from this set, false, leading to the unsoundness of ZFC and PA. We further show that both the original and the strengthened form of Goldbach's conjecture are decidable in ZFC and PA. We then examine the relationship between provability and truth for these statements and, combining the results, conclude that ZFC and PA are inconsistent.