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I am only broadly familiar with mathematics, so please have patience with me. I am wondering about the consistency of both the (a) Goldbach Conjecture and (b) its negation. If the Goldbach Conjecture would be decided, then would we see that (a) or (b) entailed/contained a contradiction on its own, or only that there would be a contradiction of either (a) or (b) together with some other mathematical axioms?

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If the Goldbach conjecture were proved, then of course it would be proved based on a certain set of axioms (such as Peano arithmetic, or maybe ZFC set theory) and the negation of the proven result would therefore be in contradiction with the axioms used. Also note that, in principle, a positive proof of Goldbach might involve more theory, but a disproof of Goldbach would ultimately boil down to exhibiting an even number not representable as sum of primes, i.e., something that could be verified "simply" by means of finitely many additions and multiplications of natural numbers.

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