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@AlexeyRomanov I think a relevant point is that Goldbachs's conjecture was always (for centuries) strictly a conjecture about the "standard" natural numbers, not about some exotic nonstandard numbers. If it is independent of first order PA, then there is no counterexample in the natural numbers, which is exactly what Goldbach's conjecture says. Which means it is true in an unqualified sense. So while "provable" might be relative to some theory here, "true" is not, since Goldbach was specifically talking about the natural numbers. Either there is a counterexample in the natural numbers or not.
The "Key Assumption" may follow from the Curry–Howard correspondence (which says that formal proofs are equivalent to programs) together with, as mentioned above, the Church–Turing thesis.
