Not for me to argue against. Let's give it a trial run. If S is consistent then G (is not provable). Do you see a bit of darkness and lightning on the horizon? The issue is not the consistency of S, but rather the provability of the consistency of S. I think that makes a difference. Because if (simply) S were consistent, then G would be provable - oops! But as an added axiom, the consistency of S (I'm guessing) becomes provable in S'. Leaving, one supposes, the consistency of S' to be provable in S''. The devil isn't just in the details; it is the details!I add "If S is consistent then G (is not provable)" to S as an axiom. — Pippen
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