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this post was submitted on 01 Oct 2025
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Ehh...
So, it's more a case that the system cannot prove it's own consistency (a system cannot prove it won't lead to a contradiction). So the proof is valid within the system, but the validity of the system is what was considered suspect (i.e. we cannot prove it won't produce a contradiction from that system alone).
These days we use relative consistency proofs - that is we assume system A is consistent and model system B in it thus giving "If A is consistent, then so too must B".
As much as I hate to admit it, classical set theory has been fairly robust - though intuitionistic logic makes better philosophical sense. Fortunately both are equiconsistent (each can be used to imply the consistency of the other).