what saves us is that proofs are text, and thus there are countably many proofs for any logic we might agree upon, where checking a proof is correct is decidable, and thus for any axioms we culturally agree on, a computer will be able to come up with all the proofs we would.
The disagreement between Gödel and Hilbert was a mathematical and not a cultural one, and it was resolved by mathematics, not by culture.
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It is only resolved if you assume every axiom in the hierarchy but doing so is inconsistent. i.e. hilbert thought mathematics was consistent and wanted a proof. Gödel thought it was impossible to know, but proved (lots of people rly) that he couldn't prove that he couldn't know
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Yes, but the point is that in principle (and to a surprisingly large degree in practice) mathematicians can know and agree that this is the case.
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