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Stuart_Armstrong comments on We won't be able to recognise the human Gödel sentence - Less Wrong Discussion
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Truth and falsity in an inconsistent system is a bit of a weird concept.
To be more specific: Gödel sentences are true within inconsistent systems (because of explosion) but if we give them a meta-interpretation then they are false about inconsistent systems.
Truth within a consistent system is a bit of a weird concept as well. Is the continuum hypothesis true within ZFC? The question isn't actually meaningful; the continuum hypothesis is true within some models of ZFC, and false within others.