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.
That’s not true. Gödel sentences are true for (most?) consistent systems, and false for all inconsistent systems.
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.