Maybe in analogy to Russell's paradox and how you can "fix" it by distinguishing sets and classes, you can "fix" a Gödel sentence by adding it as an axiom, but

Maybe in analogy to Russell's paradox and how you can "fix" it by distinguishing sets and classes, you can "fix" a Gödel sentence by adding it as an axiom, but then you'll just get a new Gödel sentence... and so on.