Indeed, the first incompleteness theorem tells us that any logical framework which can express Peano arithmetic must necessarily contain true (resp. false) facts for which no (resp. counter) proof can be given.
Sometimes you can prove that no proof exists about a specific sentence (that's what his incompleteness proof does), and I think you could extend this technique to construct sentences where no proof exists of whether it has a proof, etc...
Sometimes you can prove that no proof exists about a specific sentence (that's what his incompleteness proof does), and I think you could extend this technique to construct sentences where no proof exists of whether it has a proof, etc...