That's not quite the incompleteness theorem, which is about sentences that are unprovable but nevertheless have a truth value. I think the liar's paradox is of

That's not quite the incompleteness theorem, which is about sentences that are unprovable but nevertheless have a truth value. I think the liar's paradox is of a different kind. It's a sentence that looks well-formed but arguably has no truth value. If you were to formalise human reasoning as a logical system, such sentences would not be definable in it. Either way, for Penrose's argument to work you actually need the proof of Gödel's theorem to hold, not just the result.