This is Penrose's argument. One issue with this line of thinking is that it confuses a system which produces infallible proofs and the human brain. None of those theorems actually apply to people; we may gain quite a bit by being wrong sometimes.
We aren't infallible, but that is different than the inconsistency of a formal system. Insistency means the very concept of truth is nonsensical, since an inconsistent system can prove everything. It is fallacious to equivocate the two.