Gödel's Incompleteness Theorem applies to arbitrary programs. The AI only has to prove results about specific programs, and it can follow whatever restrictions it needs to make such proofs possible.
The physical world means bit flips are inevitable, but error correcting codes exist. You can apply as much error correction as you like and reduce the probability of failure to an arbitrarily low value.
But it can't verify that for generally intelligent copies of itself.
There is no way for it to formally verify large multi-agent dynamical systems.
It will still need a set of agreed upon behavioral protocols to keep things stable.
Grey goo or paperclip optimizers are real dangers, but super intelligence necessitates these protocols even more than lesser intelligences due to the high leverage and rapid evolution of the systems.
The physical world means bit flips are inevitable, but error correcting codes exist. You can apply as much error correction as you like and reduce the probability of failure to an arbitrarily low value.