Be careful about how you interpret the word "prove" there. Are you sure that the sense of "prove" that applies to that theorem of Gödel corresponds to the sense

Be careful about how you interpret the word "prove" there. Are you sure that the sense of "prove" that applies to that theorem of Gödel corresponds to the sense of "prove" when you are talking about proof of stake? The proofs of Gödel's theorems work by encoding the axioms and inference rules of the system in a way that can be dealt with within the system, and using that to encode statements like "this statement cannot be proven within the system" within the system. I am not sure what the analogous thing would be if you were to attempt to carry the proof over. I am not even sure what it would mean to try to do that. To prove that, there is no proof that can be carried out within proof of stake, that there is no proof of Falsum that can be carried out within proof of stake? What would that even mean? What would it even mean for proof of stake to prove Falsum? I am not asserting that proof of stake can work well, but I am not convinced by the line of reasoning that you lay out, with its current level of detail. Perhaps if you went into the details I would find it more convincing.