Isn't that equivocation on the word "proof"? In the case of "proof of stake," I don't think it's referring to a formal proof of the correctness of the system. I

Isn't that equivocation on the word "proof"? In the case of "proof of stake," I don't think it's referring to a formal proof of the correctness of the system. It's intended to be a consensus mechanic that the users of the blockchain consider "good enough" to trust. Of course there can always be collusion among parties in a blockchain or other attacks on the blockchain, but "proof of work" (and every "proof of x" mechanic I know of) is intended to be "good enough."