Yep. Just wanted to mention that every theory where you can do diagonalization, i.e. every formal one, can be defeated.

My point is that one could just make the choice be x, then express the payoff in terms of x, then solve for x that gives maximum payoff, using the methods of algebra, instead of trying to redefine algebra in some stupid sense of iteration of values of x until finding an equality (then omg it fails at x=x), and trying to reinvent already existent reasoning (in form of theorem proving).