no way

What if it's proven that "proving then possible or impossible" is impossible?

@StrayClimb But proving the problem would also prove that it's provable. So it's impossible to prove that they can't be proven possible or impossible, since, in doing so, you would prove that they are impossible, a contradiction.

@StrayClimb I'm not going to go over each problem, but eg. RH is \Pi^0_1 and one of two Turing Machines must halt with a proof or counterexample.