I'm coming off the numberphile video: https://youtu.be/O4ndIDcDSGc
In the very end, they say that 'Proving that Riemann hypothesis is unprovable would be a proof of the truth of Riemann hypothesis'
The reasoning given is: Riemann hypothesis cannot be both unprovable and false. Because if it is false, then it is provably false because a computer could find the counter example to prove it false.
But this all means that we cannot prove that Riemann Hypothesis is unprovable because proving so would prove the non-existence of any counter-example and hence would prove Riemann Hypothesis itself.
P.S. I'm not sure if I agree with their reasoning. 'Provability' is supposed to be 'provability from the axioms', right? If we just found a counter-example in the wild, then yeah, we've proven it to be false but we haven't done so from the axioms. I think it might be possible for a statement to be provable by counter-examples but unprovable by axioms, so that if we have proved that Riemann Hypothesis is unprovable from our axioms, we still haven't proved that we can't find a counter-example. Correct me if I'm wrong.