Perhaps a better explanation - it's similar to the halting problem in computer science. There are programs p where "Program p halts on input x" is unprovable (equivalently there is no computable way to determine which p,x halt), and the key is that we have no way of knowing which programs p and x this is true for, of the ones we don't yet have proofs/halt-prediction-programs about.

You may want to argue that this can be "solved" by never letting programs take other programs as input, which isn't wrong, exactly, but you're left with not very much you can compute in that paradigm for computation.