idea
The Gödel sentence that points at itself
With that machinery Gödel built a sentence that says, in effect, “this sentence cannot be proven in this system”. Now ask whether the system can prove it.
If it proves the sentence, then the sentence is false (it said it can’t be proven), so the system just proved something false and is broken. If it cannot prove it, then the sentence is true, because that is exactly what it claims. A system that isn’t broken is stuck with the second option: there is a true statement it cannot reach.