How do you find that undecidable proposition? You derive it from axioms. You can't declare it axiomatic if you can't find it because it's undecidable.
The big idea that the Incompleteness Theorem torpedoed was that it's possible to enumerate all and only those theorems that are true/provable for a given set of axioms. To get all of them, you must allow unprovable theorems; to eliminate the unprovable ones, you'll necessarily also eliminate some provable theorems. It's Heisenberg's Uncertainty Principle for logic, and it was exactly that destructive to logical determinism/positivism.
The big idea that the Incompleteness Theorem torpedoed was that it's possible to enumerate all and only those theorems that are true/provable for a given set of axioms. To get all of them, you must allow unprovable theorems; to eliminate the unprovable ones, you'll necessarily also eliminate some provable theorems. It's Heisenberg's Uncertainty Principle for logic, and it was exactly that destructive to logical determinism/positivism.