That's irrelevant, though, because a mathematical proof or a statement X is considered in the context of a specific formal system (and any isomorphic system).
Proving that S can't be proven under formal system F will always be true regardless if you are able to prove the same statement in the formal system Z.
For example, you can't say "I just disproved that triangle angles add to 180 degrees" just because you've proved they add to more than 180 degrees under a non-flat surface or non-Euclidean geometry.
People saying "triangle angles add to 180 degrees" implicitly already mean "under Euclidean geometry and in a flat plane". It's just left implied, because it's goes without saying that they mean it within the system they expressed the problem and did the proof in.
No we can't. We can only prove that certain formal systems cannot resolve certain statements. We might still invent other formal systems that are acceptable to us and that can resolve those statements.