Gödels incompleteness theorem is proven with the statement I am not a theorem of <theory> is never a theorem but always true.
The halting problem is proven because it can't be predicted if the program if I halt then loop forever halts.
In both cases you find a counter-example to "there is a consistent, complete theory" and "there is a computable halting function" respectively. It is very useful to prove the negations of those statements but does the disproven statement ever have a direct application?
We don't actually care about statements which imply there own non-theoremhood in our theories. If we could find a theory which is complete except for these statement we would still be satisfied. Likewise we don't care about programs which contradict there own halting a function, checking if any other program halts would still be very useful.
In a way we haven't learned anything we directly care about, we could still have a theory that predicts everything "useful" or a function which checks every "useful" program for halting.
Are there cases when this is not the case? When we find a counter-example we actually care very much about?
To clarify: I'm not questioning the usefulness of these proofs because statements like "There is no consistent and complete theory" or "The halting problem is undecidable" are still very useful. I am more asking if there can be direct applications of the disproved statement because it feels like you can always form a less general statement.