Are there undecidable problems for which a solution has been found? I mean are there examples of problems that have been proven to be undecidable, in the sense that it would not be possible to devise a deterministic computer program that outputs a solution for an instance of the problem. And yet human mathematicians have come up with such a solution.

It is common to find that undecidable problems often have a solution in certain special cases even though there is no general solution. The halting problem is an excellent example; certainly it is possible to prove that (most) programs written by humans intended for a practical purpose do eventually halt.
