It's important though to note that we know of no problem so far that a human mind can solve that couldn't be solved by a Turing Machine. Godel's incompleteness theorems may very well apply to human minds as well, and so far this seems more likely than not (though a lot of human thinking is actually inconsistent, so perhaps it falls to the other side of the incomplete/inconsistent "choice" than formal systems).
Well... maybe human minds are themselves as limited as Turing machines? In that case we may never be able to create a machine that can overcome our, and Turing machines', limitations.
