In comp.theory Michael Hennebry <firstname.lastname@example.org> wrote:
> An NP-hard decision problem need not be in NP, but it must be > decidable.
I am not sure what the consensus on that is. According to some definitions I have seen the Halting Problem is NP-hard. If you could solve the Halting Problem in polynomial time, you could solve any problem in NP in polynomial time.
It does not make much sense to talk about the complexity of an undecidable problem, and in discussions of complexity undecidable problems are usually ignored, but technically they do fit the definitions of some of the complexity classes.