> If there is a proven theorem that something is impossible, either the theorem is wrong or it's impossible. It can get a lot more subtle than that. I do not w

> If there is a proven theorem that something is impossible, either the theorem is wrong or it's impossible. It can get a lot more subtle than that. I do not want to elaborate, but there are classes of problems for which we can only guarantee SOME hard problems (or undecidable ones) but MOST of which are easy. There are other classes of problems where MOST problems are hard. > Your two sentences "the halting problem is solved routinely" (assuming you mean the latter case, as above) and "this does not invalidate the Halting Theorem" do not logically work. I agree it sounds illogical, but my phrasing was poor. I meant, if problem instances that arise in real life are easy to solve, yet there is a theorem that states that NOT ALL problems are easy, these two are not mutually exclusive. I agree "routinely solving the halting problem" is an exaggeration on my part.