Given an input problem P, can you construct an algorithm A to compute whether or not P is decidable or undecidable? In other words, is the undecidabiliy of a problem undecidable? My initial guess is that yes, undecidability of a given problem is undecidable, because the above algorithm seems reducible to the Halting problem which cannot be solved using a Turing machine. However, the human mind seems to be able to determine the undecidability of a given problem, and thus the above algorithm is constructable (Unless of course, there exist problems so unimaginably hard that it is impossible to determine if they are undecidable or not). So, which one of these guesses is the correct one, and why? Is undecidability undecidable or decidable?