Is The Halting Problem Undecidable?

Is The Halting Problem Undecidable? Alan Turing proved in 1936 that a general algorithm running on a Turing machine that solves the halting problem for all possible program-input pairs necessarily cannot exist. Hence, the halting problem is undecidable for Turing machines. Is halting problem recursively enumerable? The language HALT corresponding to the Halting problem is