Problems with the halting problem

Abstract Either we leave the definition of the halting function incomplete, not saying its result when applied to its own program, or we suffer inconsistency. If we choose incompleteness, we cannot require a halting program to apply to programs that invoke the halting program, and we cannot conclude that it is incomputable. If we choose inconsistency, then it makes no sense to propose a halting program. Either way, the incomputability argument is lost