Turing's Halting Problem is about the impossibility of writing a program that can figure out whether another program will or will not halt on given input.

The result that you're referring to is variously known as Church's thesis, the Church-Turing thesis, Turing's thesis and Church's conjecture. There are many minor variations, but they all boil down to, "Anything that can be computed, can be computed on a Turing machine." This statement is unproveable because "can be computed" is undefined. However it is generally accepted, and holds true for every reasonable definition of "computed" that anyone has been able to come up with.