Why Does Halting Problem Exist?

The Halting problem lets us reason about the relative difficulty of algorithms. It lets us know that, there are some algorithms that don't exist, that sometimes, all we can do is guess at a problem, and never know if we've solved it.

Why is the halting problem not solvable?

H is more gen- eral than ∆, so if H were decidable, ∆ would be also. Thus H is not decidable. This is the unsolvability of the Halting problem. Because the halting problem is not solvable on a Turing machine, it is not solvable on any computer, or by any algorithm, given the Church-Turing thesis.

What does the halting problem prove?

In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever.