Well there is much much more to say on that topic than what's already published. This story is not all that there is. Because all the Maths tricks of that kind use the same "apply on itself" system, that don't fit with reality.
Because for example you can eat anything you like, but can you eat yourself and continue to live? No! So "in reality", we don't have stuff that loop onto themselves and continue to exist. You can't time travel and change your past because that's a paradoxe and paradoxes don't exist outside of your mind, therefore time travel is impossible.
And this Maths demonstration is false because it just proves that "it" is wrong which doesn't say anything about Mathematics as a whole.
You can't put in a box, a larger box, nature prevents it. So no algorithm can answer the halting problem because if such a program could answer the problem without solving it, then it would inevitably be way bigger and more complex than the original program and therefore, the cost for solving the question would never be avoided.