Computer Science is the purest science

It's the only system that can use itself to prove itself

The only system that won't halt is a system that can counter any situation, because an infinite random walk will cross every point an infinite amount of times.
The random walk need not be "truly random", or even Turing-complete. All it needs to do is survive.
I learned this from Neon Genesis Evangelion, the best anime of our time.

Therefore the halting problem is essentially "can I survive any situation?". Gödel says that you can't tell that by formal logic.
Therefore you need transcendental logic. Maybe a quantum computer will be able to solve this interesting problem?