When you give a computer an input, it will either give a result or get stuck in a loop. Now imagine a computer, let's call it "H," that reads the blueprints of any computer, test its inputs, and always correctly determines whether it will finish or get stuck. Is the H computer possible?
One of the key components of this machine is the negator.
A reasonable question might be, “Why would anyone build such a pathological, self-contradicting machine?” This wouldn’t have any bearing on the proof, mind you. For the proof, Turing only needed to show such a machine could be made to self-contradict, not whether the design is desirable. I mean really! Who in their right mind would assemble such a thing?
And then it hit me. Our government has become such a machine.
Just install some Tea Party anarchists to act as the negator, feed the government budget into the top, and voila!
The Tea Party sees a smiley face, the rest of us see the thing shake and wobble around the floor—halting problem indeed.