Halting problem
No matter what happens, we get an impossible situation. We did something wrong, and we need to find out what it was. Most of the things we did were not wrong. We can not say "making a program look at a copy of itself is wrong", or "looking at what another program said and then going into a loop if it said one thing, and stopping if it said another thing" is wrong. It does not make sense to say that we are not allowed to do those things. The only thing that we did that looks like it could be wrong is that we pretended that a program like P exists in the first place. And since this is the only thing that could be wrong, and something must be wrong, this is it. This is the proof that a program like P does not exist. There is no program that solves the halting problem.