But that's exactly what I meant. So the brute force program proposed in the post I replied to doesn't exist, right?
Let's say, you have proposition A. You write a program to brute force proofs for A. If it finds a proof, it will halt and output Yes, if it finds a disproof it will halt and output No, otherwise it will check the next proof.
So if you say that your brute force program can proof/disproof proposition A (and A1, A2, A3, ... for that matter), you will have to proof that your program will halt for sure after a finite number of steps. Now that's proposition B.
If you find a clever proof for proposition B, then you can claim that you wrote a program that can proof or disproof proposition A - maybe in a zillion years, but it can.
If you can't find such a proof for proposition B, you can always try to write a brute force program to proof/disproof proposition B... then C, D, E, etc. :)
And here comes the halting problem: you may be able to proof that a specific program will halt on a specific input, but it can be very hard to proof. But you can't proof that a specific program will halt on every input. This is directly related to Godel's incompleteness of formal systems.
That was the original statement:
> you can encode any proof to a few hundred lines.
This type of encoding is funny of course. You can not determine from the encoding of the proof whether the encoding is the encoding of a proof (as you point out). But if you have a proof, and thus know that a proof exists you can "encode" it very succinctly in this program for whatever that is worth.
Yeah, I misread that. The sentence implies the existence of a proof. So I was clearly talking about something different. Thanks.
(This is not criticism, this is elaboration. The approach is useless in practice, but deeply understanding all the reasons why is a good exercise in computer science.)
For the same reason, for any program that halts, you can run it long enough, and see that it halts. But (unless you are lucky enough to detect a loop), if it doesn't halt, you will never witness that, you will only be able to say "it hasn't halted yet".
It definitely can take a while until that kind of asymmetry becomes intuitive.