What I mean is that while the proof for the Halting Problem is completely valid, I find it rather "weak" as it uses self-reference. That doesn't tell you anythi

What I mean is that while the proof for the Halting Problem is completely valid, I find it rather "weak" as it uses self-reference. That doesn't tell you anything about how it would work for other programs. The only thing you know is that the problem is undecidable for all input. What would happen if we were to restrict the input to a program to everything that doesn't include the entire program source code? It's quite similar to Cantor's diagonal argument. Both proofs are completely correct and use a very similar reasoning. The difference is that I'm not interested in almost denumerating the real numbers (I can do that with the rational numbers anyway), but I would be interested in almost solving the Halting Problem.