Either I'm misunderstanding your ideas or you've misunderstood the nature of the problem.
If you're being pragmatic, rest assured that for any machine/program we can construct in this universe, we can decide the halting problem, as it is bound to be finite state, i.e. it will have a bounded amount of states it can be in and repeat itself. Less technical, there is practically no infinite search space, even if theoretically there is - memory of a physical computer can only count up to a certain number, even if very large.
The halting problem, or undecidability in general is the application of a very basic intuition. If I give you infinite space, and ask you to find me a certain something in that space, there is no way I can be sure whether or not you will succeed. Diophantine equations for example, to solve them requires you to search infinitely through the complex plane, and is analogous to the halting problem.
Now in which sense of the word 'solve' could you possibly believe these can be solved? Yes - our problems and machines are finite, so in that sense maybe.
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.
If you're being pragmatic, rest assured that for any machine/program we can construct in this universe, we can decide the halting problem, as it is bound to be finite state, i.e. it will have a bounded amount of states it can be in and repeat itself. Less technical, there is practically no infinite search space, even if theoretically there is - memory of a physical computer can only count up to a certain number, even if very large.
The halting problem, or undecidability in general is the application of a very basic intuition. If I give you infinite space, and ask you to find me a certain something in that space, there is no way I can be sure whether or not you will succeed. Diophantine equations for example, to solve them requires you to search infinitely through the complex plane, and is analogous to the halting problem.
Now in which sense of the word 'solve' could you possibly believe these can be solved? Yes - our problems and machines are finite, so in that sense maybe.