> I still believe the Halting problem can be solved in some sense of the word solve. But perhaps I'm being too pragmatic.
While its true there are some forms of functions that can be found to either halt or not, some can't. For example, ask the user for a boolean, and do a while loop based on this value. You can't say whether this will halt or not, but it does contain unspecified variables.
Then consider the Collatz Conjecture. We can't prove any other number then 1 actually reaches 1 without simulating the process. Since we don't even know if the next step will reach the destination, we can't make any decision about when it will halt. If we can't even decide it for such a simple function, then I don't think we can 'solve' it, even for a general 'solve'.
>For example, ask the user for a boolean, and do a while loop based on this value. You can't say whether this will halt or not, but it does contain unspecified variables.
I don't think that's an issue. The Halting Problem ask the question whether a program will halt when supplied with a given input. Nothing is stopping us from providing a separate proof for every single program for every possible that it'll stop.
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.
While its true there are some forms of functions that can be found to either halt or not, some can't. For example, ask the user for a boolean, and do a while loop based on this value. You can't say whether this will halt or not, but it does contain unspecified variables.
Then consider the Collatz Conjecture. We can't prove any other number then 1 actually reaches 1 without simulating the process. Since we don't even know if the next step will reach the destination, we can't make any decision about when it will halt. If we can't even decide it for such a simple function, then I don't think we can 'solve' it, even for a general 'solve'.