church-turing is a a thesis, not a logical theorem. You pointed me to a proof that the halting problem is unsolvable by a Turing Machine, not that hypercomputers are impossible.
The critic Martin Davis mentioned in wikipedia has an article criticizing a kind of attempt at showing the feasibility of hypercomputers. Thats fine. If there was a well-known logical proof of its unfeasibility, his task would be much simpler though. The purely logical argument hasnt been made as far as i know and as far as you were able to show.
You would need to invent a complexity class larger than R, one that contains more than countably infinite programs. Those, too, can be diagonalised, there would still be incomputable functions. Our whole argument would repeat with that complexity class instead of R. Rinse and repeat. By induction, nothing changes, Q.E.D.
A hypercomputer has its own class of unsolvable problems, I agree. That doesnt mean that a hypercomputer cannot exist.
You know what? I’m going to plant a nuke under your ass: Turing machines can’t exist, either. Any finite machine can be expressed as a DFA. We’re nothing but a bunch of complicated regexen.
This whole time we were talking about potentiality, not reality; in terms that are convenient theory, not physics. I see no reason to extend potentiality to uncountable infinities when we can’t even exploit countable infinity.
Side note, and this might actually change my mind on things regarding “Is R all that we’ll ever need”: If people manage to get an asymptotic speedup out of quantum computers. The question is whether the parallelism inherent in operations on qbits is eaten up by noise, more or less the more states you load onto the qbits, the more fuzzy the results get, because the universe has a maximum amount of computational oomph it spends on a particle or per unit volume or whatever the right measure is. Of course, before we’d need to move past R we’d first have to load an actually infinite number of states into a qbit and I don’t really see that happening. A gazillion? Doubtful, but thinkable. An infinite number in finite time? Not while we have fat fingers typing away on macroscopic keyboards.
