Even if the universe is infinite, you can't use its infiniteness because you can't communicate partial results across infinite distances.
It is natural to think that the more time you have, the further you can travel to, potentially. But when it comes to the universe, the opposite is actually true. The more time passes, the less of the universe you can reach. A lot of universe you can see today is actually not at all reachable, meaning even if you shine the light back it will never reach the destination.
Computational capacity must not be able to travel faster than the speed of light, yes?
We are told that quantum entanglement cannot transmit information, I am unaware of how rigorously this has been proven/disproven, on the off chance there is something the scientists have missed, the requirement to travel might not be necessary. Either through some advanced entanglement (a novel approach or a not yet understood state of matter), or reaching for a more exotic theory, some additional dimension enabling warp or wormhole like behaviors.
Then computational capacity would depend on some ratio of power (to entangle, set up facilities, etc.) to time spent not doing these things, and might possibly have an upper limit, or not. Perhaps this is a halting problem in itself?
Only because our universe is currently expanding, if it wasn't or would stop in the future, given enough time you could eventually distribute the task over a large enough region to perform the computation and afterwards combine the results in one place. And one would probably have to throw in a couple of technical requirements, for example that the energy density does not decrease faster than the future light cone expands.
It can be finite, but you have to definitively prove the non halting cases are infinite (which, being infinite, can't be computed with brute force)
Look, you're arguing the collatz cobjecture can be proven by counting up by 1 & seeing when a number hits a loop without reaching 1 (which, if there is such a case, would eventually prove via refutation, but if not, you'd never know if you were about to hit an answer or not)
Mathematical uncomputability for problems with answers is a thing. Read up on Gödel's Incompleteness Theorem
Spacetime (might be) infinitely large, but that doesn't imply that there is infinite matter or energy.
That said, this is kind of a dumb argument because far lower k values have lower BB(k) values already exceed the apparent information value of the universe at any given instant. Maybe there is infinite energy and matter, but that's also irrelevant if we can't perceive more than a finite subset of it.
Edit: well, i guess what would matter would be the information value of time, multiplied by enough bits to store the machine—I'm not sure i'm literate enough in the area to compute that. But, assuming that the heat death of the universe reaches a single (possibly compressed) end-state, it should still be finite—it's seeming quantized, anyway.
It does not matter how much energy is there available in the Universe. Even if there is infinite amount of it, you can't still use it because only finite amount can ever be reached / affected by any single observer. Only finite amount of universe can ever reach any single observer.
So for example, there is a limit on the mass of the computer that can be constructed and still send its result to a single spot in space in the future. And the longer you wait, the smaller the limit because less and less of the universe is available to you to build the computer.
