In my recent discussion of 2-adic numbers I made a remark about their infiniteness, explaining that I prefer to think about it in terms of just a potential infinity. We never deal with actually infinite sequences of digits, we always have a limit, but the language of 2-adic numbers allow to considers various rules without having to worry what the particular limit we may have. We can think of it as always being sufficiently large to not be exceeded by anything we do. My assemblers in particular demonstrate this idea, as I defined them to give results of computations as if done on 2-adic numbers, even though they actually have limits on the number of possible digits.

Despite my background in theoretical mathematics, I am generally inclined to think about everything in our world as finite, as I don't think we have a way of knowing about anything that is not. If we try to keep splitting anything in half, we are going to reach Planck length after a few dozens of steps and we do not know if it would be possible to continue the process in any sense. But even if we could continue it indefinitely, how would we know if there is truly no limit there, even if always out of our reach?

In the modern science we see more and more how building blocks of the nature can be explained in terms of information. Maybe this is simply because we use the thinking tools and language that feels most natural to us nowadays, in the information age. But I also think it is related to the fact that everything that we can experience and explore is actually finite, and everything finite can be deconstructed into sequences of bits.

At this point someone is certainly going to bring quantum computing into discussion, as it seems to directly contradict everything I just said. But I do not think it does. After all, if quantum states are not actually infinitely precise, but just with high enough precision for us to not be able to notice the difference, we would never know. I should note, however, that I find it plausible that there may be limits of quantum precision that are in our reach, and I think that it could be a great opportunity to expand our knowledge. After all, if we made quantum computers powerful enough to be hit by such limitations, it would be a discovery that could pave the way for some new theories of physics.

What brought this idea back to my attention was a guest post on Backreaction blog, which references a recent publication, Discretisation of the Bloch Sphere, Fractal Invariant Sets and Bell's Theorem. It is a fascinating example of a theory that can give explanations similar to quantum mechanics while being finite in all aspects (down to states of the system being simply a finite strings of bits). It even shows how this finiteness would place upper limits on qubit entanglement, which could indeed be a testable constraint on the abilities of quantum computers. This confirms that my suspicion was not entirely ungrounded, and if there really are such limits in nature, I sincerely hope that they are in our reach to test someday.

That article also gives an excellent quotation of David Hilbert, which I am going to reproduce here, as I think it greatly summarizes what I attempted to say.

At some point there will be a boundary between the universe we exist in and whatever is beyond that. Is that, which by definition is beyond our reach, the stuff outside that boundary, an infinite abyss?

The big-bang theory suggests that the universe is finite. Then a lot of theorists go on to explain that this universe is all there is, and that there is nothing for our universe to expand into. But to me that is an unsatisfying explanation. Just because we can't ever experience anything outside of our universe* doesn't automatically mean that there is just nothing. It could be an infinite expanse of cream cheese. And for anyone claiming that that isn't possible, then go ahead and show me what is there. Indeed, there are an infinite number of possibilities of what is beyond our universe, nothingness is just one of those possibilities.

* This is the definition of the word universe. If we could experience something outside of it, then that thing is just another part of our universe. So there will always be the possibility of things existing outside of our experience.

This ties to my musings about the meaning of word "exists". Yes, obviously we cannot conclude that there is nothing outside our universe, the same as we cannot conclude that there is anything there. But I argue specifically, that when we say that something exists, we usually mean that it exists in our universe.

Moreover, even if we clarify that the thing we talk about exists in some other world (like "Gandalf exists in the world of Middle-Earth", or "number 2 exists in the set of natural numbers"), it always is an abstraction that is in some way rooted in our reality nonetheless. There are many structures in our universe that work like natural numbers (well, there is a reason why they are called "natural") and number 2 is present in each of them. Middle-Earth, although a fictional universe, exists in our universe as a collection of concepts passed from mind of the author to the minds of the readers (and then evolving further). However, for things that are completely outside of our universe we lack any anchor, so we are not able to say anything factual about them. We could assume that absolutely everything that we could think of exists there, but it would not make any difference for us (as you noted, by the very definition of "our universe") - therefore I consider such speculations pointless.