Infinity is, in a way, a simple concept. It's a magnitude; a scalar. It's just the biggest one there is. Or maybe more accurately, it's where you never get to even if you just keep going and going (to the right, in the Cartesian plane.)
I've been thinking, lately, in terms of a different kind of infinity: epistemological infinity.* This term has obvious ambiguity, but it's the best I've come up with. To really get across what I mean, I think I'll have to take the long way around. There is a well known proverb that says "The more you learn, the more you realize how much you don't know." There are two interpretations of this: the one I think is more common parses "how much you don't know" as "how much you have left to learn [and could learn, given sufficient/infinite time]"; the second is a more epistemologically nihilistic version that parses "how much you don't know" as "how little you truly can know". This second interpretation is a decent starting point for understanding what I mean by epistemological infinity. From my phrasing, it seems like it has more to do with the infinitesimal than the infinite - and I think this is how people usually think about it. However, you can flip it around to become "how much there may be that is beyond your capacity to know or understand." In a way, this is still a simple concept; I can't know the position and momentum of every fundamental particle in the universe, because that much information won't fit in my brain. This doesn't seem very deep or meaningful.
Where I think things get more interesting is if you think about unknowability in computational terms. I'm not talking about the issue of Turing computability.** Suppose you write a program, and set it running on your computer - and for now lets assume for simplicity that your computer is not connected to a network. If the program is sufficiently complex, it might have the capacity to access anything in the computer's memory, and in that sense everything in that memory is "knowable" to the program. The program cannot, however, no matter how diligently it searches, or how hard it thinks, learn something outside of its scope: how many coins are in your spare change jar, or the name of your neighbor's cat. If you give it sensors, you can expand its scope: with an aimable camera and some exceptional AI, it might be able to estimate the number of coins in the jar. An internet connection is just a special case of such a sensor. It is more to the point, however, to note that you can also restrict its scope in a fundamental way. If you create a "virtual machine" - a software instantiation, or "simulation" of a computer, that runs on your ("real") computer, and you run a program on that virtual machine, it can't get out: its scope is limited to the virtual machine. In practice, this can be very useful for some things, since it can prevent a potentially "dangerous" program from crashing your ("real") machine by, say, overwriting some bits the OS needs to function properly. In theory, you can create any number of layers of virtualization: no program running on the Nth-layer virtual machine will have access to any information on any layer M < N.
Now, if you're in the know, you're probably aware of the concept of the universe as a computation; google yielded a nice little essay on the topic on a blog with an exceptionally clever name: Gnostical Turpitude. As the blogger says, the idea is not especially new, and he dismisses it as facile and trivial (a la The Matrix: "Dude, what if we’re all living in, like, a computer?"). I think it's much more interesting than this.
Before going further, though, there's one thing I have to clear up. We generally think about computations teleologically: in terms of the purpose they serve. This makes sense, because this is how we use computers: to accomplish things***. But computation can be thought of nonteleologically, as well, and for present purposes (sorry) teleological reasoning just muddies things up. A computer, in the abstract, is simply a state machine; a program is a set of rules that govern transitions between states, and a computation is simply the sequence of states produced by that program, given some initial state. When stated (again, sorry) this abstractly, it's pretty clear that the physical universe, as we understand it, fits the definition pretty well. No purpose required.
(1) The universe may accurately be described as a computation (2) Any number of layers of virtualization are possible, and no program running on the Nth-layer virtual machine will have access to any information on any layer M < N.
Now we're getting to my concept of epistemological infinity: everything outside of the "scope" of the physical universe we have the capacity to perceive.
I'm greedy. It's not enough to think about the trillions upon trillions of teeming quarks that make up my body and everything around me, that I'll never see. I could study quantum electrodynamics and visit an accelerator, see evidence of quarks, and suppose myself to understand something about them. But beyond all of this - not at smaller or larger scales in any conventional sense, but simply outside, outside of that outside, and so on - there could be infinitely more to the world. Is there? It should be obvious that the answer is simply that we can't know. We can't even guess, since probability doesn't seem like it applies.
*Bizarre footnote: I googled the term"Epistemological Infinity" to see if it's been used before, and what came up? Gay hardcore and prescription drugs. Huh?
**If after reading this, you would argue that I haven't really gone beyond the issue of Turing computability, please leave a comment, and explain why!
***The time we waste reading news and farting around on the web notwithstanding. posted by Miles 10:50 AM
So, what immediately pops to mind for me is that you've made a good case for the possibility that all that stuff you think doesn't exist - god, life after death, the soul, etc. - might exist after all, simply outside the realm of knowable. It might even be knowable for some people, despite being unknowable for you, because within your state you have no way of knowing whether other people might have some sort of difference in them that allows knowing things you can't know.
Mmm. Pink elephants.
I also think that I'm a bit confused by the bit where you show that the universe could be a computation (did I even say that right?) Because I'm not so familiar with the terminology you're using - I got a bit twisted up in it.