I know exactly what it is you’re trying to say; but even knowing that, your description confused me! You somehow managed to make it sound even* more* complex! I don’t think it’s gonna help most people understand the point (though I should acknowledge the possibility that it’s just me, and that I’m far dumber than average!)
It’s hard to describe stuff. 
There’s no reason a hyperbolic geometry couldn’t have infinite volume, and in fact the simplest topology on a hyperbolic space is infinite in volume.
Or if you are old enough to have played one of the original space war games, which took place in a toroidal, 2D universe.
Huh?
Seconded. Mind was blown by the other thread. I’ve never heard of this idea before. And apparently, it’s not just some idea–it’s the idea.
Popularizations, AFAIK, never explain this correctly!
Okay mister smartypants, the two dimensional surface of a balloon can expand without having an edge.
Serious yet horrifically flaky sounding question:
If the universe is closed, is it necessarily true that its size is increasing, or could the physics be interpreted instead as involving everything in the universe shrinking?
(If you’re floating next to me in empty space, and we’re both shrinking, then we will appear to each other to be receding from each other in the sense that you will be a greater and greater number of armslengths away from me.)
Follow up related question:
Is it understood that there are, or must be, two particles in the universe which are further apart from each other than any other combination of particles? If there is a finite amount of matter in the universe (which seems likely), it MUST be true that there are two particles which are further away from each other than other combation of particles. If that is the case, why couldn’t the two furthest-apart particles be thought of as the “edges” of space?
Here’s what I think JWT Kottekoe is saying.
Nothing can become infinite. Things either are infinite or not. Therefore, if the universe today is infinite, then it was always infinite. Even if the universe started in a singularity, it went immediately, without any transition, to infinite. (Similar to how photons go from creation to moving at the speed of light without transition.)
If the universe is finite, OTOH, then it was always finite, and always will be. No matter how long it expands, it can only become a larger finite size.
Which is it? It all depends on the curvature of the universe. A flat universe, which as I said is what current understanding is, corresponds to an infinite universe. Therefore the universe always was infinite and never small, even though the parts of it we can see today - the observable universe - must have been smaller earlier.
This is completely contrary to common sense, because all infinities are contrary to common sense. Another reason to abandon common sense, all ye who enter physics.
Mosier, the answer to your question is that no such particles can exist. Every particle can be thought of as the center of the universe and thus is equally far away from the end of the observable part from its reference frame.
“Space” is has not been precisely defined yet by our physics knowledge.
We don’t know what “space” is.
You’ll have to rephrase your question.
I am not limiting my question to the observable universe or a particular reference frame, unless “universal” counts as a reference frame. Are there two particles in existence that are further away from each other than any other combination of two particles?
As currently understood, and very well explained above, any particle in existence will find, from its point of view (its “frame of reference”) that it is at the center of an infinite number of other particles, which go off in every direction to infinity. This applies to any particle which you can identify, so it is not possible to locate two particles which are farther apart than any other combination of particles.
This is precisely related to the inadequacy of analogies to make reality more “user-friendly.” Reality is what it is, which, unfortunately, is very counter-intuitive. We create “Models” of parts of reality, which necessarily leave out some aspects, and severely limit others, to describe some portion of reality in terms we can comprehend. The problems arise when we try to think of the Model as if it was the reality.
Most workers in technical fields run into this dilemma: to describe their field to a non-specialist, they must make compromises. Do they use words which the average well-educated person would know, even though those words may be freighted with connotations and ancillary meanings which will, inevitably, confuse the audience? Or do they use the “jargon” specialists in the field use to describe their work—words which precisely, and unambiguously, define the ideas which they wish to convey—but nobody outside their field will understand without lengthy footnotes?
Neither option is fully satisfactory. Most popularizations of technical material, cosmology in this case, have opted for the former solution—using commonly-understood words. This inexorably leads to confusion when the audience extrapolates the words beyond the purview of the model. In the “language” appropriate to the subject, mathematics, all this talk of infinities resolves neatly into well-understood concepts. There are areas which are not completely known (which is why research continues at a frenetic pace, even today) but we have a pretty good handle on the constraints within which we believe “reality” will eventually be described.
From how I understand it space is ‘infinite’ if you can’t travel faster the the speed of light, as space is being created inbetween itself at a rate that there are places you will never be able to get to because you can’t travel faster then the rate that the universe is expanding.
Space may have a similar characteristic to a black hole, with a finite ‘size’ if observed from the outside and a infinite size if observed from the inside, along with points you can never reach without exceeding the SoL. The universe as we know it may be the inside of a very large black hole.
In order for that to be true, there must be an infinite number of particles in the universe. It was my understanding that our best guess at the moment is that there is NOT an infinite number of particles in the universe. Am I wrong?
You are confusing two different issues.
It is completely possible for a finite space to have regions that are not in communication with one another. The observable universe has a radius (or better, comoving distance) of 13.7 billion light years. 13.7 billion is also considered to be the age in years since the big bang. Yet because of the expansion the size of the universe is 46 billion light years. Both numbers are finite, but we will never, even theoretically, be in communication with anything 46 billion light years away.
Infinite is a basic property of space if it exists. It is not being created by expansion. Our universe could be one of many independent universes in infinity. That’s my limit of understanding, though, and I don’t want to say anything more that is likely to be incorrect or misleading.
What you’re trying to argue is equivalent to saying that the universe has a center, a preferred reference frame from which everything is expanding from. That’s not true. (You might want to check for the many threads in which this is explained better than I can.) The universe expands equally everywhere. That’s because all reference frames are equal, and therefore every particle can look at itself as the center. That’s true no matter how many particles are in the universe, even if that number is finite. If there is no fixed center, then there can be no fixed edge. You’re trying to define a diameter. The universe doesn’t have a diameter; it has only radii. Yes, that’s not common sense.
Let me answer this for the case of a finite universe. The simplest finite but unbounded (no edges) universe is analogous to the surface of a hypersphere. In the expanding balloon analogy, every point on the balloon has an anitpode, a point that is maximally distant, all the way on the other side of the balloon. The same is true in a closed, positively curved universe. Thus, while you are correct that with a finite number of particles, barring exact coincidences, there would be two of these that are the furthest apart of any pair, these two particles will not in any sense constitute an edge.
Wouldn’t’ time as a dimension and a hard upper limit of the SoL make it infinite?
Try the balloon analogy once again. It’s a two-dimensional finite surface. You can add a third, time, dimension to it. It can have a fixed speed of light of any number. Yet it is obviously not infinite. A three-dimensional curved space would work the same way.
Again from the outside it has a fixed dimension, though inside could it be infinite as some postulate is inside a black hole?
From some models of a black hole spacetime gets ‘stretched’ to infinity, which creates infinite space inside a finite space.