Si
But none of that addresses Frylock’s questions or challenges anything he said, does it?
As for the volume argument he asked it about, it is based on the idea chrisk gave: Suppose, for the sake of contradiction, that there is some positive distance R such that arbitrary unidirectional motion from any starting point eventually causes one to acquire and maintain a minimum distance of at least R from that origin. Then, were one to carry around a ball of radius R/2 centered on oneself at all times, unidirectional motion would cause one’s ball to eventually become and remain disjoint from its starting location; and then become and remain disjoint from such a second location; and then become and remain disjoint from such a third location; etc. Adding up the volume contained in the ball during this sequence of snapshots, we would obtain an infinite quantity, presumably in contradiction to the intuitive idea of what it means to be a finite universe.
Thus, we can conclude, for arbitrarily small positive R, there are unidirectional paths which return to within R of their origin. Then you can invoke some additional smoothness principles as Chronos does to conclude that there is a unidirectional path which returns to its origin.
Of course, this argument depends on quite a few assumptions, explicitly and implicitly, and you can always choose not to impose some of them; e.g., perhaps you do not demand that the universe have a well-defined notion of volume (and, accordingly, do not take the finiteness of the universe as finiteness of volume but merely a fixed finite upper bound on distance between points), or you could deny whatever principles are required for the “smooth-scale” reasoning [which, I’ll admit, I’m not entirely clear on]. As Chronos said, you may not be inclined to call a structure a universe, but who knows? You’ll have to either off more formal guidelines to what counts, or we’ll have to toss out examples till you accept one.
[Example coming up]
Er, “offer more”, not “off more”.
[Also, that example is on hold for a second while real life intrudes]
Thanks (and OP, please forgive the hijack) but that cite discusses finite spaces with some straight lines that aren’t loops, and infinite spaces with no straight lines that aren’t loops, but does not say whether there could be a finite space with no straight lines that aren’t loops.
I can kind of picture something which might work. Suppose there is a space which can be mapped onto a two dimensional circular plane figure in the following way. The map of any straight line in the space is, on the circular plane figure, a spiral which constantly approaches the edge but never reaches it. (Going the opposite direction, the line constantly approaches the center but never reaches it.)
I don’t know if I’m actually describing something coherent, though, hence my question.
-FrL-
After reading Indistinguishable’s last post and rereading Chronos’s, I think maybe the space I described would be non-compact and so, intuitively, “infinite.” I was confusing the finitude of the map with a purported finitude of the space. Oopsie.
Now I got nothin’.
-FrL-
This’d be a great time for someone to school me about just what exactly a “space” is. 
-FrL-
Actually, one point I wanted to make was that there are a variety of options on the table for what we take “space” to mean (and, for that matter, “finiteness of a space”), and the answer to your question will depend on what you choose. Mathematics is the handmaiden of investigation, not the other way around. (Though I’m sure Chronos can offer much on the constraints offered by theories in/the paradigms of modern physics, if that’s part of what you’re looking for)
I also think the situation you’re describing can be made to work for a reasonable (if perhaps aphysical [or, rather, aphysics-al]) notion of “space” and “finiteness”, and I’ve been trying to type up something along those lines though more mechanically generated than along the lines of your intuition, but it hasn’t made its way to completion yet, alas.
I’ve hesitated to expand more on the ‘using up volume’ argument for fear that somebody will open up the relativistic barrel of monkeys - that it’s very hard to be sure if you’re remaining in place or drifting at a constant velocity in a relativistic universe, and thus to a certain extent the question of returning to the same point in space at a later time is fundamentally unanswerable. (If I get that right.)
Oh, whoops, did I just open the barrel myself? 
Well, we can imagine living in a finite universe that is inflating at faster than the speed of light. So you leave home at less than the speed of light, but your starting point receeds from you faster than the speed of light. Your starting point becomes outside your light cone and can never be accessed by you again. I’m not sure if this is possible if the universe is expanding slower than c.
Sure, and if we’re going to allow situations where we can say “You can get from A to B but you can’t retrace your steps to get back”, then it’s easy to construct “spaces” satisfying the condition. But I assume those are not the sort of thing Frylock is looking for.
Incidentally, the introduction of “compactness” into the discussion has been a red herring, I think; it’s true that this corresponds to a certain notion of finiteness, but not the relevant one (any more than having finitely many points is the relevant one). E.g., observe that the real interval (0, 1) is not compact, while the whole real line augmented by a “point at infinity” is compact. Compactness is a purely topological notion, but presumably, we are looking for a more metric notion of finiteness, which can take into account concepts like “distance”.
As a question to Frylock, while I still try to work out a concrete construction of an object like he has in mind, in your example of straight-line movement in one direction corresponding to spirals, what did you picture straight-line movement in other directions to be? If I were to start tracing along such a spiral (thus moving in a constant direction) and then decide to make a 90 degree turn and then continue my unidirectional movement, what might I then be tracing out? [Or is this question not in keeping with how you were choosing to formalize the concept of a space?]
Change “it’s easy” above to “it seems easy”. It depends on what the other constraints in the formulation of spacehood continue to be, but in any case, I can’t pretend to have an example ready.
Right, I was trying to figure this out myself. What I have in mind, I think, would involve a situation characterized by the following fact. From any point in the space, a straight line in any direction from that space traces a kind of spiral approaching the edge or center of the disk. So if you stop and turn 90 degrees, you still are going to trace a spiral of some kind.
Spirals going in different directions from a point are going to have to look pretty different from each other, I think–some will appear to go quite straight toward the edge but bend into a spiral right at the end. Some will be tightly wound spirals from the very beginning.
I am at the limits of my imagination’s visual powers here, and have absolutely none of the math to work any of this out, so my apologies at how poorly defined the question has been.
Not only this, I am feeling more sure that whatever I’m calling a “space” has little to do with actual geometrical treatments of spaces. I have been implicitly thinking of a “space” as follows: A space is a densely ordered set of points, along with something I might call a “movement function” or “translation function”, by which I mean, a function taking points in that space together with vectors and outputting new points. I have been thinking specifically about a certain subset of these “spaces,” namely, the ones in which given a shape (i.e. set of points) in a space and a vector, the result of the relevant “translation function” will be a topologically equivalent shape located somewhere in that space.
Clarification requests are welcome.
-FrL-
Would another, briefer way of defining be to ask, “Is there any finite topological space in which any straight line of infinite length will never intersect itself?”
I don’t think it works. The space would have to be closed if it is finite, which really limits your ability to turn the function for a straight line into something useful.
Wait a minute. Following a question I asked some time ago, I was led to believe that we were living in an universe inflating faster than the speed of light.
So, do we know how fast the universe is expanding, or not?
The speed of expansion is known as Hubble’s Constant
We have a guess of around 71km/sec/Mpc. The margin of error on this is around ±4km/sec/Mpc. “71 km/sec/Mpc” means that we observe a rate of expansion of 71 kilometers per second for every megaparsec of distance between us and the object we are observing. A megaparsec is 3.26 million lightyears, 2 hundred billion astronomical units, or an inconceivable number of kilometers.
There are signs that the rate of expansion has changed over the lifetime of the universe. Briefly after the big bang, there is evidence to suggest the rate of expansion exceeded the speed of light.
If you look far enough away, things are receding faster than the speed of light NOW.
The speed of light is 300,000 km/sec. If the rate of expansion is 71 km/sec/Mpc then things farther away than 4225 megaparsec are receding faster than that.
During the early inflationary stage of the Big Bang the rate of expansion was much, much greater so things much closer to each other moved out of each other’s light cones. But it’s still happening right now.
I’ve read statements that the Earth is travelling through space at 30,000 mph, or something, but if we are rotating around an object going however fast the sun is going, within a galaxy which is travelling however fast it may be travelling, how can this be so?
True. But a little clarification: the distance between the two objects is greater than the speed of light. That’s distinct from the objects themselves moving that fast. Two flashlights pointing in opposite directions are shining photons out at the speed of light, c, and the distance between two photons shot out at the same time increases at 2c. Nothing in this system is moving at 2c, though, just the gap between them.
The same thing is happening in the expansion of the universe; stuff thousands of megaparsecs away is sort of like photons shooting out in the opposite direction.
I think that this is another spot where the ‘it’s all relative’ barrel pops open. It’s very hard to determine absolute speeds in the universe - you need to pick a frame of reference. That 30,000mph may be relative to the sun - relative to another stare the velocity might be quite different, or the center of our galaxy. or a different group of galaxies etc.
An important clarification, here: Hubble’s constant isn’t really a speed; it’s a speed per distance (i.e., a frequency, if you want to look at it that way). If you want to get an actual speed, you need to plug in a distance. If the Universe is infinite and flat, as currently appears to be the case, then there’s no particular distance of interest to plug in, so you can’t really say what the “speed of the Universe’s expansion” is, or compare it to c. However, for a finite or curved universe, there is such a particular distance: The size of the universe (or at least, the characteristic scale of the curvature). So in those cases, you really could talk about the speed of the universe’s expansion, in terms of c.