[giberson]

Expanding on "There is no 'outside the universe'".

Is the universe infinite in dimension?

Is there such a path that a particle could take at infinite speed for an infinite time such that the distance between itself and a particle that had stayed in the same position is always increasing?

Is there infinite matter in the universe?

If no, is there a point at which this traveling particle will have left the "vicinity" of all matter, to never meet another particle again?

[natep]

(Edit: I'm responding to each question in its own paragraph)

What do you mean, 'infinite in dimension'? The universe has only 4 dimensions, 3 spatial, 1 in time.

Your question barely makes sense to me. Why do you need to invoke infinite speeds and times? If 2 particles stay in the same place, and do not interact with each other (gravitationally, electromagnetically, etc) then the distance between those 2 particles would increase over time, due to the expansion of the universe.

Yes, there is an infinite amount of matter in the universe [1]

Maybe. If you pick a random direction and travel really fast for a really long time, the chances of you running into any other matter is extremely remote. [2]

[1] http://www.reddit.com/r/askscience/comments/ewt9y/good_analo...

[2] http://www.youtube.com/watch?v=0fKBhvDjuy0

[giberson]

     What do you mean, 'infinite in dimension'? The universe has only 4 dimensions, 3 spatial, 1 in time.

I'm was not asking how many dimensions the universe has. I was asking if the universe is infinitely tall, infinitely wide and infinitely long.

     Why do you need to invoke infinite speeds and times? 

Simply for matters of scale. I was trying to avoid considerations like "it's so big, it would take so long ...".

     If 2 particles stay in the same place [...]

Actually, it's one particle that moves, and other is stationary, thus providing two points of reference for determining a displacement. Specifically the intention was to indicate a path through space such that you always are moving further from where you started and will never arrive back at your starting point (as opposed to the spherical theory of space such that you can only go half way around the sphere until you start getting closer to the point at which you started.

      If you pick a random direction and travel really fast for a really long time, the chances of you running into any other matter is extremely remote. 

I didn't mean physically collide with other matter, I simply meant to pass by it in the "vicinity". Similar to how traveling along a highway you pass by or through towns, you don't physically collide with them though. If matter in the universe was finite, then regardless of how sparse that matter is, there's an imaginary bounding box you could draw around all matter in the universe. A particle moving really fast for a long time would eventually end up outside of that bounding box. At which point its distance not just from the stationary particle at it's original, but all particles of matter would forever be increasing. However, if matter is in fact infinite, then such a scenario can not occur.

Luckily, despite my questions barely making any sense, you've managed to provide in your reference [1] a comment that answers exactly my questions so thank you for that.

[smeg]

The problem is you are applying linear Earth-scale concepts of space to the whole Universe.

In reality space (or space-time) is curved - as you approach the "edge" of the universe you actually are traversing a loop around a 4-dimensional hyper-sphere.

The common analogy of an ant on the surface of a balloon is quite effective. As the balloon expands, the distance between points on the surface of the balloon certainly grow and they become further apart. The ant can measure this, see this, but for the life of him, no matter how far he walks, he cant find the "edge of the universe".

Of course it is hard to visualise this analogy when projected into our 4-dimensional universe, but it's the best we've got.

[natep]

No, you do not 'loop around' the universe. It is flat, not a hyper-sphere. The 4th dimension of the universe is time, and not spatial.