Member Rara Avis
Basically all things in motion exist in a space, so the corrolary would be that space can't exist without motion occurring within it.
Uh? I don't see any logical reason for one to follow the other. You could just as easily say "All coffee exists in space, so the corollary would be that space can't exist without coffee." I certainly can't exist without coffee, but I suspect space would do just fine.
But can motion occur without time? I mean any moving object must have a set time it reaches a point in that space, and a set time it passes through that point.
I guess that depends on your notion of motion. Let's say we have points A, B, and C all lined up in a neat little row. Those three points are the only three in our entire Universe. If I am standing at A and then instantly appear at C, does that fit your definition of motion? Did I "move" from A to C? Did I necessarily "move through" B to get there? According to Quantum Theory, these kinds of goings on are going on constantly, giving us the term Quantum Leap long before television warped the real meaning. It's fairly clear that time is not a factor, as the transitions from one point to another are instant. It is far less clear whether space is a factor (or whether it is, instead, ignored).
There certainly exists a clear and definite correlation between motion and time, but what you've described is only a limited geometrical view. At relativistic speeds, time slows down to a crawl. A radioactive particle traveling at the speed of light (were that possible) would have a half-life of eternity.
Your question pitting motion against space, in a sense, falls within the scope of a much larger question. Can space exist without something to occupy it? Mathematically, it seems possible that our Universe has an "edge," a place where it stops and something else (or nothing else) begins. Since mass and motion are both just different aspects of the same thing - energy - we can perhaps rephrase the question. Can energy and space exist separately, one without the other? In at least one theoretical instances, the answer seems to be a qualified yes.
The Big Bang theory postulates an explosion (for lack of a better term) that flung monstrous quantities of mass and energy from a central point, thus signaling the formation of our Universe. Prior to the Big Bang there existed neither space nor time (making the phrase "prior to" completely meaningless). If the Big Bang was an exploding singularity, however, it was likely an infinitely dense concentration of energy - which apparently existed apart from space.
That's my long answer. The short answer is much, much easier.
Time is just another word for entropy.