The apparent paradox is what happens when the way we look at things is too one-sided. We could say that it should be impossible to travel through an infinite number of points for any finite distance. But we could also say that since each point would then be infinitesimal, we should travel each point instantaneously, so that travelling through any number of such points should also be instantaneous, regardless of how far we've actually travelled. So if we are instantaneously travelling through an infinite number of points of zero dimension, how far can we possibly get? Well, zero distance divided by zero time gives us 0/0 (or (1/infinity)/(1/infinity), analogous to snarkophilus' post earlier). 0/0 gives us any real number, since any number can be multiplied by zero to give us zero (0/0=x, x*0=0), and we can therefore travel at any finite speed through any finite distance.

In fact, the idea of infinitesimal points is only a mathematical convenience anyway, as with expressing the gravitation of a spherical body as the representation of a single point. This is only possible because the actual radius of the sphere does not figure into the formula for gravity. Similarly, the idea of space as the compilation of infinitesimal points in a "connect the dots" sort of way is not valid. It is much more than that. Points really only signify a beginning and an end, as with boundaries and such. What lies between is something very different.

I'm not sure how it should really be thought about, though. I try mental experiments like this a lot. For instance, instead of just empty space, let's think about something substantial, like particles. We know that bodies are made up of atoms, which are made up of particles. Those particles can also be broken down into more particles, so what would the ultimate particle be? It would have to be something that cannot be broken down any further, so infinitely rigid. But if it is rigid, then it has an edge. If it has an edge, then it has structure. If it has structure, then something else must form this structure, so it is not the ultimate particle. If it has no edge, then it's concentration must taper off, and it is penetrable, both of which means it still has internal structure. If it is pliable, as a pure energy or a wave, then its structure changes, and so it must still be comprised of some more basic substance in order to do so, which would itself undergo the same dilemma.

It would appear, then, that our ideas about everything having a beginning and an end, whereby we define points in space and time, and that everything is built up something fundamental, may not be the case. It may be more in compliance to the way our brains operate, by starting with the most basic concepts we can comprehend and building up from this, with a beginning point and an end, while in reality no such boundaries might even exist.

__________________
Let's put together the pieces of The Grand Puzzle . (website)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."