Even if space and time are continuous, there's still no paradox, really.

Let's get our use of "infinity" straight. "Infinity," for our purposes, will be a very big number... something bigger in magnitude than all the real numbers. "Infinitesimal" will refer to something smaller in magnitude than any real number (but bigger than zero). We'll also say that you can multiply infinitesimal by infinity to get some real number.

Say you want to move one meter in one second. Now, Zeno might argue that you can never get there, because first you have to go half a meter, et cetera. The answer is that it only takes half a second to go that half meter. It only takes a quarter second to go a quarter meter. And so on, down to infinitesimal times and distances.

In fact, by the time you get down to infinitesimal distances (dividing by 2 infinity times), you're also dealing with infinitesimal lengths of time. In essence, it takes no real amount of time to move that tiny (unreal) distance. And if you move that unreal distance infinity times, then you get a real distance... and it takes infinity times infinitesimal = some real number amount of time.

Looking at it in slightly mathy terms, you move 1/infinity meters in 1/infinity seconds. Those are both infinitesimals. And when you solve for how long it takes to go 1 meter, you get (1/infinity seconds)/(1/infinity meters) x 1 meter = (1 * infinity seconds / infinity * 1 meters) x 1 meter = (1 second/meter) x 1 meter = 1 second, just as predicted. The infinities cancel nicely.

(The assumption is that those are the same infinity, of course. Properly, you'd do that with limits....)

So yes, you can certainly move infinite distance in infinite time, but even more importantly, you can move infinitesimal distance in infinitesimal time. If it took real time to move an infinitesimal distance, then there would be a problem. But as it stands, the infinities cancel out: the time gets shorter at some real multiple of the rate that the distance gets shorter, so the two reach infinitesimal at the same time.

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