Zeno’s Paradox, specifically the one about a race between Achilles and a tortoise (rabbit, hare), has annoyed me for years, because so many folks casually dismiss it as a simple mathematical limit issue, i.e., “the infinite step process takes a finite amount of time, so stop with the Zeno’s Paradox bit already!”
As Dr. Grime, above, points out, excellently, how can an infinite process have a last step, so how can it terminate in a finite amount of time, or more to the physics side of the question, can space, or equivalently time, be subdivided an infinite number of times?
I would submit that Zeno’s Paradox strongly suggests that space and time cannot be infinitely subdivided, that they must be quantized. Of course, suggesting this immediately runs up against the problem of Heisenberg’s Uncertainty Principle, which does not state that space cannot be infinitely subdivided, only that the description of the motion of a particle becomes increasingly meaningless, because the tolerance for a particle’s momentum blows up the more we localize it in space.
My vote is for quantization of both space and time as a reasonable explanation for Zeno’s Achilles and Tortoise Paradox, namely that a finite number of minimum sized steps allows Achilles to pass the tortoise in a finite amount of time.
A similar issue that pops up is whether space and time have fundamental meaning, or whether the truly fundamental measurable variable is motion. We are conditioned to think of motion as derivative of space and time, but I have come to the opposite conclusion, that we infer time’s passage, and the immediacy of space, by the observation of motion.