Interesting discussion from NOVA:
Zeno’s paradox is solved, but the question of whether there is a smallest unit of length hasn’t gone away. Today, some physicists think that the existence of an absolute minimum length could help avoid another kind of logical nonsense; the infinities that arise when physicists make attempts at a quantum version of Einstein’s General Relativity, that is, a theory of “quantum gravity.” When physicists attempted to calculate probabilities in the new theory, the integrals just returned infinity, a result that couldn’t be more useless. In this case, the infinities were not mistakes but demonstrably a consequence of applying the rules of quantum theory to gravity. But by positing a smallest unit of length, just like Zeno did, theorists can reduce the infinities to manageable finite numbers. And one way to get a finite length is to chop up space and time into chunks, thereby making it discrete: Zeno would be pleased.
He would also be confused. While almost all approaches to quantum gravity bring in a minimal length one way or the other, not all approaches do so by means of “discretization”—that is, by “chunking” space and time. In some theories of quantum gravity, the minimal length emerges from a “resolution limit,” without the need of discreteness. Think of studying samples with a microscope, for example. Magnify too much, and you encounter a resolution-limit beyond which images remain blurry. And if you zoom into a digital photo, you eventually see single pixels: further zooming will not reveal any more detail. In both cases there is a limit to resolution, but only in the latter case is it due to discretization. More.