Zeno's Paradox

For more than a thousand years, philosophers and mathematician have tried to understand Zeno's paradoxes, a set of riddles that suggest that motion should be impossible or that it is an illusion. Zeno was a pre-Socratic Greek philosopher from southern Italy. His most famous paradox involves the Greek hero Alchilles and a slow tortoise that Achilles can never overtake during a race once the tortoise is given a head start. In fact, the paradox seems to imply that you can never leave the room you are in.  In order to reach the door, you must first travel half the distance there. You;ll also need to continue to half the remaining distance, and half again, and so on. You won't reach the door in a finite number of jumps! Mathematically one can represent this limit of an infinite sequence of actions as the sum of the series (1/2 + 1/4 + 1/8 + ...). One modern tendency is to attempt to resolve Zeno's paradox by insisting that this sum of infinite series 1/2 + 1/4 + 1/8 is equal to 1. If each step is done in half as much time, the actual time to complete the infinite series is no different than the real time required to leave the room.
However, this approach may not provide a satisfying resolution because it does not explain how one is able to finish going through an infinite number of points, one after another. Today, mathematicians make use of infinitesimal (unimaginably tiny quantities that are almost but not quite zero) to provide a microscopic analysis of the paradox. Coupled with a branch of mathematics called nonstandard analysis and, in particular, internal set theory, we may have resolved the paradox, but debates continues. Some have also argued that if space and time are discrete, the total number of jumps in going from one point to another must be finite.