Setup: Achilles and the Tortoise engage in a race, but the tortoise is given a head start, while Achilles is the faster racer.

  1. Before Achilles can catch the Tortoise, he must reach the point halfway between his current location and the Tortoise's current location.
  2. While Achilles traverses the interval to the halfway point, the Tortoise can traverse another interval.
  3. If (1) and (2), then Achilles can never reach the Tortoise.
  4. So, Achilles can never reach the Tortoise.

Premise 3 is false because looking at all the points when Achilles reaches a midpoint is only looking at a sequence of points that get closer together, and thus correspond to points in time that get closer together (by proportionality through velocity). Since they are midpoints, it is a standard convergent geometric series. By thinking about the situation, you can realize that the points in time converge to the point where Achilles passes the Tortoise. Clearly if we only look at points in time before the passing ocurrs, we will never observe the passing itself, but this does not imply that the passing never occurs in the real world.