Achilles and the Tortoise: Imagine that Achilles meets a Tortoise, who challenges him to a foot race. Achilles is amused when the Tortoise asks merely for a modest head start. But then the Tortoise explains that by agreeing to this demand, Achilles has already lost! The logic, says the Tortoise, is that if he starts ahead of Achilles at point A, Achilles will have to run to point A before he can overtake the Tortoise (which is, of course, obvious enough). Meanwhile, the Tortoise will have moved ahead slightly to point B. Again, Achilles must advance to point B before he can push ahead, by which time the Tortoise will have traveled farther (if only by inches), to point C. And so on. Although with each successive point in the race the Tortoise moves smaller and smaller distances, Achilles never quite catches up, always remaining one segment behind. And thus, says Zeno, the faster can never overtake the slower.

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