Idioms, Fallacies, and Paradoxes

Achilles and the Tortoise

If the tortoise is ahead of Achilles, by the time Achilles reaches the tortoise's current position, the toroise will have moved a bit further ahead, which goes on indefinitely.

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

Achilles allows the tortoise a head start of one hundred metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run one hundred metres, bringing him to the tortoise's starting point.

During this time, the tortoise has run a much shorter distance, say, ten metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise.