Before Achilles can catch the tortoise he must reach the point where the tortoise started. But in the time he takes to do this the tortoise crawls a little further forward. So next Achilles must reach this new point. But in the time it takes Achilles to achieve this the tortoise crawls forward a tiny bit further. And so on to infinity: every time that Achilles reaches the place where the tortoise was the tortoise has had enough time to get a little bit further, and so Achilles has another run to make, and so Achilles has in infinite number of finite catch-ups to do before he can catch the tortoise, and so, Zeno concludes, he never catches the tortoise.

 

The following argument against motion explicitly turns on a particular kind of assumption of plurality: that time is composed of moments (or ‘nows’) and nothing else. Consider an arrow, apparently in motion, at any instant. First, Zeno assumes that it travels no distance during that moment -- ‘it occupies an equal space’ for the whole instant. But the entire period of its motion contains only instants, all of which contain an arrow at rest, and so, Zeno concludes, the arrow cannot be moving.

 

Tantalus was the son of Zeus and was the king of Sipylos. He was uniquely favored among mortals since he was invited to share the food of the gods. However, he abused the guest-host relationship and was punished by being "tantalized" with hunger and thirst in Tartarus: he was immersed up to his neck in water, but when he bent to drink, it all drained away; luscious fruit hung on trees above him, but when he reached for it the winds blew the branches beyond his reach.

 

Sisyphus is the son of Aeolus (the king of Thessaly) and Enarete, and founder of Corinth. He instituted, among others, the Isthmian Games. According to tradition he was sly and evil and used to way-lay travelers and murder them. He betrayed the secrets of the gods and chained the god of death, Thanatos, so the deceased could not reach the underworld. Hades himself intervened and Sisyphus was severely punished.

In the realm of the dead, he is forced to roll a block of stone against a steep hill, which tumbles back down when he reaches the top. Then the whole process starts again, lasting all eternity. His punishment was depicted on many Greek vases. He is represented as a naked man, or wearing a fur over his shoulders, pushing a boulder.