Zeno, Russell, and Borges on Tortoises

Point at Infinity

There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite.

-Jorge Luis Borges, “Avatars of the Tortoise”

This week, we will consider two classical paradoxes of infinity. Today, we turn to Zeno of Elea, a pre-Socratic Greek philosopher. Zeno, who lived during the 5th century BC, is best known today for his four paradoxes of motion, the most famous of which is that of Achilles and the tortoise.

The paradox goes as follows. Achilles and the tortoise are engaged to run a 100 meter race. Achilles runs ten meters per second, the tortoise runs one meter per second,  and Achilles gives the tortoise a head start of ten meters. An easy algebraic calculation reveals that Achilles will catch up with the tortoise after seconds, and Achilles will then quickly pull ahead and win the…

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