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Zeno's paradoxes
Zeno of Elea, around 450 BC, as reported and answered by Aristotle in the Physics.
Achilles, the fastest runner in Greece, races a tortoise and gives it a head start of 100 metres. Say he runs ten times as fast as the tortoise. By the time he has covered the 100 metres, the tortoise has crawled 10 metres further on. By the time he covers those 10 metres, it has moved 1 metre. When he covers that metre, it has moved 10 centimetres. Every time Achilles reaches the spot where the tortoise was, the tortoise has moved on, and there is a new gap to close.
Zeno of Elea put this argument, and a simpler one called the dichotomy, about 2,450 years ago. The dichotomy says that before you can cross a room you must reach the halfway point, before that the quarter point, before that the eighth, and so on with no first step. Each argument seems to show that finishing requires completing infinitely many steps, and infinitely many steps can never be finished. Does Achilles ever catch the tortoise?