MATH EXPLORERS' CLUB Cornell Department of Mathematics 

Puzzles and Paradoxes: Infinity in Finite Terms


Motivation: Zeno's Paradox

Zeno of Elea asked a simple question that, to this day, can still be bothersome. He asked: How is movement possible?

Not one for posing idle questions, no matter how deep and philosophical they may have sounded, Zeno backed up his query with a simple thought experiment. Imagine that you are running a race, say the 100 meter dash. Certainly (unless you cheat) before you reach the finish line you'll have to pass the 50 meter mark. But, then again, before you run the remaining 50 meters, you'll have to go half that---you'll have to reach the 75 meter mark. Of course, this line of reasoning can continue forever: before you go any given distance, you have to go half that, and before you go the remaining distance, you'll have to go half again that distance, and so on. So it seems that in order to complete a simple race, you're going to have to accomplish infinitely many things along the way (first run 50 meters, then run 25 more meters, then run 12.5 more meters, and so on): an impossibility!

But, we all know, it can't really be impossible, since the 100 meter dash is routinely completed. What's wrong here? The division of the 100 meter dash into infinitely many shorter runs is strange, to be sure, but it's hard to point to anything fundamentally wrong about it. The only other substantive claim made in the presentation of Zeno's paradox is that it's impossible to complete infinitely many tasks in a finite period of time. Could this be mistaken?

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