> > It's the classic story of crossing half a distance, then half the > remaining distance, on and on. The limit is 1. But you can never reach > the other side, even should you attempt to do so infinitely. I'm > always boggled why people say that a journey taken in nine-tenths the > remaining distance would in fact reach 1...
Zeno's Paraodx of Achilles and the Tortoise is perfectly resolved using the standard definitions of convergence and limit. If one sums of the times of the partial crossings one gets a limit hence a finite time for the crossing. The limit is reached in a finite time even though an infinite number terms are summed up.
1/2 + 1/4 + 1/8 + etc = 1 exactly. This is the sum of the times necessary to cross the remaining portion exactly.
Zeno's error was to assume the infinite sum must be either undefined or infinite. Not so. The system of (standard) real numbers is a resolution to Zeno's Paradox.
