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Zeno's Paradox Two

  1. Sep 7, 2004 #1
    Achilles and the Tortoise
    Suppose the swift Greek warrior Achilles is to run a race with a tortoise. Because the tortoise is the
    slower of the two, he is allowed to begin at a point some distance ahead. Once the race has started
    however, Achilles can never overtake his opponent. For to do so, he must first reach the point from
    where the tortoise began. But by the time Achilles reaches that point, the tortoise will have advanced
    further yet. It is obvious, Zeno maintains, that the series is never ending: there will always be some
    distance, however small, between the two contestants. More specifically, it is impossible for Achilles to
    preform an infinite number of acts in a finite time.
    Distance behind the Tortoise:
    ------5, 2.5, 1.25, 0.625, 0.3125, 0.015625, ….
    Time: 1, 0.5, 0.25, 0.125, 0.625, 0.03125,

    extended version

    In the modern analysis, the paradox is resolved with the fundamental insight of calculus that a sum of infinitely many terms can yield a finite result. Adding the (infinitely many) times together that Achilles needs to reach the previous positions of the tortoise results in a finite total time, and that is indeed the time when Achilles overtakes the tortoise.
  2. jcsd
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