Zeno's Paradox (Math Lair)

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Zeno's Racecourse Paradox involves the story of a race between Achilles and a tortoise. In this race, Achilles, being much faster, gives the tortoise a head start. Zeno's assertion is that Achilles can never overtake the tortoise, since when Achilles reaches the point where the tortoise started, the tortoise has moved ahead somewhat, say to point A. When Achilles reaches point A, the tortoise has moved ahead to point B. When Achilles reaches point B, the tortoise has moved further. Therefore, the tortoise must always hold a lead. This is quite similar to Zeno's bisection paradox, which is examined in detail below. This conclusion is very counter-intuitive. For example, everyone can remember overtaking someone while walking, driving or biking. If Zeno's assertions were true, motion would be impossible.

Zeno's Bisection Paradox:

Zeno's Assertion:

A runner can never reach the end of a racecourse in a finite time.

Statement: Reason:
1. The runner must first pass the point ½ located halfway between himself and the finish line before he can finish the race. ½ is between the runner and the finish line.
2. It will take a finite time to reach the point ½. It is a finite distance from the start (1) to ½.
3. Once reached, there is another halfway point ½ which the runner must reach before he can finish. The remaining interval is divided in half.
4. There are an infinite number of such halfway points which the runner must reach. Each of these points will take a finite time. Statements 1, 2, and 3 can be repeated an infinite number of times.
5. The total time for the race is infinite. The sum of an infinite series of finite terms is infinite.

	Statement:	Reason:
1.	The runner must first pass the point ½ located halfway between himself and the finish line before he can finish the race.	½ is between the runner and the finish line.
2.	It will take a finite time to reach the point ½.	It is a finite distance from the start (1) to ½.
3.	Once reached, there is another halfway point ½ which the runner must reach before he can finish.	The remaining interval is divided in half.
4.	There are an infinite number of such halfway points which the runner must reach. Each of these points will take a finite time.	Statements 1, 2, and 3 can be repeated an infinite number of times.
5.	The total time for the race is infinite.	The sum of an infinite series of finite terms is infinite.

The problem with this reasoning is in step 5. The sum of an infinite series of finite terms is not necessarily infinite. Some infinite series, such as the harmonic series, are infinite, but not this series. The sum of the series ½ + ¼ + ¹/₈ + ... is equal to 1. The total time is finite because each step is done in half as much time as the previous step. Many of the ancient Greeks had problems with infinite concepts like this.