Zeno's paradox

Among the most famous of Zeno's "paradoxes" involves Achilles and the tortoise, who are going to run a race. Achilles, being confident of victory, gives the tortoise a head start. Zeno supposedly proves that Achilles can never overtake the tortoise.

Achilles gives the tortoise a head start, in a race. Before he can overtake the tortoise, he must run to the place where the tortoise began, and the tortoise has move on to some other point. From there, before he can overtake the tortoise, he must run to the place where the tortoise had move on to. This goes on forever, and Achilles can never pass the tortoise.

Here, I paraphrase Zeno's argument:
Before Achilles can overtake the tortoise, he must first run to point A, where the tortoise started. But then the tortoise has crawled to point B. Now Achilles must run to point B. But the tortoise has gone to point C, etc. Achilles is stuck in a situation in which he gets closer and closer to the tortoise, but never catches him.

A finite length can be divided up into an infinite number of pieces, all of zero length. You can imagine that, can't you. Just divide a length into halves, then fourths, then eighths, etc. But, in the Zeno story above, we find that none of the pieces is of zero length.


Here is an animation which counts to infinity in 8 seconds. Actually, I cheated. There are not an infinite number of frames in this animation. I skipped 1/100 second (approximately). "1" takes 4 seconds, "2" takes 2 seconds, "3" is 1 sec., then .5 sec., .25 sec., etc. If our computers were infinitely faster, we could get in infinitely many frames in that 8 seconds.
Adapted from http://www.jimloy.com/physics/zeno.htm

Therefore, there must be a finite number of pieces. Hence, this finite number of pieces is coverable in a finite amount of time, since the nett speed (subtract the tortoise's speed from Achilles' speed) is obviously greater than zero i.e. Achilles is faster than the tortoise. This proves that Achilles can overtake the tortoise eventually. Effectively, Achilles will overtake the tortoise when t = headstart/nett speed.

The question is when, and whether he'll do so before running out of time or distance to do so. There are therefore three different outcomes to this farcical race. To quote a gameshow, it is Achilles' race to win, lose or draw, interestingly all dependent on the finishing line's position. Even with so many unknowns, it is actually possible to establish mathematical inequalities to demonstrate the three outcomes.

Basically, there is a tie when headstart of tortoise + speed of tortoise x time given = speed of Achilles x time given. Here, time is obviously the time taken for the tortoise to reach the finishing line, since he is in effect winning from the start of the race.

Making the time given the subject of the equation,
The tortoise wins if: t > nett difference of speed divided by headstart
Achilles wins if: t < nett difference of speed divided by headstart
Draw if: t = nett difference of speed divided by headstart

I end this discussion with Jim Loy's little anecdote:

Little-known story: Achilles didn't win the above race. So, he challenged the tortoise to a pole vault competition, double or nothing. The tortoise's pole bent impressively, before it catapulted him out of Greece, never to be seen again. I made that one up.