Zeno's Paradox

I must say, I did intend to complete a discussion of Accelaration and Newtons laws today, but in trying to figure out the principle of differentiation, I came across this pretty interesting one which is a sort of introduction to limits and infinity (and beyond too!).
I promise that this will help while we discuss differentiation.

This one is about the greeks. There was this philosopher named Zeno, who was generally regarded as hot stuff in the philosophical circles of Greece. What did he come up with? Well, mainly he was of the school of thought that said that all motion is impossible, or some such thing. In order to prove it, he came up with these paradoxes, which at first sight seem terribly convincing. For example, there is his famous one about Achilles and the tortoise. (For those who came in late, Achilles was a mighty greek hero. He was a great runner. And showed a clean pair of heels to all his competitors. And was shot in the heel, causing his death. For the unabridged version, read the Illiad, by Homer). Below is a quote from the Wiki (always useful).

In the paradox of Achilles and the tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.

Now, try to spot the hole in that logic. Its actually quite tough. But I shall spare you, the reader from thinking this through. The problem with Zeno's logic is this. He assumes that for there are an infinite number of steps for Achilles to take to overtake the tortoise (Infinitely small steps). And since it takes a finite time to complete each step, it will take infinite time to complete infinite steps. In functions, this is called a one-to-one correspondence. But this is not necessarily true. Let us quantify the tortoise and Achilles problem like this.

In the case of Achilles and the tortoise, suppose that the tortoise runs at a constant speed of v metres per second (ms-1) and gets a head start of distance d metres (m), and that Achilles runs at constant speed xv ms-1 with x > 1. It takes Achilles time d/xv seconds (s) to reach the point where the tortoise started, at which time the tortoise has travelled d/x m. After further time d/x2v s, Achilles has another d/x m, and so on. Thus, the time taken for Achilles to catch up is


The above value is a finite number, which means that Achilles will eventually catch up with the tortoise. Notice, that without a definition that implies that distance and time can be related, this problem cannot be solved at all.

p.s: Almost all the material is lifted from the Wiki. The link is given below for those who wish to have a full explanation (and to get thoroughly confused...I was).
here