# Zeno's Paradox of Achilles and the Tortoise

The Wikipedia article tells us

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. — as recounted by Aristotle, Physics VI:9, 239b15

In the paradox of Achilles and the tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. Suppose that each racer starts running at some constant speed, one faster than the other. After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say 2 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise.

Here is the resolution to the paradox. I doubt it is original. It is true that the number of intervals before Achilles catches up to the tortoise is infinite. This does not, however, imply that the sum of these intervals is an infinite amount of time. However you divide it up, the intervals gets shorter and shorter, and they get shorter faster than they increase in number. Thus, it is quite possible for Achilles to pass an infinite number of benchmarks in finite time.

I think this is what has been called The Standard Solution. See the Internet Encyclopedia of Philosophy. It is what a lot of solutions on the Web are groping towards. Usually, discussions say that this solution was impossible until the concept of "mathematical limit" was rigorously defined in the 19th century. I disagree. What I said above is essentially the same as what is going on with the limit argument, but is understandable without previous mathematical knowledge. We don't have to define limits. All we have to show is that if the intervals get shorter as specified, it won't take Achilles much time to get through all of them.