This paradox turns on much the same considerations as the last. Imagine Achilles chasing a tortoise, and suppose that Achilles is running at 1 m/s, that the tortoise is crawling at 0.1 m/s and that the tortoise starts out 0.9 m ahead of Achilles. On the face of it Achilles should catch the tortoise after 1s, at a distance of 1m from where he starts (and so 0.1m from where the Tortoise starts). We could break Achilles' motion up as we did Atalanta's, into halves, or we could do it as follows: before Achilles can catch the tortoise he must reach the point where the tortoise started. But in the time he takes to do this the tortoise crawls a little further forward. So next Achilles must reach this new point. But in the time it takes Achilles to achieve this the tortoise crawls forward a tiny bit further. And so on to infinity: every time that Achilles reaches the place where the tortoise was, the tortoise has had enough time to get a little bit further, and so Achilles has another run to make, and so Achilles has in infinite number of finite catch-ups to do before he can catch the tortoise, and so, Zeno concludes, he never catches the tortoise.

One aspect of the paradox is thus that Achilles must traverse the following infinite series of distances before he catches the tortoise: first 0.9m, then an additional 0.09m, then 0.009m, … . These are the series of distances ahead that the tortoise reaches at the start of each of Achilles' catch-ups. Looked at this way the puzzle is identical to the Dichotomy, for it is just to say that ‘that which is in locomotion must arrive [nine tenths of the way] before it arrives at the goal’. And so everything we said above applies here too.

But what the paradox in this form brings out most vividly is the problem of completing a series of actions that has no final member—in this case the infinite series of catch-ups before Achilles reaches the tortoise. But just what is the problem? Perhaps the following: Achilles' run to the point at which he should reach the tortoise can, it seems, be completely decomposed into the series of catch-ups, none of which take him to the tortoise. Therefore, nowhere in his run does he reach the tortoise after all. But if this is what Zeno had in mind it won't do. Of course Achilles doesn't reach the tortoise at any point of the sequence, for every run in the sequence occurs before we expect Achilles to reach it! Thinking in terms of the points that Achilles must reach in his run, 1m does not occur in the sequence 0.9m, 0.99m, 0.999m, … , so of course he never catches the tortoise during that sequence of runs! The series of catch-ups does not after all completely decompose the run: the final point—at which Achilles does catch the tortoise—must be added to it. So is there any puzzle? Arguably yes.

Achilles run passes through the sequence of points 0.9m, 0.99m, 0.999m, … , 1m. But does such a strange sequence—comprised of an infinity of members followed by one more—make sense mathematically? If not then our mathematical description of the run cannot be correct, but then what is? Fortunately the theory of transfinites pioneered by Cantor assures us that such a series is perfectly respectable. It was realized that the order properties of infinite series are much more elaborate than those of finite series. Any way of arranging the numbers 1, 2 and 3 gives a series in the same pattern, for instance, but there are many distinct ways to order the natural numbers: 1, 2, 3, … for instance. Or … , 3, 2, 1. Or … , 4, 2, 1, 3, 5, … . Or 2, 3, 4, … , 1, which is just the same kind of series as the positions Achilles must run through. Thus the theory of the transfinites treats not just ‘cardinal’ numbers—which depend only on how many things there are—but also ‘ordinal’ numbers which depend further on how the things are arranged. Since the ordinals are standardly taken to be mathematically legitimate numbers, and since the series of points Achilles must pass has an ordinal number, we shall take it that the series is mathematically legitimate. (Again, see ‘Supertasks’ below for another kind of problem that might arise for Achilles’.)