 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’.)