ABSTRACT
The progressive version of the Racecourse is essentially the same as Achilles and the Tortoise. Indeed, it is slightly simpler, in that, instead of having to catch a moving tortoise, Achilles has only to reach the stationary end of the course; but that makes no essential difference to the paradox or its resolution. Achilles can traverse the infinitely many intervals in a finite time because each successive interval is half as long as the last. The sum of these intervals is the sum of the infinite series
which is 1. (This is briefly explained in the entry on Achilles and the Tortoise.)
The regressive version introduces a further paradoxical feature. Achilles seems to be prevented from even starting to run, since he cannot move beyond the start without having first traversed
infinitely many intervals. And in any case there is no first interval for him to run. The sequence of intervals he needs to run is given by taking the terms in the series displayed above in the reverse order:
But if Achilles can get to the end of the course in the progressive version above, he can get to any point after the start by traversing an infinite sequence of intervals. For example, he can get 1⁄64 of the way by traversing intervals which can be represented in a series which sums to 1⁄64. However short the distance from the start, he will have traversed infinitely many of these intervals, but they will always have a sum. It is true that in the sequence of ever-increasing intervals there is no first interval. But all this means is that we should avoid analysing his run in terms of a sequence with no first term. There are plenty of other ways of analysing it so that the question makes sense, either using finite sequences or infinite sequences with a beginning.
