Notes from the underground
barang dot sg
Updated 2 December 2023
4. Darn infinity

There is an endless sequence of points between Achilles and the doorway, marked 1, 2, 3, and so on.

Since Achilles can get through the doorway, he must somehow be able to move through the entire sequence of points. But how is that possible if the sequence is endless?

Let’s start by noticing two things.

The problem is not simply that Achilles must move through an endless sequence of points. For, given an infinite amount of time, he could arguably manage that! The trouble is that he does not have the luxury of infinite time.

It also matters that it takes time for Achilles to move from point to point. For if each movement took no time at all, he would also have no trouble completing the endless sequence. This would be the case, for example, if he walked at infinite speed!

But, of course, he doesn’t, and so each movement in the sequence is bound to occupy some of his time. Thus, if he originally stands one meter from the wall, and walks at a speed of one meter per second, he would need ½ a second to reach point 2, another ¼ of a second to reach point 3, and so on, the extra time taken being halved each time:

MovementSeconds to complete
1 to 212
2 to 314
3 to 418
4 to 5116
5 to 6132

So each movement takes less time to complete, the further along the sequence he is, but the amount of time taken in each case is never zero.

We may now express the difficulty more precisely.

Achilles must undertake an infinite sequence of movements. But each movement will occupy some of his time. So it seems that, to complete the entire sequence of movements, he needs an infinite amount of time.

In terms of the table, the charge is that the numbers on the right, when listed without end, will sum to infinity:

12 + 14 + 18 + 116 + 132 + …  =   ∞

The implication is that Achilles will be unable to complete the entire sequence of movements because he does not have the luxury of infinite time.

This is a nice way to put it, because, where we previously just had a rhetorical question (“How can one possibly move through an endless sequence of points?”), we now have a clear line of reasoning for why this is supposed to be impossible.

So our difficulty is more clearly in focus. Something must be wrong with this line of reasoning, since we know that Achilles can move through the entire sequence of Zeno’s points.

Indeed, the obvious point of criticism is the given mathematical equation. If that equation were correct, Achilles would need an infinity of seconds to complete Zeno’s sequence. For the equation just adds the individual amounts of time needed for each movement in the sequence.

But the equation also looks very natural. A list of positive quantities is being added and the list goes on without end. So the sum will keep on getting larger, without end. What could the result be except infinity?

But here at last we attain some relief from our difficulties. For, as natural as the equation looks, it is not hard to see that it is wrong. The result of the infinite sum in question is decidedly not infinity!