Notes from the underground
barang dot sg
Updated 2 December 2023
6. Paradox regained

We have considered one reason why Achilles cannot complete Zeno’s sequence – supposedly, he would need an infinite amount of time for the feat, a luxury he clearly does not have.

But this reason has proven to be unjustified and we might thereby conclude that the paradox has been satisfactorily resolved. (Some people do!)

In reality, though, many people are unsatisfied with this diagnosis because they have a different reason for why Achilles cannot complete Zeno’s sequence, one which has nothing to do with time.

These people feel that the mere endlessness of Zeno’s sequence is (by itself) sufficient to prevent Achilles from completing the entire sequence. This goes against what we said before but let’s consider it all the same. Why would the mere endlessness of Zeno’s sequence be a barrier for Achilles?

Well, the idea is presumably that, no matter how many of Zeno’s points Achilles crosses, there’s always another one ahead. But if there’s always another one ahead, Achilles can never be done!

The idea is easy to appreciate if we think of counting. There are an infinity of positive integers, 1, 2, 3, ... . Can one count them all? Even given the luxury of infinite time, the answer may seem to be no, since no matter how many numbers you count, there’s always another one to count! So you can never be done.

Nor does it help to count faster as you go along, hoping to get it all done in, say, one second. (Using the technique of the previous section.) For, on the current view, time is irrelevant. No matter how fast you count, there’s always another number to count. So it’s impossible for anyone to count them all.

Exactly the same is true of Achilles.

This sort of reason strikes a chord with many people. What could be wrong with it?

Well, let’s start by seeing what’s right about the objection.

We should not dispute that, no matter how many points Achilles crosses, there’s always another one ahead. For this just says that Zeno’s sequence of points is endless, which we have granted.

Accordingly, we should concede that Achilles cannot reach the end of Zeno’s sequence. (Because it has no end.) In this sense, it’s clearly true that Achilles cannot “complete” Zeno’s sequence.

But, curiously enough, there’s another sense in which Achilles can complete Zeno’s sequence. For there is no point in the endless sequence which Achilles will fail to cross. In other words, no matter which point you pick from Zeno’s endless lot, Achilles will eventually cross it. In this sense, he can certainly cross them all. (None will be left out.)

So the question of whether Achilles can “complete” Zeno’s sequence is actually ambiguous. In one sense, he cannot, but in another sense, he can. He certainly cannot reach the last point in Zeno’s endless sequence, because no such point exists. Nevertheless, he can leave no point unvisited!

It’s strange to see this ambiguity, because it does not normally arise. Thus, if I’m trying to read a book from cover to cover, these statements would normally mean the same:
I’ll reach the last page.
I’ll leave no page unread.
Speaking of “completing” the book, or reading “all” the pages, would mean either of the above, indifferently.

But if the book contains infinitely many pages, the statements are no longer the same. The first can no longer be true, because the book no longer has a last page. But the second could still be true!

Now, in our case, we know that Achilles can get through the doorway. So, in some sense, he must be able to cross “all” of the points, 1, 2, 3, ... , without end. But what do we mean by this?

Well, one thing we certainly mean is that Achilles should leave no point uncrossed. That is, no matter which point you pick, he must eventually cross it. The reason is obvious. Since each point stands between him and the doorway, it must be crossed before the doorway can be breached.

But do we also require the existence of a last point in Zeno’s sequence? Shall we say that, in addition to the above, Achilles also cannot breach the doorway unless the sequence contains a final point to be crossed?

This question is rather tricky! The requirement clearly differs from the previous one, whose rationale was plain to see. But why should Achilles be unable to breach the doorway simply because Zeno’s sequence has no last point? If there’s a reason, it’s not obvious what it might be.

Nevertheless, this is presumably the heart of the present objection. Recall that the endlessness of the sequence was supposed to be a problem for Achilles. But, as we saw, two concerns may be distinguished, the first of which seems harmless. So the question is if there is anything to the second, trickier one.

Let’s summarize our situation before we proceed.

We have conceded that Zeno’s sequence has no last point. But our antagonist must also concede that there is no point in Zeno’s sequence that Achilles will fail to cross. So the only question is whether the non-existence of the last point somehow prevents Achilles from getting through the doorway.