We have considered

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

These people feel that the mere

Well, the idea is presumably that, no matter how many of Zeno’s points Achilles crosses, there’s always

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

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

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

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

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.Speaking of “completing” the book, or reading “all” the pages, would mean either of the above, indifferently.I’ll leave no page unread.

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

But do we also require the existence of a

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.

Achilles & the tortoise

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