Written in 2009.

What Is the Answer to Zeno’s Paradox? (2014), by Brian Palmer.

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As we saw, before Achilles can reach the wall, he must first walk half the distance to the wall, and then half the remaining distance, and then half the new remaining distance, and so on, supposedly without end.

But does this process really have

This question comes naturally to many and we should address it first.

Let’s call Achilles’s starting point 1, the first halfway point 2, the next halfway point 3, and so on, where each point stands midway between its predecessor and the wall.

Clearly, Achilles must first get to point 2, and then point 3, and then point 4, and so on. But why is Zeno so confident that this sequence of points has no end? Couldn’t there be a

Well, no, unfortunately Zeno is quite right that the sequence of points has no end. Due to the clever way in which the points are laid down, the numbers just go on forever, without reaching the wall. This is not hard to see, because each point is stipulated to lie

This ensures two things. First, that point 5 is not

We can also see this with some simple numbers. Suppose that Achilles begins

Point | Meters from wall |

1 | 1 |

2 | ^{1}⁄_{2} |

3 | ^{1}⁄_{4} |

4 | ^{1}⁄_{8} |

5 | ^{1}⁄_{16} |

6 | ^{1}⁄_{32} |

7 | ^{1}⁄_{64} |

8 | ^{1}⁄_{128} |

9 | ^{1}⁄_{256} |

10 | ^{1}⁄_{512} |

. . . | . . . |

The table clearly has no end. For that to happen, the number on the right would have to reach

So infinitely many points indeed stand between Achilles and the wall and Zeno is quite right to say that, to reach the wall, Achilles must first get to point 2, and then point 3, and then point 4, and so on, without end.

Naturally, this “without end” is annoying since it threatens to put the wall beyond reach. If Achilles must move from one point to another in a sequence that has

At this point, a second thought comes naturally to some people’s minds. For even though Zeno’s points number an infinity, it is obvious that they edge ever

Thus, the table shows that point 10 is only

Now if Achilles is just a hair’s breadth from the wall, it may seem pedantic to insist that he is not yet there! Surely he has

What should we make of this?

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