# Achilles Paradox

## Paradox

Achilles and a tortoise are to have a race.

Achilles (not surprisingly) runs considerably faster than the tortoise. So, to make it marginally more fair, he gives the tortoise a head start.

But it is apparent that Achilles can not actually catch up with the tortoise.

Suppose he gives the tortoise a headstart of $x_0$.

By the time he has got to $x_0$, the tortoise has moved on, to $x_1$, say.

But by the time Achilles has reached $x_1$, the tortoise has moved on, to $x_2$, say.

You can continue this indefinitely.

## Resolution

It is clear that there is a problem with this reasoning, as it is tantamountly clear that someone running faster than another will overtake, sooner or later.

The solution depends on the concept of a limit.

The sum of the distances run by Achilles in catching up the tortoise is an infinite series whose sequence of partial sums is bounded above.

As such, once Achilles reaches that limit, any further distance he travels will bring him further than the tortoise.

The problem lies in the assumption that Achilles is bounded to points only previously set by the tortoise, however in all practicality this is not the case (Achilles' step size).

## Also known as

The **Achilles paradox** is also known as the **racehorse paradox**.

## Historical Note

The **Achilles Paradox** is one of Zeno's Paradoxes, as famously raised by Zeno of Elea.

Some attention was given to this problem by René Descartes in around $1646$. However, his solution would not be accepted by today's standards of mathematical rigor.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.1$: Achilles and the tortoise - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Achilles paradox**:**1.** - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Zeno's paradoxes** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Zeno's paradoxes** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Achilles paradox** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Zeno of Elea**(5th century bc)