Achilles and the Tortoise

The paradox

Achilles is ten times faster than a tortoise. He gives it a 100-meter head start. To catch the tortoise, Achilles must first reach the 100-meter mark. By the time he does, the tortoise has moved to 110 meters. He must reach 110. By then the tortoise is at 111. He must reach 111. By then the tortoise is at 111.1.

This sequence never ends. At every stage, the tortoise is still ahead by a tenth of the previous gap. There are infinitely many such stages. Achilles must complete infinitely many tasks. So he never catches up.

Yet Achilles obviously does catch up. You could time the race. This is the paradox: a valid-seeming argument leads to a conclusion that contradicts what we can plainly observe.

The mathematical resolution

Modern mathematics dissolves the arithmetic problem. The distances Achilles must cover form a convergent infinite series: 100 + 10 + 1 + 0.1 + 0.01 + ... = 111.11... meters. The times form a similar series that sums to a finite value. An infinite number of terms can sum to a finite total. Achilles covers the infinite sequence of distances in a finite time and catches the tortoise.

This was not available to Zeno. The rigorous theory of infinite series was not developed until the seventeenth and nineteenth centuries. From Zeno's vantage, the conclusion that infinitely many additions could yield a finite result was not obvious.

The residual philosophical problem

The mathematical answer shows that the numbers work out. It does not obviously settle whether supertasks, the completion of infinitely many distinct actions in finite time, are physically or conceptually coherent.

The worry is not about arithmetic. It is about what it means to say a process has infinitely many stages that are each genuinely completed. Can a physically real race consist of an actually infinite sequence of sub-races? Philosophers like J. F. Thomson and Adolf Grünbaum have debated whether supertasks are possible even if the mathematics is consistent. The series converges, but whether the convergence reflects a real fact about motion or merely a useful fiction about intervals is a question mathematics alone cannot answer.