The Fallacy in the Paradox of Achilles and the Tortoise

TL;DR

This paper resolves Zeno's paradox of Achilles and the Tortoise by demonstrating that an infinite sequence of events can sum to a finite time and distance, clarifying the fallacy in the original argument.

Contribution

It models the paradox using converging geometric series to show the infinite steps occur within finite time, resolving the paradox.

Findings

The infinite series of events converges to a finite sum.

Achilles overtakes the tortoise after finite time.

The paradox arises from misinterpreting infinite events as infinite time.

Abstract

Zeno's ancient paradox depicts a race between swift Achilles and a slow tortoise with a head start. Zeno argued that Achilles could never overtake the tortoise, as at each step Achilles arrived at the tortoise's former position, the tortoise had already moved ahead. Though Zeno's premise is valid, his conclusion that Achilles can "never" pass the tortoise relies on equating infinite steps with an infinite amount of time. By modeling the sequence of events in terms of a converging geometric series, this paper shows that such an infinite number of events sum up to a finite distance traversed in finite time. The paradox stems from confusion between an infinite number of events, which can happen in a finite time interval, and an infinite amount of time. The fallacy is clarified by recognizing that the infinite number of events can be crammed into a finite time interval. At a given speed difference after a finite amount of time, Achilles will have completed the infinite series of gaps at the "catch-up time" and passed the tortoise. Hence this paradox of Achilles and the tortoise can be resolved by simply adding "before the catch-up time" to the concluding statement of "Achilles would never overtake the tortoise".