The most accurate explanation of Zeno’s Achilles and the Tortoise paradox

paradoxessequences-and-series

I heard Zeno's "Achilles and the Tortoise" paradox is resolved using the limit concept of mathematics, i.e., the time cost in every sub-race is added up, which has a limit, not limitless, so Achilles can catch up the Tortoise finally.

That resolution is based on two assumptions:

  1. the catch-up time equals to the sum of a series
  2. the series is convergent and the sum of the series is a limited number

The second assumption can be strictly proved in math. But how to prove the first assumption?

Best Answer

I think the assumption follows from the analysis of the motion before the catch. We see that it splits into stages.

Stage 1: Achilles is running towards the Tortoise starting point $A_1$, the Tortoise is running forward. At the moment when Achilles achieves $A_1$, Stage 1 ends and begins Stage 2. At this moment the Tortoise is at point $A_2$.

Stage 2: Achilles is running towards $A_2$, the Tortoise is running forward. At the moment when Achilles achieves $A_2$, Stage 2 ends and begins Stage 3. At this moment the Tortoise is at point $A_3$.

And so forth...

The construction of the stages show that each of them happens before the catch. This is a surprising view on a usual motion, but it sounds reasonable. So, we have that catch-up time $T_C $ is at least the sum $T_\infty=\sum T_n $ of the series of the durations of the stages $T_n$.

To show the assumption it remains to show that a strict inequality $T_C>T_\infty$ is impossible. Indeed, suppose to the contrary that after time $T_\infty$ Achilles is still behind the Tortoise. This means that a distance $d_\infty$ between them is non-zero. Let $d_0$ be the initial distance between Achilles and the Tortoise. Since Achilles is faster than the Tortoise, the $q=v_A/v_T$ of their velocities is bigger than $1$ and the distance between them decreases. But induction we can easily show that the distance $d_n$ between Achilles and the Tortoise after Stage $n$ is $d_0/q^n$. Since $q>1$, there exists $n$ such that $d_0/q^n<d_\infty$. But this is impossible, because the distance between Achilles and the Tortoise decreases and the end of Stage $n$ happens before $T_\infty$.

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