How does Zeno's paradox of Achilles and the Tortoise generate an infinite sequence of sub-intervals?

Conditions

• $v_A = 2$ (Achilles' speed)
• $v_T = 1$ (Tortoise's speed)

Reasoning, step by step

1. Achilles runs to close the initial gap $s_1 = 2$, taking time $t_1 = 1$.
2. During $t_1$, the tortoise advances by distance $s_2 = 1$.
3. Achilles spends another fraction of time $t_2 = 1/2$ to reach this new position.
4. By the time Achilles arrives, the tortoise has moved again by $s_3 = 1/2$.
5. This recursive pattern continues, generating infinitely many sub-intervals.

Example

First, Achilles covers $s_1=2$ in $t_1=1$. The tortoise moves $s_2=1$. Then Achilles covers $s_2=1$ in $t_2=1/2$. The tortoise moves $s_3=1/2$. This repeats with halving distances and times.

Common misconceptions

• Believing that there is a final step where Achilles catches the tortoise within the paradoxical logic itself without summing the series.
• Confusing the physical motion with the mathematical subdivision; the video notes these are not infinitely many separate physical actions.

Watch the explanation

Explore next

• What is the total distance Achilles travels according to the geometric series?
• Does the infinite number of steps imply infinite time?