# Zeno's Paradoxes

**Type:** Problem  
**Domain:** Named Results · Analysis  
**Proposed:** c. 450 BCE, Zeno of Elea  
**Resolved:** 19th century, via rigorous theory of infinite series/limits  
**Codex URL:** /mathematics/named-results/zenos-paradoxes/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Ancient arguments that motion is logically impossible. Resolved by convergent infinite series (finite sum, finite time).

## Achilles and the Tortoise
Gap-closing distances (10, 5, 2.5, 1.25, ...) sum to exactly 20 — a convergent geometric series. Achilles catches the tortoise at 20m, finite time.

## The dichotomy paradox
Room-crossing distances ½,¼,⅛,... sum to exactly 1 — resolved by convergent series theory.

## The Arrow Paradox — distinct and deeper
Requires the rigorous derivative concept (instantaneous velocity as limit) for full resolution — not just convergent series.

## Aristotle's distinction
Potential infinity (ongoing process) vs actual infinity (completed totality) — Zeno's error was treating space/time as actually-infinite point collections.

## Modern philosophical status
Grünbaum (1967) and others: deeper question of whether space/time are genuinely continuously divisible in physical reality remains philosophically debated, beyond the purely mathematical resolution.

## Sources
### Tier 1
- Aristotle. *Physics*, Book VI.
- Huggett, N. (2010/2019). Zeno's Paradoxes. Stanford Encyclopedia of Philosophy.
### Tier 2
- Grünbaum, A. (1967). *Modern Science and Zeno's Paradoxes*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*