---
tag: 0ZG61TFBES
kind: definition
name: Countable set
msc: ["03E10"]
defines: 01TQB88H4C
uses: [0PVG1CRAJN]
sources:
  - url: https://en.wikipedia.org/wiki/Countable_set
    locator: lead section
---

## Definition

A [[set|0SJ3RX9TXA]] $A$ is countable when there is an [[injective function|0PVG1CRAJN]] $f \colon A \to \mathbb{N}$.

## Remarks

Equivalently, $A$ is [[finite|0BBTH3YSZA]] or [[countably infinite|0SEZ3JGZP7]].
A nonempty set is countable exactly when there is a [[surjection|0XKH881BZ7]] from $\mathbb{N}$ to it.
Some authors use "countable" only for countably infinite sets, and "at most countable" for this notion.
Subsets of countable sets are countable.

## Explanations

### L1 human

A countable set is one that you can list: first element, second element, and so on.
The list may stop, or it may go on forever, but each element appears at some finite position.