---
tag: 0SEZ3JGZP7
kind: definition
name: Countably infinite set
msc: ["03E10"]
defines: 00CR00C9DN
uses: [0SNQEF7Z53]
sources:
  - url: https://en.wikipedia.org/wiki/Countable_set
    locator: Definition
---

## Definition

A [[set|0SJ3RX9TXA]] $A$ is countably infinite when there is a [[bijective function|0SNQEF7Z53]] $f \colon \mathbb{N} \to A$.

## Remarks

Another name is denumerable. Its cardinality is [[aleph-null|0V2H5PGJ9B]].
Examples are $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$, and also the set of primes.
A set is countably infinite exactly when it is [[countable|0ZG61TFBES]] and [[infinite|01MKW4R283]].

## Explanations

### L1 human

A countably infinite set can be listed as $a_0, a_1, a_2, \dots$ without end, and each element appears exactly one time.
The integers have such a list: $0, 1, -1, 2, -2, \dots$.