---
tag: 01MKW4R283
kind: definition
name: Infinite set
msc: ["03E10"]
defines: 0PWE3BCHVJ
uses: [0BBTH3YSZA]
sources:
  - url: https://en.wikipedia.org/wiki/Infinite_set
    locator: lead section
---

## Definition

A [[set|0SJ3RX9TXA]] $A$ is infinite when it is not [[finite|0BBTH3YSZA]], that is, when $A$ is not equinumerous with $\{1, \dots, n\}$ for any natural number $n$.

## Remarks

Examples are $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$.
With the axiom of choice, a set is infinite exactly when it is equinumerous with a proper subset of itself. This is the Dedekind definition.
Infinite sets can have different sizes. See [[Cantor's theorem|0WTEMGYV2C]].

## Explanations

### L1 human

An infinite set never runs out of elements when you count them.
The natural numbers are infinite: after each number there is a next one.