---
tag: 0V2H5PGJ9B
kind: object
name: Aleph-null
aliases: [aleph-zero, aleph-nought]
msc: ["03E10"]
instance_of: [0EBACQFAZW]
uses: [0QKPNZKNEA, 0SEZ3JGZP7]
sources:
  - url: https://en.wikipedia.org/wiki/Aleph_number
    locator: Aleph-zero
same_as:
  wikidata: Q908627
  wikipedia: Aleph number
---

## Description

Aleph-null, written $\aleph_0$, is the [[cardinality|0JTGRV4WT4]] of the set $\mathbb{N}$ of natural numbers.
It is the cardinality of each [[countably infinite set|0SEZ3JGZP7]].
It is the least infinite [[cardinal number|0QKPNZKNEA]]: each infinite set has cardinality at least $\aleph_0$, with the axiom of choice.

## Remarks

Other names are aleph-zero and aleph-nought.
$\aleph_0 + \aleph_0 = \aleph_0 \cdot \aleph_0 = \aleph_0$, but $2^{\aleph_0} > \aleph_0$ by [[Cantor's theorem|0WTEMGYV2C]].
As an ordinal, $\aleph_0$ is $\omega$, the set of the finite ordinals.

## Explanations

### L1 human

$\aleph_0$ is the size of the infinity of counting.
The sets $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ all have this size.