---
tag: 0QKPNZKNEA
kind: definition
name: Cardinal number
msc: ["03E10"]
defines: 0EBACQFAZW
uses: [0JTGRV4WT4, 0RDJK7EQYY]
sources:
  - url: https://en.wikipedia.org/wiki/Cardinal_number
    locator: lead section
---

## Definition

A cardinal number is the [[cardinality|0JTGRV4WT4]] $|A|$ of some set $A$.
In ZFC, the von Neumann definition takes $|A|$ to be the least [[ordinal|0RDJK7EQYY]] $\kappa$ such that $\kappa$ is [[equinumerous|08EJHHMB5R]] with $A$. A cardinal number is then an ordinal that is not equinumerous with any smaller ordinal.

## Remarks

The finite cardinals are the natural numbers $0, 1, 2, \dots$.
The least infinite cardinal is [[aleph-null|0V2H5PGJ9B]], the cardinality of $\mathbb{N}$.
Cardinals have an order, an addition, a multiplication and an exponentiation.
The sum $|A| + |B|$ is the cardinality of a disjoint union, and $|A| \cdot |B| = |A \times B|$.
The power $|B|^{|A|}$ is the cardinality of the set of functions from $A$ to $B$.

## Explanations

### L1 human

A cardinal number answers "how many?".
For finite sets, the answers are $0, 1, 2, \dots$.
For infinite sets, there are many different answers, such as $\aleph_0$ for $\mathbb{N}$ and $\mathfrak{c}$ for $\mathbb{R}$.