Cardinal number

Definition

A cardinal number is the cardinality $|A|$ of some set $A$. In ZFC, the von Neumann definition takes $|A|$ to be the least ordinal $\kappa$ such that $\kappa$ is equinumerous 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, 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 $\mathbb{R}$.