cardinality

The cardinality of a set is the count of the elements.

Formally,
a finite set $S$ is said to have cardinality $n$ if there exists a bijection $f:S\to \{1,2,3,\dots ,n\}$, i.e. from $S$ to a subset of sequential natural numbers with least element 1. We write $|S|=n$, where $n$ is the cardinal number of $S$.

Given an equivalence relation $\sim$ between sets $A,B$, the cardinal function is a function $A\mapsto |A|$ such that $A\sim B⟺|A|=|B|$.

The aleph numbers are a sequence of cardinal numbers that represent the size of infinite sets. They are denoted ${\mathrm{\aleph }}_i$, where ${\mathrm{\aleph }}_0$ is the cardinality of the natural numbers, and ${\mathrm{\aleph }}_1$ is the next largest cardinality, and so on.