# Cardinality

Cardinality is a property of sets. For finite sets, the cardinality of is the number of elements in that set, i.e. the size of the set. The cardinality of $\{3, 4\}$ is 2, the cardinality of $\{1, \{2, 3\}, \{1, 2, 3\}\}$ is 3, and the cardinality of the empty set is 0.

## Notation

The cardinality of a set $A$ is denoted by $|A|$. In the above example, the cardinality of $\{3, 4\}$ is $|\{3, 4\}| = 2$. Sometimes, the notations $n(A)$ and $\# (A)$ are used.

## Infinite

For infinite sets, cardinality also measures (in some sense) the "size" of the set, but an explicit formulation is more complicated: the cardinality of a set $S$ is the least cardinal that can be put in bijection with $S$. With the Axiom of choice ( $\sf{AC}$), each set is well-orderable, and since the class of well-orderable cardinals is well-ordered, we can reasonably talk about the least cardinal in bijection with a set $S$. In the absence of $\sf{AC}$, one can define cardinals using equivalence classes, formed via the relation $X\sim Y\Leftrightarrow|X|=|Y|$ (there is a bijection between $X$ and $Y$).

The notion of cardinalities for infinite sets is due to Georg Cantor and is one aspect of the field of set theory. Most significantly, Cantor showed that there are multiple infinite cardinalities. In other words, not all infinite sets are the same size.