Cardinality

The cardinality of a set is a formalization of the "number of elements" in the set.

Set cardinality is an equivalence relation. Two sets have the same cardinality if (and only if) there exists a bijection between them.

Definition of equivalence classes

Finite sets

A set $S$ has a cardinality of a natural number $n$ if there exists a bijection between $S$ and the set of natural numbers from $1$ to $n$. For example, the set $\{9,15,12,20\}$ has a bijection with $\{1,2,3,4\}$, which is simply mapping the $m$th element in the first set to $m$; therefore it has a cardinality of $4$.

We can see that this equivalence class is well-defined — if there exist two sets $S$ and $T$, and there exist bijective functions $f:S\to \{1,2,3,\dots ,n\}$ and $g:\{1,2,3,\dots ,n\}\to T$, then $g\circ f$ is a bijection between $S$ and $T$, and so the two sets also have the same cardinality as each other, which is $n$.

The cardinality of a finite set is always a natural number, never a fraction or decimal.

Infinite sets

Assuming the axiom of choice, the cardinalities of infinite sets are represented by the Aleph_numbers. A set has a cardinality of ${\aleph }_0$ if there exists a bijection between that set and the set of all natural numbers. This particular class of sets is also called the class of countably_infinite_sets.

Larger infinities (which are uncountable) are represented by higher Aleph numbers, which are ${\aleph }_1,{\aleph }_2,{\aleph }_3,$ and so on through the ordinals.

In the absence of the Axiom of Choice

Without the axiom of choice, not every set may be well-ordered, so not every set bijects with an ordinal, and so not every set bijects with an aleph. Instead, we may use the rather cunning Scott_trick.