[summary(brief): The cardinality of a Set is a formalization of the "number of elements" in the set.]
[summary: If $~$A$~$ is a finite set then the cardinality of $~$A$~$, denoted $~$|A|$~$, is the number of elements $~$A$~$ contains. When $~$|A| = n$~$, we say that $~$A$~$ is a set of cardinality $~$n$~$. There exists a bijection from any finite set of cardinality $~$n$~$ to the set $~$\{0, …, (n-1)\}$~$ containing the first $~$n$~$ natural numbers.
Two infinite sets have the same cardinality if there exists a bijection between them. Any set in bijective correspondence with [45h $~$\mathbb N$~$] is called countably infinite, while any infinite set that is not in bijective correspondence with $~$\mathbb N$~$ is call uncountably infinite. All countably infinite sets have the same cardinality, whereas there are multiple distinct uncountably infinite cardinalities.]
[summary(technical): The cardinality (or size) $~$|X|$~$ of a set $~$X$~$ is the number of elements in $~$X.$~$ For example, letting $~$X = \{a, b, c, d\}, |X|=4.$~$
[todo: technical summary of infinite 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, \ldots, n\}$~$ and $~$g : \{1, 2, 3, \ldots, 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 [ordinal 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].
%%todo: Examples and exercises (possibly as lenses) %%
%%todo: Split off a more accessible cardinality page that explains the difference between finite, countably infinite, and uncountably infinite cardinalities without mentioning alephs, ordinals, or the axiom of choice.%%