Cardinality is a measure of the size of a set. Two sets have the same cardinality - they are said to be *equinumerous* - when there is a one-to-one correspondence between their elements. The cardinality assignment problem is the problem of assigning to each equinumerosity class a cardinal number to represent it. In [[ZFC]], this problem can be solved via the [[Well-ordering principle]] which asserts that every set can be well-ordered and therefore admits a bijection with a unique smallest [ordinal](Ordinal.md "Ordinal"), an *initial ordinal*. By these means, in ZFC we are able to assing to every set $X$ a canonical representative of its equinumerosity class, the smallest ordinal bijective with $X$. We therefore adopt the following definition: >[!info] Definition >$\kappa$ is a *cardinal* if it is an *initial ordinal*, an [ordinal](Ordinal.md "Ordinal") that is not equinumerous with any smaller ordinal. ## Finite and infinite cardinals The set $\omega$ is defined as the smallest *inductive set*, that is, the smallest set $X$ for which $\varnothing \in X$ and whenever $x \in X$ then also $x \cup \{x\} \in X$. We call the elements of $\omega$ *natural numbers*. Thus defined, $\omega$ and its elements are all ordinals, one can prove they are in fact cardinals. $\omega$ is provably the least [[Ordinal#Limit ordinals|limit ordinal]]. A set is *finite* if it is equinumerous with a natural number, and otherwise it is is *infinite*. Thus rhe finite cardinals are precisely the natural numbers, and $\omega$ is the least infinite cardinal. In ZFC, the finite sets are the same as the [](Dedekind_finite.md) sets, but in ZF, these concepts may differ. In ZFC, [$\aleph$](Aleph.md "Aleph") is a unique [order-isomorphism](Order-isomorphism.md "Order-isomorphism") between the ordinals and the infinite cardinal numbers with respect to membership. ## Countable and uncoutable cardinals A set is *countable* when it is equinumerous with a subset of $\omega$. This includes all finite sets, including the empty set, and the infinite countable sets are said to be *countably infinite*. An <a href="Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> set is a set that is not countable. The existence of uncountable sets is a consequence of Cantor's observationt that the set of reals is uncountable. ## Successor cardinals and limit cardinals Hartog established that for every set $X$, there is a smallest ordinal that does not have an injection into $X$, and this ordinal is now known as the **[](Hartog_number.md)** of $X$. When $\kappa$ is a cardinal, then the **successor cardinal** of $\kappa$, denoted $\kappa^+$, is the Hartog number of $\kappa$, the smallest ordinal of strictly larger cardinality than $\kappa$. The existence of successor cardinals can be proved in ZF without the axiom of choice. Iteratively taking the successor cardinal leads to the [](Aleph.md). Although ZF proves the existence of successor cardinals for every cardinal, ZF also proves that there exists some cardinals which are not the successor of any cardinal. These cardinals are known as **limit cardinals**. Cardinals which are not limit cardinals are known as **successor cardinals**. The limit cardinals are precisely those which are limit points in the topology of cardinals (hence the name). That is, for any cardinal $\lambda<\kappa$, there is some $\nu>\lambda$ with $\nu<\kappa$. The limit cardinals share an incredible affinity towards the singular cardinals; there does not exist a [](Inaccessible.md) cardinal if and only if the singular cardinals are precisely the limit cardinals. If inaccessibility is inconsistent (which is thought "untrue" by most set theorists, although possible), then ZFC actually proves that any cardinal is singular if and only if it is a limit cardinal. ## Regular and singular cardinals A cardinal $\kappa$ is *regular* when $\kappa$ not the union of fewer than $\kappa$ many sets of size each less than $\kappa$. Otherwise, when $\kappa$ is the union of fewer than $\kappa$ many sets of size less than $\kappa$, then $\kappa$ is said to be *singular*. The <a href="Axiom_of_choice" class="mw-redirect" title="Axiom of choice">axiom of choice</a> implies that every successor cardinal $\kappa^+$ is regular, but it is known to be consistent with ZF that successor cardinals may be singular. The *cofinality* of an infinite cardinal $\kappa$, denoted $\text{cof}(\kappa)$, is the smallest size family of sets, each smaller than $\kappa$, whose union is all of $\kappa$. Thus, $\kappa$ is regular if and only if $\text{cof}(\kappa)=\kappa$, and singular if and only if $\text{cof}(\kappa)\lt\kappa$. ## Cardinals in ZF See [](Cardinal_general.md) for an account of the cardinality concept arising without the axiom of choice. When the axiom of choice is not available, the concept of cardinality is somewhat more subtle, and there is in general no fully satisfactory solution of the cardinal assignment problem. Rather, in ZF one works directly with the equinumerosity relation. In ZF, the <a href="Axiom_of_choice" class="mw-redirect" title="Axiom of choice">axiom of choice</a> is equivalent to the assertion that the cardinals are linearly ordered. This is because for every set $X$, there is a smallest ordinal $\alpha$ that does not inject into $X$, the [](Hartog_number.md) of $X$, and conversely, if $X$ injects into $\alpha$, then $X$ would be well-orderable. ## Dedekind finite sets The *Dedekind finite* sets are those not equinumerous with any proper subset. Although in ZFC this is an equivalent characterization of the finite sets, in ZF the two concepts of finite differ: every finite set is Dedekind finite, but it is consistent with ZF that there are infinite Dedekind finite sets. An *amorphous* set is an infinite set, all of whose subsets are either finite or co-finite.