Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

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- Finite and infinite cardinals
- Countable and uncoutable cardinals
- Successor cardinals and limit cardinals
- Regular and singular cardinals
- Cardinals in ZF
- Dedekind finite sets

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,
an *initial ordinal*. By this 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 definition that $\kappa$ is a *cardinal* if it
is an *initial ordinal*, an
ordinal
that is not equinumerous with any smaller ordinal.

The set
$\omega$
of natural
numbers is
the smallest inductive set, that is, the smallest set for which
$0\in\omega$ and whenever $n\in\omega$ then also $n+1\in\omega$,
where $n+1=n\cup\{n\}$ is the successor
ordinal
of $n$. A set is *finite* if it is equinumerous with a natural number,
and otherwise it is is *infinite*. In ZFC, the finite sets are the same
as the Dedekind
finite
sets, but in ZF, these concepts may differ. In ZFC,
$\aleph$
is a unique
order-isomorphism
between the ordinals and the cardinal numbers with respect to
membership.

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
uncountable
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.

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**
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
hierarchy.

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 weakly inaccessible 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.

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 axiom of choice 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 $\text{cof}(\kappa)\lt\kappa$.

See general cardinal 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 axiom of choice 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 of $X$, and conversely, if $X$ injects into $\alpha$, then $X$ would be well-orderable.

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.