TheIn ZFC all sets are well-orderable and only definition (1) above is needed.cardinal(orcardinality) of x, denoted card(x), is:

- the least ordinal
aequinumerous to x, if x is well-orderable, and- the set of all sets y of least rank which are equinumerous to x, otherwise.

The purpose of this page is to describe the different types of cardinal numbers. For an introduction to cardinal and ordinal numbers see the ``Mathematics of CH'' section in my Continuum Hypothesis article.

*aleph*_{0}is used for aleph-zero, the zero'th well-ordered infinite cardinal;*aleph*_{0}=*w*= {0, 1, 2, ...}*aleph*_{1}=*w*_{1}= {0, 1, 2, ...,*w*,*w*`+`1,...,*w*`+`*w*,...,*w*`x`*w*,...} = set of all countable ordinals*aleph*_{a}is used for aleph-alpha, the alpha'th (or*a*'th) well-ordered infinite cardinal.

- first infinite cardinal
- first infinite limit cardinal, i.e.
first infinite cardinal that
cannot be reached from cardinals below it using the
cardinal successor function.
(
*k*`->`*k*^{+}). - first infinite strong limit cardinal, i.e.
first infinite cardinal that
cannot be reached from cardinals below it using the power set function
(
*k*`->`2^{k}). - only infinite regular limit cardinal that is `constructable' in ZFC
- only infinite regular strong limit cardinal that is `constructable' in ZFC

- finite and infinite
- successor and limit
- 0, successor, and infinite limit
- regular and singular
- finite, infinite regular successor, infinite singular limit, and infinite regular limit

**Finite Cardinal**- A cardinal is finite iff it is a natural number.
**Infinite Cardinal**- A cardinal that is not a finite cardinal is
an infinite cardinal.
**Dedekind-Finite Cardinal**- The cardinal of a Dedekind-finite set (a set
that does not include a countably infinite subset) is a
Dedekind-finite cardinal.
__Note__: Every finite cardinal is a Dedekind-finite cardinal. In ZFC every Dedekind-finite cardinal is a finite cardinal, i.e., the Dedekind-finite cardinals are exactly the finite cardinals, i.e., no infinite cardinal is a Dedekind-finite cardinal. In ZF it is possible to have a Dedekind-finite cardinal that is not a finite cardinal. **Infinite Well-Ordered Cardinal**- Any infinite cardinal that is also an ordinal
is an infinite well-ordered cardinal.
The infinite well-ordered cardinals are called alephs
since they are exactly the
*aleph*_{a}'s where*a*is an ordinal.Examples:

- Any cardinal of the form
*aleph*_{a}where*a*is an ordinal. *aleph*_{0}=*w*is the least infinite well-ordered cardinal.

__Note__: Every infinite well-ordered cardinal is an infinite cardinal. In ZFC every infinite cardinal is an infinite well-ordered cardinal, i.e., the infinite cardinals are exactly the infinite well-ordered cardinals. In ZF it is possible to have an infinite cardinal which is not a well-ordered cardinal. - Any cardinal of the form
or*k*^{+}**The Cardinal Successor of***k*- The least well-ordered cardinal
*l*such that*l*is not less than or equal to card(*k*) is called*k*^{+}or the cardinal successor of*k*.__Note__: In ZFC we can use the following simpler definition: The least cardinal greater than*k*is the cardinal successor of*k*. **Successor Cardinal***l*is a successor cardinal iff*l*=*k*^{+}for some well-ordered cardinal*k*.*k*is called the**cardinal predecessor**of*k*^{+}.Examples:

- any nonzero finite cardinal, i.e., in {1, 2, 3, ....}.
- Any cardinal of the form
aleph
_{a+1}, where*a*is an ordinal and*a*`+`1 is the ordinal successor of*a*, is the cardinal successor of aleph_{a}. *aleph*_{1}=*w*_{1}is the least infinite successor cardinal. It is the cardinal successor of*aleph*_{0}.

__Note__: Every successor cardinal is a well-ordered cardinal.**Limit Cardinal**(Hausdorff 1908)*k*is a limit cardinal iff for every*l*<*k*,*l*^{+}<*k*. Every well-ordered cardinal that is not a successor cardinal is a limit cardinal.Examples:

- 0
*aleph*_{0}is the least infinite limit cardinal.*aleph*_{w}is the least uncountable limit cardinal.- Any cardinal of the form
*aleph*_{a}, where*a*is a limit ordinal, is a limit cardinal.

__Note About the Definition__: The definition of*limit cardinal*is not consistent in the literature. The inconsistency is about whether to consider 0 and*aleph*_{0}limit cardinals. Some authors include both 0 and*aleph*_{0}, some do not include 0, and some do Some authors require limit cardinals to be uncountable. The reason that I prefer the definition I give is that it makes the partitions of cardinals much nicer, since we do not need to consider 0 and*aleph*_{0}special cases, i.e., well-ordered cardinals which are neither successors nor limits. Also, with the definition I use*aleph*_{0}is considered a regular limit cardinal and a regular strong limit cardinal. These special properties of*aleph*_{0}are what inspired the definition of weakly inaccessible and inaccessible cardinals.Authors who use the definition I give include Jech [1978,p.25], Levy [1979,p.90], and Devlin [1993,p.88]. Authors who require

*k*to be uncountable include Kunen [1980,p.30].*(Please send me info about other authors...)***Strong Limit Cardinal***k*is a strong limit cardinal iff for every*l*<*k*, 2^{l}<*k*.Examples:

- 0
*aleph*_{0}*k*=sup{*l*, 2^{l}, 2^{2l}, ...} where*l*is any cardinal`>`0. If*l*is finite and`>`0 then*k*=*aleph*_{0}, otherwise*k*is really big!

__Note__: Every strong limit cardinal is a limit cardinal. In ZF+GCH every limit cardinal is a strong limit cardinal (since*l*^{+}=2^{l}), i.e., the limit cardinals are exactly the strong limit cardinals.**Regular Cardinal**(Hausdorff 1908)*k*is regular iff cf(*k*)=*k*. Another way to say this is to say that it is not possible to represent*k*as the supremum of fewer than*k*smaller ordinals, or*k*cannot be written as the sum of fewer smaller cardinals.Examples:

- 0
- 1
*aleph*_{0}=*w*- (AC) any infinite successor cardinal, i.e., every
cardinal of the form
*aleph*_{a+1}(e.g.:*aleph*_{1},*aleph*_{2}, ...,*aleph*_{w+1}, etc.)

**Singular Cardinal**(Hausdorff 1908)*k*is singular iff cf(*k*)<*k*. Another way to say this is to say that it is possible to represent*k*as the supremum of fewer than*k*smaller ordinals.Examples:

- any finite cardinal greater than 1, i.e., in {2, 3, 4, ....}, has cofinality 1.
- any limit cardinal
*aleph*_{a}where*a*is a countable limit ordinal. E.g.,*aleph*_{w}and*aleph*_{w+w}both have cofinality*aleph*_{0}. - the least infinite singular cardinal is
*aleph*_{w}. - (AC)
*aleph*_{w1}has cofinality*aleph*_{1}. *k*=sup{*w*,*w*,_{w}*w*, ... } has cofinality_{ww}*aleph*_{0}. This*k*is the least fixed point of the aleph sequence, i.e., the least*k*such that*k*=*aleph*_{k}.*k*=sup{*l*, 2^{l}, 2^{2l}, ...}, where*l*is any infinite cardinal, has cofinality*aleph*_{0}.

__Note__: In ZFC every infinite singular cardinal is a limit cardinal. In ZF+~AC it is possible to have an infinite singular successor cardinal since Feferman and Levy [1963] constructed a model in which*aleph*_{1}had cofinality*aleph*_{0}, i.e.:Con(ZF)

`==>`Con(cf(*aleph*_{1})=*aleph*_{0})**Weakly Inaccessible Cardinal**(Hausdorff 1908)*k*is weakly inaccessible iff*k*is uncountable, regular, and a limit cardinal.__Note__: If*k*is weakly inaccessible then*k*=*aleph*_{k}, i.e.,*k*is the*k*'th well-ordered infinite cardinal, i.e.,*k*is a fixed point of the aleph function.__Note About Existence__: In ZFC, it is not possible to prove that weak inaccessibles exist.**Inaccessible Cardinal**(Tarski, 1930)*k*is inaccessible iff*k*is uncountable, regular, and a strong limit cardinal.__Note__: Every inaccessible cardinal is a weakly inaccessible cardinal and the least inaccessible cardinal is not less than the least weakly inaccessible cardinal. In ZF+GCH every weakly inaccessible cardinal is an inaccessible cardinal, i.e., the weakly inaccessible cardinals are exactly the inaccessible cardinals.__Note About Existence__: In ZFC, it is not possible to prove that inaccessibles exist.__Note About the Definition__:*Inaccessible cardinals*are also called*strongly inaccessible cardinals*.

Con(ZFC+a type m cardinal exists)
`==>`
Con(ZFC+a type n cardinal exists)

- rank-into-rank
- n-huge (n<
*w*) - huge
- supercompact
- superstrong
- Shelah
- Woodin
- strong
*o*(*k*)=*k*^{++}*o*(*k*)=2- measurable
- Ramsey
- 0
^{#} - weakly compact
- Mahlo
- inaccessible
*aleph*_{0}- 0

*Thanks to James Cummings of the
Carnegie Mellon Math Department
for giving me feedback on this page.*