A cardinal number is one way to measure the size of a set. Here is the definition used in Zermelo Fraenkel set theory (ZF):

The cardinal (or cardinality) of x, denoted card(x), is:

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

In ZFC all sets are well-orderable and only definition (1) above is needed.

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.

 

Notation

k, l, and µ are used for the Greek letters kappa, lambda, and mu and represent cardinals; a is used for the Greek letter alpha and represents an ordinal; w is used for the Greek letter omega; aleph is used for the Hebrew letter aleph; and:

• aleph₀ is used for aleph-zero, the zero'th well-ordered infinite cardinal; aleph₀=w = {0, 1, 2, ...}
  
• aleph₁=w₁ = {0, 1, 2, ..., w, w+1,..., w+w,..., wxw,...} = set of all countable ordinals
• aleph_a is used for aleph-alpha, the alpha'th (or a'th) well-ordered infinite cardinal.

Since all well-ordered cardinals are ordinals, sometimes I use ordinal notation for a cardinal or vice versa. For example, when I say k=aleph_k, the k on the left is being used as a cardinal and the k on the right is being used as an ordinal.

 

Special Properties of aleph₀

If we restrict our attention to the well-ordered cardinals, which in ZFC is all the cardinals, then aleph₀ is the:

• 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

Many of the notions that are defined below were motivated by the desire to recognize these properties of aleph₀, and to look for similar properties, or their negation, in other cardinals.

 

Partitioning Cardinals

A partition of cardinal numbers divides them up into subsets so that each cardinal is a member of one and only one of the subsets. In ZFC the most common partitions of cardinals are:

• 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

The definitions below help to explain these partitions and the relations between them.

 

Definitions

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.
• aleph₀=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.

 

k⁺ or 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.
• aleph₁=w₁ is the least infinite successor cardinal. It is the cardinal successor of aleph₀.

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
• aleph₀ 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 aleph₀ limit cardinals. Some authors include both 0 and aleph₀, 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 aleph₀ special cases, i.e., well-ordered cardinals which are neither successors nor limits. Also, with the definition I use aleph₀ is considered a regular limit cardinal and a regular strong limit cardinal. These special properties of aleph₀ 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
• aleph₀
• k=sup{l, 2^l, 2^2l, ...} where l is any cardinal > 0. If l is finite and > 0 then k=aleph₀, 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
• aleph₀=w
• (AC) any infinite successor cardinal, i.e., every cardinal of the form aleph_a+1 (e.g.: aleph₁, aleph₂, ..., 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 aleph₀.
• the least infinite singular cardinal is aleph_w.
• (AC) aleph_w1 has cofinality aleph₁.
• k=sup{w, w_w, w_ww, ... } has cofinality aleph₀. 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 aleph₀.

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 aleph₁ had cofinality aleph₀, i.e.:

Con(ZF)   ==>   Con(cf(aleph₁)=aleph₀)

 

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.

 

Large Cardinals

Large cardinals are cardinals whose existence cannot be proved in ZFC. Examples include weakly inaccessible cardinals and inaccessible cardinals. There are many other large cardinals, some of which are listed in the hierarchy below. Note that the cardinals at the bottom of this hierarchy -- 0 and aleph₀ -- are not large cardinals per se, but each has the same flavor as a large cardinal, i.e., it cannot be constructed using the methods used to construct the cardinals listed below it.

 

Types of Cardinal Numbers Ordered By Consistency Strength

If cardinal type m is listed above cardinal type n in the list below then:

  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
• aleph₀
• 0

 
 

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