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CARDINAL NUMBERS 
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:In ZFC all sets are wellorderable and only definition (1) above is needed.
 the least ordinal a equinumerous to x, if x is wellorderable, 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.
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:Since all wellordered 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.
 aleph_{0} is used for alephzero, the zero'th wellordered infinite cardinal; aleph_{0}=w = {0, 1, 2, ...}
 aleph_{1}=w_{1} = {0, 1, 2, ..., w, w+1,..., w+w,..., wxw,...} = set of all countable ordinals
 aleph_{a} is used for alephalpha, the alpha'th (or a'th) wellordered infinite cardinal.
Special Properties of aleph_{0}
If we restrict our attention to the wellordered cardinals, which in ZFC is all the cardinals, then aleph_{0} is the:Many of the notions that are defined below were motivated by the desire to recognize these properties of aleph_{0}, and to look for similar properties, or their negation, in other cardinals.
 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
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:The definitions below help to explain these partitions and the relations between them.
 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
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.
 DedekindFinite Cardinal
 The cardinal of a Dedekindfinite set (a set that does not include a countably infinite subset) is a Dedekindfinite cardinal.
Note: Every finite cardinal is a Dedekindfinite cardinal. In ZFC every Dedekindfinite cardinal is a finite cardinal, i.e., the Dedekindfinite cardinals are exactly the finite cardinals, i.e., no infinite cardinal is a Dedekindfinite cardinal. In ZF it is possible to have a Dedekindfinite cardinal that is not a finite cardinal.
 Infinite WellOrdered Cardinal
 Any infinite cardinal that is also an ordinal is an infinite wellordered cardinal. The infinite wellordered 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 wellordered cardinal.
Note: Every infinite wellordered cardinal is an infinite cardinal. In ZFC every infinite cardinal is an infinite wellordered cardinal, i.e., the infinite cardinals are exactly the infinite wellordered cardinals. In ZF it is possible to have an infinite cardinal which is not a wellordered cardinal.
 k^{+} or The Cardinal Successor of k
 The least wellordered 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 wellordered 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 wellordered cardinal.
 Limit Cardinal (Hausdorff 1908)
 k is a limit cardinal iff for every l<k, l^{+}<k. Every wellordered 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., wellordered 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_{ww}, ... } has cofinality 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 wellordered 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 aleph_{0}  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)
 rankintorank
 nhuge (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.
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19921997
Infinite Ink
and
Nancy McGough Last significant update on June 3, 1997 Last tweak on June 5, 1997 
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