In mathematics, cardinal numbers, or cardinals for short, are a generalization of the

_{0}, ''a''_{1}, ..., ''a_{n}''), ''a_{i}'' ∈ Z together with a pair of rationals (''b''_{0}, ''b''_{1}) such that ''z'' is the unique root of the polynomial with coefficients (''a''_{0}, ''a''_{1}, ..., ''a_{n}'') that lies in the interval (''b''_{0}, ''b''_{1}).
In his 1874 paper "

^{+}, where κ^{+} > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, using Hartogs' theorem, it can be shown that for any cardinal number κ, there is a minimal cardinal κ^{+} such that $\backslash kappa^+\backslash nleq\backslash kappa.$) For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the

^{Y}'' is the set of all functions from ''Y'' to ''X''.
:κ^{0} = 1 (in particular 0^{0} = 1), see ^{''μ''} = 0.
:1^{''μ''} = 1.
:''κ''^{1} = ''κ''.
:''κ''^{''μ'' + ''ν''} = ''κ''^{''μ''}·''κ''^{''ν''}.
:κ^{''μ'' · ''ν''} = (''κ''^{''μ''})^{''ν''}.
:(''κ''·''μ'')^{''ν''} = ''κ''^{''ν''}·''μ''^{''ν''}.
Exponentiation is non-decreasing in both arguments:
:(1 ≤ ''ν'' and ''κ'' ≤ ''μ'') → (''ν''^{''κ''} ≤ ''ν''^{''μ''}) and
:(''κ'' ≤ ''μ'') → (''κ''^{''ν''} ≤ ''μ''^{''ν''}).
2^{, ''X'', } is the cardinality of the ^{, ''X'', } > , ''X'', for any set ''X''. This proves that no largest cardinal exists (because for any cardinal ''κ'', we can always find a larger cardinal 2^{''κ''}). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.)
All the remaining propositions in this section assume the axiom of choice:
:If ''κ'' and ''μ'' are both finite and greater than 1, and ''ν'' is infinite, then ''κ''^{''ν''} = ''μ''^{''ν''}.
:If ''κ'' is infinite and ''μ'' is finite and non-zero, then ''κ''^{''μ''} = ''κ''.
If 2 ≤ ''κ'' and 1 ≤ ''μ'' and at least one of them is infinite, then:
:Max (''κ'', 2^{''μ''}) ≤ ''κ''^{''μ''} ≤ Max (2^{''κ''}, 2^{''μ''}).
Using König's theorem, one can prove ''κ'' < ''κ''^{cf(''κ'')} and ''κ'' < cf(2^{''κ''}) for any infinite cardinal ''κ'', where cf(''κ'') is the cofinality of ''κ''.

^{''μ''}. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of

natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...

s used to measure the cardinality (size) of sets. The cardinality of a finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. T ...

is a natural number: the number of elements in the set. The '' transfinite'' cardinal numbers, often denoted using the Hebrew
Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved ...

symbol $\backslash aleph$ ( aleph) followed by a subscript, describe the sizes of infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only s ...

s.
Cardinality is defined in terms of bijective function
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

s. Two sets have the same cardinality if, and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicon ...

, there is a one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

(bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...

s. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set—something that cannot happen with proper subsets of finite sets.
There is a transfinite sequence of cardinal numbers:
:$0,\; 1,\; 2,\; 3,\; \backslash ldots,\; n,\; \backslash ldots\; ;\; \backslash aleph\_0,\; \backslash aleph\_1,\; \backslash aleph\_2,\; \backslash ldots,\; \backslash aleph\_,\; \backslash ldots.\backslash $
This sequence starts with the natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...

s including zero (finite cardinals), which are followed by the aleph number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named a ...

s (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least ...

includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.
Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra and mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...

. In category theory, the cardinal numbers form a skeleton of the category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...

.
History

The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884. Cardinality can be used to compare an aspect of finite sets. For example, the sets and are not ''equal'', but have the ''same cardinality'', namely three. This is established by the existence of a bijection (i.e., a one-to-one correspondence) between the two sets, such as the correspondence . Cantor applied his concept of bijection to infinite sets (for example the set of natural numbers N = ). Thus, he called all sets having a bijection with N ''denumerable (countably infinite) sets'', which all share the same cardinal number. This cardinal number is called $\backslash aleph\_0$,aleph-null
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named af ...

. He called the cardinal numbers of infinite sets transfinite cardinal numbers
In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to qua ...

.
Cantor proved that any unbounded subset of N has the same cardinality as N, even though this might appear to run contrary to intuition. He also proved that the set of all ordered pairs of natural numbers is denumerable; this implies that the set of all rational numbers is also denumerable, since every rational can be represented by a pair of integers. He later proved that the set of all real algebraic numbers is also denumerable. Each real algebraic number ''z'' may be encoded as a finite sequence of integers, which are the coefficients in the polynomial equation of which it is a solution, i.e. the ordered n-tuple (''a''On a Property of the Collection of All Real Algebraic Numbers
Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is his "revolutionary discovery" that the set of all real numbers is unc ...

", Cantor proved that there exist higher-order cardinal numbers, by showing that the set of real numbers has cardinality greater than that of N. His proof used an argument with nested intervals
In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals I_n on the real number line with natural numbers n=1,2,3,\dots as an index. In order for a sequence of intervals to be considered n ...

, but in an 1891 paper, he proved the same result using his ingenious and much simpler diagonal argument A diagonal argument, in mathematics, is a technique employed in the proofs of the following theorems:
*Cantor's diagonal argument (the earliest)
*Cantor's theorem
* Russell's paradox
*Diagonal lemma
** Gödel's first incompleteness theorem
**Tarski ...

. The new cardinal number of the set of real numbers is called the cardinality of the continuum and Cantor used the symbol $\backslash mathfrak$ for it.
Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number ($\backslash aleph\_0$, aleph-null), and that for every cardinal number there is a next-larger cardinal
:$(\backslash aleph\_1,\; \backslash aleph\_2,\; \backslash aleph\_3,\; \backslash ldots).$
His continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

is the proposition that the cardinality $\backslash mathfrak$ of the set of real numbers is the same as $\backslash aleph\_1$. This hypothesis is independent of the standard axioms of mathematical set theory, that is, it can neither be proved nor disproved from them. This was shown in 1963 by Paul Cohen
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician. He is best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was award ...

, complementing earlier work by Kurt Gödel in 1940.
Motivation

In informal use, a cardinal number is what is normally referred to as a '' counting number'', provided that 0 is included: 0, 1, 2, .... They may be identified with the natural numbers beginning with 0. The counting numbers are exactly what can be defined formally as thefinite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...

cardinal numbers. Infinite cardinals only occur in higher-level mathematics and logic.
More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can construct a set that has exactly the right size. For example, 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set , which has 3 elements.
However, when dealing with infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only s ...

s, it is essential to distinguish between the two, since the two notions are in fact different for infinite sets. Considering the position aspect leads to ordinal numbers
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least ...

, while the size aspect is generalized by the cardinal numbers described here.
The intuition behind the formal definition of cardinal is the construction of a notion of the relative size or "bigness" of a set, without reference to the kind of members which it has. For finite sets this is easy; one simply counts the number of elements a set has. In order to compare the sizes of larger sets, it is necessary to appeal to more refined notions.
A set ''Y'' is at least as big as a set ''X'' if there is an injective mapping from the elements of ''X'' to the elements of ''Y''. An injective mapping identifies each element of the set ''X'' with a unique element of the set ''Y''. This is most easily understood by an example; suppose we have the sets ''X'' = and ''Y'' = , then using this notion of size, we would observe that there is a mapping:
: 1 → a
: 2 → b
: 3 → c
which is injective, and hence conclude that ''Y'' has cardinality greater than or equal to ''X''. The element d has no element mapping to it, but this is permitted as we only require an injective mapping, and not necessarily a bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

mapping. The advantage of this notion is that it can be extended to infinite sets.
We can then extend this to an equality-style relation. Two sets ''X'' and ''Y'' are said to have the same ''cardinality'' if there exists a bijection between ''X'' and ''Y''. By the Schroeder–Bernstein theorem, this is equivalent to there being ''both'' an injective mapping from ''X'' to ''Y'', ''and'' an injective mapping from ''Y'' to ''X''. We then write , ''X'', = , ''Y'', . The cardinal number of ''X'' itself is often defined as the least ordinal ''a'' with , ''a'', = , ''X'', . This is called the von Neumann cardinal assignment; for this definition to make sense, it must be proved that every set has the same cardinality as ''some'' ordinal; this statement is the well-ordering principle. It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.
The classic example used is that of the infinite hotel paradox, also called Hilbert's paradox of the Grand Hotel
Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely m ...

. Supposing there is an innkeeper at a hotel with an infinite number of rooms. The hotel is full, and then a new guest arrives. It is possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:
: 1 → 2
: 2 → 3
: 3 → 4
: ...
: ''n'' → ''n'' + 1
: ...
With this assignment, we can see that the set has the same cardinality as the set , since a bijection between the first and the second has been shown. This motivates the definition of an infinite set being any set that has a proper subset of the same cardinality (i.e., a Dedekind-infinite set
In mathematics, a set ''A'' is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset ''B'' of ''A'' is equinumerous to ''A''. Explicitly, this means that there exists a bijective function from ''A'' onto ...

); in this case is a proper subset of .
When considering these large objects, one might also want to see if the notion of counting order coincides with that of cardinal defined above for these infinite sets. It happens that it does not; by considering the above example we can see that if some object "one greater than infinity" exists, then it must have the same cardinality as the infinite set we started out with. It is possible to use a different formal notion for number, called ordinals, based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers.
It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described. This can be visualized using Cantor's diagonal argument
In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a m ...

;
classic questions of cardinality (for instance the continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals. In more recent times, mathematicians have been describing the properties of larger and larger cardinals.
Since cardinality is such a common concept in mathematics, a variety of names are in use. Sameness of cardinality is sometimes referred to as ''equipotence'', ''equipollence'', or ''equinumerosity''. It is thus said that two sets with the same cardinality are, respectively, ''equipotent'', ''equipollent'', or ''equinumerous''.
Formal definition

Formally, assuming the axiom of choice, the cardinality of a set ''X'' is the least ordinal number α such that there is a bijection between ''X'' and α. This definition is known as the von Neumann cardinal assignment. If the axiom of choice is not assumed, then a different approach is needed. The oldest definition of the cardinality of a set ''X'' (implicit in Cantor and explicit in Frege andPrincipia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...

) is as the class 'X''of all sets that are equinumerous with ''X''. This does not work in ZFC or other related systems of axiomatic set theory because if ''X'' is non-empty, this collection is too large to be a set. In fact, for ''X'' ≠ ∅ there is an injection from the universe into 'X''by mapping a set ''m'' to × ''X'', and so by the axiom of limitation of size
In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes.; English translation: . It formalizes the limitation of size principle, which avoids the paradoxes encountered in earli ...

, 'X''is a proper class. The definition does work however in type theory and in New Foundations and related systems. However, if we restrict from this class to those equinumerous with ''X'' that have the least rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...

, then it will work (this is a trick due to Dana Scott
Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, Ca ...

: it works because the collection of objects with any given rank is a set).
Von Neumann cardinal assignment implies that the cardinal number of a finite set is the common ordinal number of all possible well-orderings of that set, and cardinal and ordinal arithmetic (addition, multiplication, power, proper subtraction) then give the same answers for finite numbers. However, they differ for infinite numbers. For example, $2^\backslash omega=\backslash omega<\backslash omega^2$ in ordinal arithmetic while $2^>\backslash aleph\_0=\backslash aleph\_0^2$ in cardinal arithmetic, although the von Neumann assignment puts $\backslash aleph\_0=\backslash omega$. On the other hand, Scott's trick implies that the cardinal number 0 is $\backslash $, which is also the ordinal number 1, and this may be confusing. A possible compromise (to take advantage of the alignment in finite arithmetic while avoiding reliance on the axiom of choice and confusion in infinite arithmetic) is to apply von Neumann assignment to the cardinal numbers of finite sets (those which can be well ordered and are not equipotent to proper subsets) and to use Scott's trick for the cardinal numbers of other sets.
Formally, the order among cardinal numbers is defined as follows: , ''X'', ≤ , ''Y'', means that there exists an injective function from ''X'' to ''Y''. The Cantor–Bernstein–Schroeder theorem states that if , ''X'', ≤ , ''Y'', and , ''Y'', ≤ , ''X'', then , ''X'', = , ''Y'', . The axiom of choice is equivalent to the statement that given two sets ''X'' and ''Y'', either , ''X'', ≤ , ''Y'', or , ''Y'', ≤ , ''X'', .Enderton, Herbert. "Elements of Set Theory", Academic Press Inc., 1977.
A set ''X'' is Dedekind-infinite if there exists a proper subset ''Y'' of ''X'' with , ''X'', = , ''Y'', , and Dedekind-finite if such a subset doesn't exist. The finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...

cardinals are just the natural numbers, in the sense that a set ''X'' is finite if and only if , ''X'', = , ''n'', = ''n'' for some natural number ''n''. Any other set is infinite
Infinite may refer to:
Mathematics
* Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...

.
Assuming the axiom of choice, it can be proved that the Dedekind notions correspond to the standard ones. It can also be proved that the cardinal $\backslash aleph\_0$ (aleph null
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named a ...

or aleph-0, where aleph is the first letter in the Hebrew alphabet, represented $\backslash aleph$) of the set of natural numbers is the smallest infinite cardinal (i.e., any infinite set has a subset of cardinality $\backslash aleph\_0$). The next larger cardinal is denoted by $\backslash aleph\_1$, and so on. For every ordinal α, there is a cardinal number $\backslash aleph\_,$ and this list exhausts all infinite cardinal numbers.
Cardinal arithmetic

We can define arithmetic operations on cardinal numbers that generalize the ordinary operations for natural numbers. It can be shown that for finite cardinals, these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with ordinary arithmetic.Successor cardinal

If the axiom of choice holds, then every cardinal κ has a successor, denoted κsuccessor ordinal In set theory, the successor of an ordinal number ''α'' is the smallest ordinal number greater than ''α''. An ordinal number that is a successor is called a successor ordinal. Properties
Every ordinal other than 0 is either a successor ordin ...

.
Cardinal addition

If ''X'' and ''Y'' are disjoint, addition is given by theunion
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...

of ''X'' and ''Y''. If the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace ''X'' by ''X''× and ''Y'' by ''Y''×).
:$,\; X,\; +\; ,\; Y,\; =\; ,\; X\; \backslash cup\; Y,\; .$
Zero is an additive identity ''κ'' + 0 = 0 + ''κ'' = ''κ''.
Addition is associative (''κ'' + ''μ'') + ''ν'' = ''κ'' + (''μ'' + ''ν'').
Addition is commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

''κ'' + ''μ'' = ''μ'' + ''κ''.
Addition is non-decreasing in both arguments:
:$(\backslash kappa\; \backslash le\; \backslash mu)\; \backslash rightarrow\; ((\backslash kappa\; +\; \backslash nu\; \backslash le\; \backslash mu\; +\; \backslash nu)\; \backslash mbox\; (\backslash nu\; +\; \backslash kappa\; \backslash le\; \backslash nu\; +\; \backslash mu)).$
Assuming the axiom of choice, addition of infinite cardinal numbers is easy. If either ''κ'' or ''μ'' is infinite, then
:$\backslash kappa\; +\; \backslash mu\; =\; \backslash max\backslash \backslash ,.$
Subtraction

Assuming the axiom of choice and, given an infinite cardinal ''σ'' and a cardinal ''μ'', there exists a cardinal ''κ'' such that ''μ'' + ''κ'' = ''σ'' if and only if ''μ'' ≤ ''σ''. It will be unique (and equal to ''σ'') if and only if ''μ'' < ''σ''.Cardinal multiplication

The product of cardinals comes from the Cartesian product. :$,\; X,\; \backslash cdot,\; Y,\; =\; ,\; X\; \backslash times\; Y,$ ''κ''·0 = 0·''κ'' = 0. ''κ''·''μ'' = 0 → (''κ'' = 0 or ''μ'' = 0). One is a multiplicative identity ''κ''·1 = 1·''κ'' = ''κ''. Multiplication is associative (''κ''·''μ'')·''ν'' = ''κ''·(''μ''·''ν''). Multiplication iscommutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

''κ''·''μ'' = ''μ''·''κ''.
Multiplication is non-decreasing in both arguments:
''κ'' ≤ ''μ'' → (''κ''·''ν'' ≤ ''μ''·''ν'' and ''ν''·''κ'' ≤ ''ν''·''μ'').
Multiplication distributes over addition:
''κ''·(''μ'' + ''ν'') = ''κ''·''μ'' + ''κ''·''ν'' and
(''μ'' + ''ν'')·''κ'' = ''μ''·''κ'' + ''ν''·''κ''.
Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy. If either ''κ'' or ''μ'' is infinite and both are non-zero, then
:$\backslash kappa\backslash cdot\backslash mu\; =\; \backslash max\backslash .$
Division

Assuming the axiom of choice and, given an infinite cardinal ''π'' and a non-zero cardinal ''μ'', there exists a cardinal ''κ'' such that ''μ'' · ''κ'' = ''π'' if and only if ''μ'' ≤ ''π''. It will be unique (and equal to ''π'') if and only if ''μ'' < ''π''.Cardinal exponentiation

Exponentiation is given by :$,\; X,\; ^\; =\; \backslash left,\; X^Y\backslash ,$ where ''Xempty function
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...

.
:If 1 ≤ ''μ'', then 0power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...

of the set ''X'' and Cantor's diagonal argument
In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a m ...

shows that 2Roots

Assuming the axiom of choice and, given an infinite cardinal ''κ'' and a finite cardinal ''μ'' greater than 0, the cardinal ''ν'' satisfying $\backslash nu^\backslash mu\; =\; \backslash kappa$ will be $\backslash kappa$.Logarithms

Assuming the axiom of choice and, given an infinite cardinal ''κ'' and a finite cardinal ''μ'' greater than 1, there may or may not be a cardinal ''λ'' satisfying $\backslash mu^\backslash lambda\; =\; \backslash kappa$. However, if such a cardinal exists, it is infinite and less than ''κ'', and any finite cardinality ''ν'' greater than 1 will also satisfy $\backslash nu^\backslash lambda\; =\; \backslash kappa$. The logarithm of an infinite cardinal number ''κ'' is defined as the least cardinal number ''μ'' such that ''κ'' ≤ 2cardinal invariant In mathematics, a cardinal function (or cardinal invariant) is a function (mathematics), function that returns cardinal numbers.
Cardinal functions in set theory
* The most frequently used cardinal function is a function that assigns to a Set (m ...

s of topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

s, though they lack some of the properties that logarithms of positive real numbers possess.D. A. Vladimirov, Boolean Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.
The continuum hypothesis

Thecontinuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

(CH) states that there are no cardinals strictly between $\backslash aleph\_0$ and $2^.$ The latter cardinal number is also often denoted by $\backslash mathfrak$; it is the cardinality of the continuum (the set of real numbers). In this case $2^\; =\; \backslash aleph\_1.$
Similarly, the generalized continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

(GCH) states that for every infinite cardinal $\backslash kappa$, there are no cardinals strictly between $\backslash kappa$ and $2^\backslash kappa$. Both the continuum hypothesis and the generalized continuum hypothesis have been proved independent of the usual axioms of set theory, the Zermelo–Fraenkel axioms together with the axiom of choice ( ZFC).
Indeed, Easton's theorem shows that, for regular cardinals $\backslash kappa$, the only restrictions ZFC places on the cardinality of $2^\backslash kappa$ are that $\backslash kappa\; <\; \backslash operatorname(2^\backslash kappa)$, and that the exponential function is non-decreasing.
See also

*Aleph number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named a ...

* Beth number
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written \beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots, where \beth is the second H ...

* The paradox of the greatest cardinal
* Cardinal number (linguistics)
* Counting
Counting is the process of determining the number of elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every ele ...

* Inclusion–exclusion principle
In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as
: , A \cu ...

* Large cardinal
* Names of numbers in English
* Nominal number
* Ordinal number
* Regular cardinal
References

Notes Bibliography * * Hahn, Hans, ''Infinity'', Part IX, Chapter 2, Volume 3 of ''The World of Mathematics''. New York: Simon and Schuster, 1956. * Halmos, Paul, '' Naive set theory''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition). *External links

* {{DEFAULTSORT:Cardinal Number