In

_{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 ^{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 ^{''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

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

^{, ''X'' , }. The continuum hypothesis is independent of the usual axioms of set theory, the Zermelo–Fraenkel axioms together with the axiom of choice ( ZFC).

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, cardinal numbers, or cardinals for short, are a generalization of the natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s used to measure the cardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(size) of sets. The cardinality of a finite set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is a natural number: the number of elements in the set. The '' transfinite'' cardinal numbers, often denoted using the Hebrew symbol $\backslash aleph$ (aleph
Aleph (or alef or alif, transliterated ʾ) is the first letter
Letter, letters, or literature may refer to:
Characters typeface
* Letter (alphabet)
A letter is a segmental symbol
A symbol is a mark, sign, or word that indicates, sig ...

) followed by a subscript, describe the sizes of infinite set
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...

s.
Cardinality is defined in terms of bijective function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (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
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...

shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s 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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s. It is also possible for a proper subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s including zero (finite cardinals), which are followed by the aleph number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal number
In set theory
illustrating the intersection of two sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a s ...

s. Under the assumption of the axiom of choice
In , the axiom of choice, or AC, is an of equivalent to the statement that ''a of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object ...

, this transfinite sequence
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...

includes every cardinal number. If one that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.
Cardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is studied for its own sake as part of set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...

. It is also a tool used in branches of mathematics including model theory
In mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...

, combinatorics
Combinatorics is an area of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...

, abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...

and mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...

. In category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

, the cardinal numbers form a skeleton
A skeleton is a structural frame that supports an animal
Animals (also called Metazoa) are multicellular eukaryotic organisms that form the Kingdom (biology), biological kingdom Animalia. With few exceptions, animals Heterotroph, consu ...

of the category of sets In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

.
History

The notion of cardinality, as now understood, was formulated byGeorg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...

, the originator of set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...

, 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

(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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. He called the cardinal numbers of infinite sets transfinite cardinal numbers.
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 pair
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s of natural numbers is denumerable; this implies that the set of all rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s is also denumerable, since every rational can be represented by a pair of integers. He later proved that the set of all real algebraic number
An algebraic number is any complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...

s 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
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor e ...

", 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, but in an 1891 paper, he proved the same result using his ingenious but simpler diagonal argumentDiagonal argument in mathematics may refer to:
*Cantor's diagonal argument (the earliest)
*Cantor's theorem
*Halting problem
*Diagonal lemma
See also
* Diagonalization (disambiguation)
{{mathdab ...

. The new cardinal number of the set of real numbers is called the cardinality of the continuum
In set theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ar ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

is the proposition that $\backslash mathfrak$ is the same as $\backslash aleph\_1$. This hypothesis has been found to be independent of the standard axioms of mathematical set theory; it can neither be proved nor disproved from the standard assumptions.
Motivation

In informal use, a cardinal number is what is normally referred to as a ''counting number
In mathematics, the natural numbers are those used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, words coll ...

'', provided that 0 is included: 0, 1, 2, .... They may be identified with the natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

beginning with 0. The counting numbers are exactly what can be defined formally as the finite
Finite is the opposite of Infinity, 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 ...

cardinal numbers. Infinite cardinals only occur in higher-level mathematics and logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

.
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
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...

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
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...

, 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

mapping
Mapping may refer to:
* Mapping (cartography), the process of making a map
* Mapping (mathematics), a synonym for a mathematical function and its generalizations
** Mapping (logic), a synonym for functional predicate
Types of mapping
* Animated ...

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 an injective and onto
In , a surjective function (also known as surjection, or onto function) is a that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's is the of one element of its ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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
The von Neumann cardinal assignment is a cardinal assignment which uses ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly ...

; 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
In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered by its "natural" or "magnitude" order in which x precedes y if ...

. 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. 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 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 number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s is greater than that of the natural numbers just described. This can be visualized using Cantor's diagonal argument
250px, An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above.
In set theory, Cantor's diagonal argument, also cal ...

;
classic questions of cardinality (for instance the continuum hypothesis
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

) 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 theaxiom of choice
In , the axiom of choice, or AC, is an of equivalent to the statement that ''a of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object ...

, the cardinality of a set ''X'' is the least ordinal number
In set theory
illustrating the intersection of two sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a s ...

α such that there is a bijection between ''X'' and α. This definition is known as the von Neumann cardinal assignment
The von Neumann cardinal assignment is a cardinal assignment which uses ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly ...

. 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 and Principia Mathematica
Image:Principia Mathematica 54-43.png, 500px, ✸54.43:
"From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." – Volume I, 1st editionp. 379(p. 362 in 2nd edition; p. 360 in abridged v ...

) is as the class 'X''of all sets that are equinumerous with ''X''. This does not work in or other related systems of axiomatic set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...

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 Set (mathematics), sets and Class (set theory), classes.; English translation: . It formalizes the limitation of size principle, which avoi ...

, 'X''is a proper class. The definition does work however in type theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

and in New FoundationsIn mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...

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
A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...

, 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
Computer science deals with the theoretical foundations of information, algorithms and the architecture ...

: it works because the collection of objects with any given rank is a set).
Formally, the order among cardinal numbers is defined as follows: , ''X'', ≤ , ''Y'', means that there exists an injective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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
In , the axiom of choice, or AC, is an of equivalent to the statement that ''a of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object ...

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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

''Y'' of ''X'' with , ''X'', = , ''Y'', , and Dedekind-finiteIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

if such a subset doesn't exist. The finite
Finite is the opposite of Infinity, 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 ...

cardinals are just the natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

, 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 (band), a South Korean boy band
*''Infinite'' (EP), debut EP of American musi ...

.
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 or aleph-0, where aleph is the first letter in the Hebrew alphabet
The Hebrew alphabet ( he, wikt:אלפבית, אָלֶף־בֵּית עִבְרִי, ), known variously by scholars as the Ktav Ashuri, Jewish script, square script and block script, is an abjad script used in the writing of the Hebrew language ...

, 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
Ordinal may refer to:
* Ordinal data, a statistical data type consisting of numerical scores that exist on an arbitrary numerical scale
* Ordinal date, a simple form of expressing a date using only the year and the day number within that year
* O ...

α, there is a cardinal number $\backslash aleph\_,$ and this list exhausts all infinite cardinal numbers.
Cardinal arithmetic

We can definearithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...

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 ordinalIn set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any typ ...

.
Cardinal addition

If ''X'' and ''Y'' are , addition is given by the union 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 isassociative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

(''κ'' + ''μ'') + ''ν'' = ''κ'' + (''μ'' + ''ν'').
Addition is commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

''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 theCartesian product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

.
:$,\; X,\; \backslash cdot,\; Y,\; =\; ,\; X\; \backslash times\; Y,$
''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 is commutative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

''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 ''Xfunctions
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

from ''Y'' to ''X''.
:κempty function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

.
:If 1 ≤ ''μ'', then 0power set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of the set ''X'' and Cantor's diagonal argument
250px, An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above.
In set theory, Cantor's diagonal argument, also cal ...

shows that 2class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differently f ...

of cardinals is a proper class
Proper may refer to:
Mathematics
* Proper map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and qu ...

. (This proof fails in some set theories, notably New FoundationsIn mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...

.)
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''cofinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of ''κ''.
Roots

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'' ≤ 2topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

(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
In set theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ar ...

(the set of real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s). In this case $2^\; =\; \backslash aleph\_1.$ The generalized continuum hypothesis
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(GCH) states that for every infinite set ''X'', there are no cardinals strictly between , ''X'' , and 2See also

*Aleph number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

* Beth number
In mathematics, the beth numbers are a certain sequence of infinite set, infinite cardinal numbers, conventionally written \beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots, where \beth is the second Hebrew alphabet, Hebrew letter (bet (letter), beth). ...

* The paradox of the greatest cardinal
* Cardinal number (linguistics)
In linguistics
Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them.
The traditional areas of linguistic analysis include ...

* Counting
Counting is the process of determining the number of Element (mathematics), elements of a finite set of objects, i.e., determining the size (mathematics), size of a set. The traditional way of counting consists of continually increasing a (mental ...

* Inclusion–exclusion principle
In combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely relat ...

* Large cardinal
In the mathematical field of set theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly co ...

* Names of numbers in English
English number words include numerals and various words derived from them, as well as a large number of words borrowed from other languages.
Cardinal numbers
Cardinal numbers refer to the size of a group. In English, these words are numerals ...

* Nominal number
Nominal numbers are categorical, which means that these are numerals used as labels to identify items uniquely. Importantly, the actual values of the number
A number is a mathematical object used to counting, count, measurement, measure, and ...

* Ordinal number
In set theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...

* Regular cardinalIn set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that \kappa is a regular cardinal if and only if every unbounded subset C \subseteq \kappa has cardinality \kappa. Infinite wel ...

Notes

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
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sens ...

''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition).
External links

* {{DEFAULTSORT:Cardinal Number