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OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 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 In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
(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 $\aleph$ (
aleph Aleph (or alef or alif, transliterated ʾ) is the first letter of the Semitic abjads, including Phoenician , Hebrew , Aramaic , Syriac , Arabic ʾ and North Arabian 𐪑. It also appears as South Arabian 𐩱 and Ge'ez . These lett ...
) 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 functions. 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 Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
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, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 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 In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
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, \ldots, n, \ldots ; \aleph_0, \aleph_1, \aleph_2, \ldots, \aleph_, \ldots.\$ 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 number 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 leas ...
s. Under the assumption of the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, 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 In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
is studied for its own sake as part of
set theory 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 set, set theory, as a branch of mathematics, is mostly concern ...
. It is also a tool used in branches of mathematics including
model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
,
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 structures. It is closely related to many other areas of mathematics and has many a ...
,
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
and
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
. In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, the cardinal numbers form a
skeleton A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
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 Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
, the originator of
set theory 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 set, set theory, as a branch of mathematics, is mostly concern ...
, 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, 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 ...
(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 $\aleph_0$, aleph-null. 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, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
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 a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of th ...
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''0, ''a''1, ..., ''an''), ''ai'' ∈ 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, ..., ''an'') that lies in the interval (''b''0, ''b''1). In his 1874 paper "
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, 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 In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \math ...
and Cantor used the symbol $\mathfrak$ 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 ($\aleph_0$, aleph-null), and that for every cardinal number there is a next-larger cardinal :$\left(\aleph_1, \aleph_2, \aleph_3, \ldots\right).$ 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 $\mathfrak$ of the set of real numbers is the same as $\aleph_1$. 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 Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...
in 1940.

# Motivation

In informal use, a cardinal number is what is normally referred to as a ''
counting 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 ...
'', 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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
beginning with 0. The counting numbers are exactly what can be defined formally as 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 ...
cardinal numbers. Infinite cardinals only occur in higher-level mathematics and
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
. 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 In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
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 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 ...
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 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 ...
. 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 number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s 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 In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, the cardinality of a set ''X'' is the least
ordinal number 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 leas ...
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 and
Principia 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 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 set, set theory, as a branch of mathematics, is mostly concern ...
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 In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...
and in
New Foundations In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of '' Principia Mathematica''. Quine first proposed NF in a 1937 article titled "New Foundatio ...
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, $2^\omega=\omega<\omega^2$ in ordinal arithmetic while $2^>\aleph_0=\aleph_0^2$ in cardinal arithmetic, although the von Neumann assignment puts $\aleph_0=\omega$. On the other hand, Scott's trick implies that the cardinal number 0 is $\$, 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 In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
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 mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
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 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 ...
if there exists a
proper subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
''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 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 ...
, 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 $\aleph_0$ ( aleph null or aleph-0, where aleph is the first letter in the
Hebrew alphabet The Hebrew alphabet ( he, 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 and other Jewi ...
, represented $\aleph$) of the set of natural numbers is the smallest infinite cardinal (i.e., any infinite set has a subset of cardinality $\aleph_0$). The next larger cardinal is denoted by $\aleph_1$, and so on. For every ordinal α, there is a cardinal number $\aleph_,$ and this list exhausts all infinite cardinal numbers.

# Cardinal arithmetic

We can define
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
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 κ+, where κ+ > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, using
Hartogs' theorem In mathematics, Hartogs's theorem is a fundamental result of Friedrich Hartogs in the theory of several complex variables. Roughly speaking, it states that a 'separately analytic' function is continuous. More precisely, if F:^n \to is a functi ...
, it can be shown that for any cardinal number κ, there is a minimal cardinal κ+ such that $\kappa^+\nleq\kappa.$) For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the
successor 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 ...
.

If ''X'' and ''Y'' are disjoint, 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''×). :$, X, + , Y, = , X \cup Y, .$ Zero is an additive identity ''κ'' + 0 = 0 + ''κ'' = ''κ''. Addition is
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
(''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: :$\left(\kappa \le \mu\right) \rightarrow \left(\left(\kappa + \nu \le \mu + \nu\right) \mbox \left(\nu + \kappa \le \nu + \mu\right)\right).$ Assuming the axiom of choice, addition of infinite cardinal numbers is easy. If either ''κ'' or ''μ'' is infinite, then :$\kappa + \mu = \max\\,.$

### 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 In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
. :$, X, \cdot, Y, = , X \times Y,$ ''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, 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 :$\kappa\cdot\mu = \max\.$

### 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 :$, X, ^ = \left, X^Y\,$ where ''XY'' is the set of all functions from ''Y'' to ''X''. :κ0 = 1 (in particular 00 = 1), see empty function. :If 1 ≤ ''μ'', then 0''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
power 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 2, ''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 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 ...
of cardinals is a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
. (This proof fails in some set theories, notably
New Foundations In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of '' Principia Mathematica''. Quine first proposed NF in a 1937 article titled "New Foundatio ...
.) 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 In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''. This definition of cofinality relies on the axiom of choice, as it uses t ...
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 $\nu^\mu = \kappa$ will be $\kappa$.

### 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 $\mu^\lambda = \kappa$. However, if such a cardinal exists, it is infinite and less than ''κ'', and any finite cardinality ''ν'' greater than 1 will also satisfy $\nu^\lambda = \kappa$. The logarithm of an infinite cardinal number ''κ'' is defined as the least cardinal number ''μ'' such that ''κ'' ≤ 2''μ''. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants 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 poin ...
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

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 ...
(CH) states that there are no cardinals strictly between $\aleph_0$ and $2^.$ The latter cardinal number is also often denoted by $\mathfrak$; it is the
cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \math ...
(the set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s). In this case $2^ = \aleph_1.$ Similarly, the generalized continuum hypothesis (GCH) states that for every infinite cardinal $\kappa$, there are no cardinals strictly between $\kappa$ and $2^\kappa$. 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 In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. (extending a result of Robert M. Solovay) showed via forcing that the only constraints on permissible values for 2''κ'' when ''κ'' is a regular cardinal ...
shows that, for
regular cardinal In 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. Infinit ...
s $\kappa$, the only restrictions ZFC places on the cardinality of $2^\kappa$ are that $\kappa < \operatorname\left(2^\kappa\right)$, and that the exponential function is non-decreasing.

*
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) In linguistics, and more precisely in traditional grammar, a cardinal numeral (or cardinal number word) is a part of speech used to count. Examples in English are the words ''one'', ''two'', ''three'', and the compounds ''three hundred ndfo ...
*
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 In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...
* Names of numbers in English * Nominal number *
Ordinal number 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 leas ...
*
Regular cardinal In 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. Infinit ...

# 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. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It ...
''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition). *