cardinal number

TheInfoList

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 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 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 t ...
is a natural number: the number of elements in the set. The '' transfinite'' cardinal numbers, often denoted using the Hebrew symbol $\aleph$ (
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 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 ...
s. Cardinality is defined in terms of
bijective function In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is paired with exactly on ...
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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...
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 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 a ...
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 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 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 ...
s (infinite cardinals of well-ordered sets). The aleph numbers are indexed by
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 ...
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#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

, 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 ...
,
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 by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...
, 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 $\aleph_0$,
aleph-null 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 ...
. 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''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 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 In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of Interval (mathematics), intervals I_n on the Interval (mathematics), real number line with natural number, natural numbers n=1,2,3,\dots as an ...
, 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 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 ...
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 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 $\mathfrak$ is the same as $\aleph_1$. 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 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 a ...
'', provided that 0 is included: 0, 1, 2, .... They may be identified with the
natural numbers 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 a ...
beginning with 0. The counting numbers are exactly what can be defined formally as the
finite Finite is the opposite of 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 *''Infin ...
cardinal numbers. Infinite cardinals only occur in higher-level mathematics and
logic Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact A fact is something that is true True most commonly refers to truth Truth is the property of being in accord with fac ...
. 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 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 ...
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 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 ...
, 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 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 mathematics, a surjective function (also known as surjection, or onto function) is a Function (mathematics), function that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the ...
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 Von Neumann may refer to: * John von Neumann (1903–1957), a Hungarian American mathematician * Von Neumann family * Von Neumann (surname), a German surname * Von Neumann (crater), a lunar impact crater See also * Von Neumann al ...
; 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 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 so ...
); 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 In 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 bran ...
; 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 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#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

, the cardinality of a set ''X'' is the least
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 ...
α such that there is a bijection between ''X'' and α. This definition is known as the
von Neumann cardinal assignment The von Neumann Von Neumann may refer to: * John von Neumann (1903–1957), a Hungarian American mathematician * Von Neumann family * Von Neumann (surname), a German surname * Von Neumann (crater), a lunar impact crater See also * Von Neumann al ...
. 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 Foundations of mathematics is the study of the philosophical Philosophy (from , ) is the study of general and fun ...
) 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 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 ...
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 avo ...
, '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 Foundations In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the Type theory, theory of types of ''Principia Mathematica''. Quine first proposed NF in a 1937 article titled "New ...
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 mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

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 so ...
if there exists a
proper subset 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 a ...
''Y'' of ''X'' with , ''X'', = , ''Y'', , and
Dedekind-finite 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 ...
if such a subset doesn't exist. The
finite Finite is the opposite of 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 *''Infin ...
cardinals are just the
natural numbers 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 a ...
, 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 mus ...
. 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 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 ...
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 Ktav Ashuri ( he, כְּתָב אַשּׁוּרִי, ' "Assyrian script"; also Ashurit) is the traditional Hebrew language ...
, 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 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 $\aleph_,$ and this list exhausts all infinite cardinal numbers.

# Cardinal arithmetic

We can define
arithmetic Arithmetic (from the Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is app ...
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, 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 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 ...
.

If ''X'' and ''Y'' are disjoint, addition is given by the
union Union commonly refers to: * Trade union A trade union (or a labor union in American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of variety (linguistics), ...
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 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 a ...
(''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: :$\left(\kappa \le \mu\right) \rightarrow \left(\left(\kappa + \nu \le \mu + \nu\right) \mbox \left(\nu + \kappa \le \nu + \mu\right)\right).$ 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 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 ...
. :$, X, \cdot, Y, = , X \times Y,$ ''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 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 :$\kappa\cdot\mu = \max\.$

### 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''.0 = 1 (in particular 00 = 1), see
empty function 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 ...
. :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 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 In 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 bran ...
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 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 Foundations In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the Type theory, theory of types of ''Principia Mathematica''. Quine first proposed NF in a 1937 article titled "New ...
.) 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 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 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 invariant In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers. Cardinal functions in set theory * The most frequently used cardinal function is a function which assigns to a Set (mathematics), set "A" it ...
s of
topological 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

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 ...
(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 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 ...
(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 $2^ = \aleph_1.$ The
generalized continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in a ...
(GCH) states that for every infinite set ''X'', there are no cardinals strictly between ,  ''X'' , and 2,  ''X'' , . The continuum hypothesis is independent of the usual axioms of set theory, the Zermelo–Fraenkel axioms together with the axiom of choice ( ZFC).

*
Aleph number 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 ...
*
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 scientific study of language A language is a structured system of communication Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic ...
*
Counting Counting is the process of determining the number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be ...
*
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 (set theory), union of two finite sets; symbolically exp ...
*
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 numeral A numeral is a figure, symbol, or group of figures or symbols denoting a number. It may refer to: * Numeral system used in mathematics * Numeral (linguistics), a part of speech denoting numbers (e.g. ''one'' ...
*
Nominal number Nominal numbers are numeral A numeral is a figure, symbol, or group of figures or symbols denoting a number. It may refer to: * Numeral system used in mathematics * Numeral (linguistics), a part of speech denoting numbers (e.g. ''one'' and ''fir ...
*
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 cardinal In 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 ...

# 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 Philosophy (from , ) is the study of general and fundamental quest ...
''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition).