In

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 denoted by "$\backslash mathfrak\; c$" (a lowercase fraktur script "c"), and is also referred to as 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, or , ''X'' , < , N , , is said to be a

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. In the late nineteenth century

Reprinted in: Here: p.413 bottom One example of this is Hilbert's paradox of the Grand Hotel. Indeed, Dedekind defined an infinite set as one that can be placed into a one-to-one correspondence with a strict subset (that is, having the same size in Cantor's sense); this notion of infinity is called Dedekind infinite. Cantor introduced the cardinal numbers, and showed—according to his bijection-based definition of size—that some infinite sets are greater than others. The smallest infinite cardinality is that of the natural numbers ($\backslash aleph\_0$).

^{R}
:* the set R^{R} of all functions from R to R
Both have cardinality
:$2^\backslash mathfrak\; =\; \backslash beth\_2\; >\; \backslash mathfrak\; c$
:(see Beth two).
The cardinal equalities $\backslash mathfrak^2\; =\; \backslash mathfrak,$ $\backslash mathfrak\; c^\; =\; \backslash mathfrak\; c,$ and $\backslash mathfrak\; c\; ^\; =\; 2^$ can be demonstrated using

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 ...

, the cardinality of a set is a measure of the "number of elements" of the set. For example, the set $A\; =\; \backslash $ contains 3 elements, and therefore $A$ has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to 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, which allows one to distinguish between the different types of infinity, and to perform arithmetic
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 ...

on them. There are two approaches to cardinality: one which compares sets directly using 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 ...

s and injection
Injection or injected may refer to:
Science and technology
* Injection (medicine)
An injection (often referred to as a "shot" in US English, a "jab" in UK English, or a "jag" in Scottish English and Scots Language, Scots) is the act of adminis ...

s, and another which uses cardinal 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.
The cardinality of a set is also called its size, when no confusion with other notions of size is possible.
The cardinality of a set $A$ is usually denoted $,\; A,$, with a vertical bar
The vertical bar, , is a glyph
The term glyph is used in typography
File:metal movable type.jpg, 225px, Movable type being assembled on a composing stick using pieces that are stored in the type case shown below it
Typography is the ...

on each side; this is the same notation as absolute value
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 ...

, and the meaning depends on context
Context may refer to:
* Context (language use)
In semiotics, linguistics, sociology and anthropology, context refers to those objects or entities which surround a ''focal event'', in these disciplines typically a communication, communicative event ...

. The cardinality of a set $A$ may alternatively be denoted by $n(A)$, $A$, $\backslash operatorname(A)$, or $\backslash \#A$.
Comparing sets

While the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets (some of which are possibly infinite).Definition 1: =

:Two sets ''A'' and ''B'' have the same cardinality if there exists abijection
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 ...

(a.k.a., one-to-one correspondence) from ''A'' to ''B'', that is, a function
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 ''A'' to ''B'' that is both 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 ...

and surjective
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 ...

. Such sets are said to be ''equipotent'', ''equipollent'', or ''equinumerous
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 ...

''. This relationship can also be denoted ''A'' ≈ ''B'' or ''A'' ~ ''B''.
:For example, the set ''E'' = of non-negative even 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 ge ...

s has the same cardinality as the set N = of 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 ...

, since the function ''f''(''n'') = 2''n'' is a bijection from N to ''E'' (see picture).
Definition 2: ≤

:''A'' has cardinality less than or equal to the cardinality of ''B'', if there exists an injective function from ''A'' into ''B''.Definition 3: <

:''A'' has cardinality strictly less than the cardinality of ''B'', if there is an injective function, but no bijective function, from ''A'' to ''B''. :For example, the set N of allnatural 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 ...

has cardinality strictly less than its 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 ...

''P''(N), because ''g''(''n'') = is an injective function from N to ''P''(N), and it can be shown that no function from N to ''P''(N) can be bijective (see picture). By a similar argument, N has cardinality strictly less than the cardinality of the set R of all 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. For proofs, see 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 ...

or Cantor's first uncountability proof
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 ...

.
If ≤ and ≤ , then = (a fact known as Schröder–Bernstein theoremIn set theory, the Schröder–Bernstein theorem states that, if there exist injective functions and between the Set (mathematics), sets and , then there exists a bijection, bijective function .
In terms of the cardinality of the two sets, this c ...

). 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 ≤ or ≤ for every ''A'', ''B''.
Cardinal numbers

In the above section, "cardinality" of a set was defined functionally. In other words, it was not defined as a specific object itself. However, such an object can be defined as follows. The relation of having the same cardinality is calledequinumerosity
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 ...

, and this is an equivalence relation
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 ...

on 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 f ...

of all sets. The equivalence class
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 a set ''A'' under this relation, then, consists of all those sets which have the same cardinality as ''A''. There are two ways to define the "cardinality of a set":
#The cardinality of a set ''A'' is defined as its equivalence class under equinumerosity.
#A representative set is designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal 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 ...

in 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 ...

.
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 cardinalities of the 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 are denoted
:$\backslash aleph\_0\; <\; \backslash aleph\_1\; <\; \backslash aleph\_2\; <\; \backslash ldots\; .$
For each 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 ...

$\backslash alpha$, $\backslash aleph\_$ is the least cardinal number greater than $\backslash aleph\_\backslash alpha$.
The cardinality 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 is denoted 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 ...

($\backslash aleph\_0$), while the cardinality of 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 ...

. Cantor showed, using the 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 ...

, that $>\backslash aleph\_0$. We can show that $\backslash mathfrak\; c\; =\; 2^$, this also being the cardinality of the set of all subsets of the natural numbers.
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 ...

says that $\backslash aleph\_1\; =\; 2^$, i.e. $2^$ is the smallest cardinal number bigger than $\backslash aleph\_0$, i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independen ...

of ZFC, a standard axiomatization of set theory; that is, it is impossible to prove the continuum hypothesis or its negation from ZFC—provided that ZFC is consistent. For more detail, see § Cardinality of the continuum below.
Finite, countable and uncountable sets

If theaxiom 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 ...

holds, the law of trichotomy holds for cardinality. Thus we can make the following definitions:
*Any set ''X'' with cardinality less than that of the 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 ...

.
*Any set ''X'' that has the same cardinality as the set of the natural numbers, or , ''X'' , = , N , = $\backslash aleph\_0$, is said to be a countably infinite
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 ...

set.
*Any set ''X'' with cardinality greater than that of the natural numbers, or , ''X'' , > , N , , for example , R , = $\backslash mathfrak\; c$ > , N , , is said to be uncountable
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 ...

.
Infinite sets

Our intuition gained fromfinite 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 ...

s breaks down when dealing with 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 ...

, Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...

, Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For citiz ...

and others rejected the view that the whole cannot be the same size as the part.Reprinted in: Here: p.413 bottom One example of this is Hilbert's paradox of the Grand Hotel. Indeed, Dedekind defined an infinite set as one that can be placed into a one-to-one correspondence with a strict subset (that is, having the same size in Cantor's sense); this notion of infinity is called Dedekind infinite. Cantor introduced the cardinal numbers, and showed—according to his bijection-based definition of size—that some infinite sets are greater than others. The smallest infinite cardinality is that of the natural numbers ($\backslash aleph\_0$).

Cardinality of the continuum

One of Cantor's most important results was that thecardinality 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 ...

($\backslash mathfrak$) is greater than that of the natural numbers ($\backslash aleph\_0$); that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that $\backslash mathfrak\; =\; 2^\; =\; \backslash beth\_1$ (see Beth one) satisfies:
:$2^\; >\; \backslash aleph\_0$
:(see 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 ...

or Cantor's first uncountability proof
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 ...

).
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 ...

states that there is no cardinal 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 ...

between the cardinality of the reals and the cardinality of the natural numbers, that is,
:$2^\; =\; \backslash aleph\_1$
However, this hypothesis can neither be proved nor disproved within the widely accepted 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 ...

, if ZFC is consistent.
Cardinal arithmetic can be used to show not only that the number of points in a real number line
Real may refer to:
Currencies
* Brazilian real
The Brazilian real ( pt, real, plural, pl. '; currency symbol, sign: R$; ISO 4217, code: BRL) is the official currency of Brazil. It is subdivided into 100 centavos. The Central Bank of Brazil ...

is equal to the number of points in any segment
Segment or segmentation may refer to:
Biology
*Segmentation (biology), the division of body plans into a series of repetitive segments
**Segmentation in the human nervous system
*Internodal segment, the portion of a nerve fiber between two Nodes of ...

of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space. These results are highly counterintuitive, because they imply that there exist 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 ...

s and proper supersets of an infinite set ''S'' that have the same size as ''S'', although ''S'' contains elements that do not belong to its subsets, and the supersets of ''S'' contain elements that are not included in it.
The first of these results is apparent by considering, for instance, the tangent function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

, which provides a one-to-one correspondence
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 ...

between the interval (−½π, ½π) and R (see also Hilbert's paradox of the Grand Hotel).
The second result was first demonstrated by Cantor in 1878, but it became more apparent in 1890, when Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

introduced the space-filling curve
In 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 (mathemat ...

s, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

, or finite-dimensional space. These curves are not a direct proof that a line has the same number of points as a finite-dimensional space, but they can be used to obtain such a proof.
Cantor also showed that sets with cardinality strictly greater than $\backslash mathfrak\; c$ exist (see his generalized diagonal argument and theorem
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 ge ...

). They include, for instance:
:* the set of all subsets of R, i.e., 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 R, written ''P''(R) or 2cardinal arithmetic
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 ...

:
:$\backslash mathfrak^2\; =\; \backslash left(2^\backslash right)^2\; =\; 2^\; =\; 2^\; =\; \backslash mathfrak,$
:$\backslash mathfrak\; c^\; =\; \backslash left(2^\backslash right)^\; =\; 2^\; =\; 2^\; =\; \backslash mathfrak,$
:$\backslash mathfrak\; c\; ^\; =\; \backslash left(2^\backslash right)^\; =\; 2^\; =\; 2^.$
Examples and properties

* If ''X'' = and ''Y'' = , then , ''X'' , = , ''Y'' , because is a bijection between the sets ''X'' and ''Y''. The cardinality of each of ''X'' and ''Y'' is 3. * If , ''X'' , ≤ , ''Y'' , , then there exists ''Z'' such that , ''X'' , = , ''Z'' , and ''Z'' ⊆ ''Y''. *If , ''X'' , ≤ , ''Y'' , and , ''Y'' , ≤ , ''X'' , , then , ''X'' , = , ''Y'' , . This holds even for infinite cardinals, and is known as Cantor–Bernstein–Schroeder theorem. * Sets with cardinality of the continuum include the set of all real numbers, the set of allirrational 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 ge ...

s and the interval $$, 1
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...

/math>.
Union and intersection

If ''A'' and ''B'' aredisjoint sets
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 ...

, then
:$\backslash left\backslash vert\; A\; \backslash cup\; B\; \backslash right\backslash vert\; =\; \backslash left\backslash vert\; A\; \backslash right\backslash vert\; +\; \backslash left\backslash vert\; B\; \backslash right\backslash vert.$
From this, one can show that in general, the cardinalities of unions and intersections are related by the following equation:Applied Abstract Algebra, K.H. Kim, F.W. Roush, Ellis Horwood Series, 1983, (student edition), (library edition)
:$\backslash left\backslash vert\; C\; \backslash cup\; D\; \backslash right\backslash vert\; +\; \backslash left\backslash vert\; C\; \backslash cap\; D\; \backslash right\backslash vert\; =\; \backslash left\backslash vert\; C\; \backslash right\backslash vert\; +\; \backslash left\backslash vert\; D\; \backslash right\backslash vert.$
See also

*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). ...

* Cantor's paradox
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 ...

* Cantor's theorem
In elementary 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. ...

* Countable 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 ...

* 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 ...

* Ordinality
* Pigeonhole principle
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 ...

References

{{Authority control Basic concepts in infinite set theory