In

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

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 ...

, 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, 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 set t ...

s, which allows one to distinguish between different types of infinity, and to perform 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 19t ...

on them. There are two approaches to cardinality: one which compares sets directly using bijections and injection
Injection or injected may refer to:
Science and technology
* Injective function, a mathematical function mapping distinct arguments to distinct values
* Injection (medicine), insertion of liquid into the body with a syringe
* Injection, in broadca ...

s, and another which uses cardinal numbers.
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 with various uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke (in logic), pipe, bar, or (literally the word "or"), vbar, and others.
Usage
...

on each side; this is the same notation as absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

, and the meaning depends on context
Context may refer to:
* Context (language use), the relevant constraints of the communicative situation that influence language use, language variation, and discourse summary
Computing
* Context (computing), the virtual environment required to s ...

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

A crude sense of cardinality, an awareness that groups of things or events compare with other groups by containing more, fewer, or the same number of instances, is observed in a variety of present-day animal species, suggesting an origin millions of years ago. Human expression of cardinality is seen as early as years ago, with equating the size of a group with a group of recorded notches, or a representative collection of other things, such as sticks and shells. The abstraction of cardinality as a number is evident by 3000 BCE, in Sumerianmathematics
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 ...

and the manipulation of numbers without reference to a specific group of things or events.
From the 6th century BCE, the writings of Greek philosophers show the first hints of the cardinality of infinite sets. While they considered the notion of infinity as an endless series of actions, such as adding 1 to a number repeatedly, they did not consider the size of an infinite set of numbers to be a thing. The ancient Greek notion of infinity also considered the division of things into parts repeated without limit. In Euclid's '' Elements'', commensurability was described as the ability to compare the length of two line segments, ''a'' and ''b'', as a ratio, as long as there were a third segment, no matter how small, that could be laid end-to-end a whole number of times into both ''a'' and ''b''. But with the discovery of irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

s, it was seen that even the infinite set of all rational numbers was not enough to describe the length of every possible line segment. Still, there was no concept of infinite sets as something that had cardinality.
To better understand infinite sets, a notion of cardinality was formulated circa 1880 by Georg Cantor, the originator of set theory. He examined the process of equating two sets with bijection, a one-to-one correspondence between the elements of two sets based on a unique relationship. In 1891, with the publication of Cantor's diagonal argument, he demonstrated that there are sets of numbers that cannot be placed in one-to-one correspondence with the set of natural numbers, i.e. uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...

s that contain more elements than there are in the infinite set of natural numbers.
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 a bijection (a.k.a., one-to-one correspondence) from ''A'' to ''B'', that is, a function from ''A'' to ''B'' that is bothinjective
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 ...

and surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...

. Such sets are said to be ''equipotent'', ''equipollent'', or ''equinumerous
In mathematics, two sets or classes ''A'' and ''B'' are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from ''A'' to ''B'' such that for every element ''y'' of ''B'', th ...

''. This relationship can also be denoted ''A'' ≈ ''B'' or ''A'' ~ ''B''.
:For example, the set ''E'' = of non-negative even number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because
\begin
-2 \cdot 2 &= -4 \\
0 \cdot 2 &= 0 \\
41 ...

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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...

, since the function ''f''(''n'') = 2''n'' is a bijection from N to ''E'' (see picture).
:For finite sets ''A'' and ''B'', if ''some'' bijection exists from ''A'' to ''B'', then ''each'' injective or surjective function from ''A'' to ''B'' is a bijection. This is no longer true for infinite ''A'' and ''B''. For example, the function ''g'' from N to ''E'', defined by ''g''(''n'') = 4''n'' is injective, but not surjective, and ''h'' from N to ''E'', defined by ''h''(''n'') = ''n'' - (''n'' mod 2) is surjective, but not injective. Neither ''g'' nor ''h'' can challenge = , which was established by the existence of ''f''.
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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...

has cardinality strictly less than its 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 pos ...

''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, 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. For proofs, see Cantor's diagonal argument or Cantor's first uncountability proof.
If ≤ and ≤ , then = (a fact known as Schröder–Bernstein theorem). 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 o ...

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 called equinumerosity, and this is anequivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...

on the class of all sets. The equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...

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 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 concer ...

.
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 o ...

, the cardinalities of the 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 set t ...

s are denoted
:$\backslash aleph\_0\; <\; \backslash aleph\_1\; <\; \backslash aleph\_2\; <\; \backslash ldots\; .$
For each ordinal $\backslash alpha$, $\backslash aleph\_$ 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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...

s is denoted aleph-null ($\backslash aleph\_0$), while 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 denoted by "$\backslash mathfrak\; c$" (a lowercase fraktur script "c"), and is also referred to as the cardinality of the continuum. Cantor showed, using the 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
**Tarsk ...

, 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 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 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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection o ...

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 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 n ...

s, or , ''X'' , < , N , , is said to be 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. ...

.
*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 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.
Infinite sets

Our intuition gained fromfinite 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. ...

s breaks down 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 set t ...

s. In the late nineteenth century Georg Cantor, Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phil ...

, Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...

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 the cardinality of the continuum ($\backslash mathfrak$) 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 or Cantor's first uncountability proof). The continuum hypothesis states that there is no cardinal number 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 ZFCaxiomatic 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 concer ...

, if ZFC is consistent.
Cardinal arithmetic can be used to show not only that the number of points in a real number line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...

is equal to the number of points in any segment 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, 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 o ...

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, which provides 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 ...

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 and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...

introduced the space-filling curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, sp ...

s, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube
In geometry, a hypercube is an N-dimensional space, ''n''-dimensional analogue of a Square (geometry), square () and a cube (). It is a Closed set, closed, Compact space, compact, Convex polytope, convex figure whose 1-N-skeleton, skeleton consis ...

, 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, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...

). They include, for instance:
:* the set of all subsets of R, i.e., 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 pos ...

of R, written ''P''(R) or 2Beth two
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 ...

).
The cardinal equalities $\backslash mathfrak^2\; =\; \backslash mathfrak,$ $\backslash mathfrak\; c^\; =\; \backslash mathfrak\; c,$ and $\backslash mathfrak\; c\; ^\; =\; 2^$ can be demonstrated using cardinal arithmetic
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...

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

* If ''X'' = and ''Y'' = , where ''a'', ''b'', and ''c'' are distinct, 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, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

s and the interval $;\; href="/html/ALL/s/,\_1.html"\; ;"title=",\; 1">,\; 1$Union and intersection

If ''A'' and ''B'' are disjoint sets, 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, 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
* Cantor's paradox
* Cantor's theorem
* Countable set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...

* 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 el ...

* Ordinality
* Pigeonhole principle
In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there mu ...

References

{{Authority control Basic concepts in infinite set theory