In

^{R} of

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

, the cardinality of the continuum is the cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

or "size" of the set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

of real numbers
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 re ...

$\backslash mathbb\; R$, sometimes called the continuum. It is an infinite cardinal number
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. The ...

and is denoted by $\backslash mathfrak\; c$ (lowercase fraktur "c") or $,\; \backslash mathbb\; R,$.
The real numbers $\backslash mathbb\; R$ are more numerous than the natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...

$\backslash mathbb\; N$. Moreover, $\backslash mathbb\; R$ has the same number of elements as 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 $\backslash mathbb\; N.$ Symbolically, if the cardinality of $\backslash mathbb\; N$ is denoted as $\backslash aleph\_0$, the cardinality of the continuum is
This was proven by Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...

in his uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his diagonal argument in 1891. Cantor defined cardinality in terms of bijective function
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 s ...

s: two sets have the same cardinality if, and only if, there exists a bijective function between them.
Between any two real numbers ''a'' < ''b'', no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. In other words, the open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...

(''a'',''b'') is equinumerous with $\backslash mathbb\; R.$ This is also true for several other infinite sets, such as any ''n''-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

$\backslash mathbb\; R^n$ (see space filling curve). That is,
The smallest infinite cardinal number is $\backslash aleph\_0$ ( aleph-null). The second smallest is $\backslash aleph\_1$ ( aleph-one). The continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

, which asserts that there are no sets whose cardinality is strictly between $\backslash aleph\_0$ and means that $\backslash mathfrak\; c\; =\; \backslash aleph\_1$. The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC).
Properties

Uncountability

Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...

introduced the concept of cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

to compare the sizes of infinite sets. He famously showed that the set of real numbers is uncountably infinite. That is, $$ is strictly greater than the cardinality of the natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...

, $\backslash aleph\_0$:
In practice, this means that there are strictly more real numbers than there are integers. Cantor proved this statement in several different ways. For more information on this topic, see Cantor's first uncountability proof and Cantor's diagonal argument
In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a m ...

.
Cardinal equalities

A variation of Cantor's diagonal argument can be used to prove Cantor's theorem, which states that the cardinality of any set is strictly less than that of itspower 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 ...

. That is, $,\; A,\; <\; 2^$ (and so that the power set $\backslash wp(\backslash mathbb\; N)$ 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 $\backslash mathbb\; N$ is uncountable). In fact, one can show that the cardinality of $\backslash wp(\backslash mathbb\; N)$ is equal to $$ as follows:
#Define a map $f:\backslash mathbb\; R\backslash to\backslash wp(\backslash mathbb\; Q)$ from the reals to the power set of the rationals
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...

, $\backslash mathbb\; Q$, by sending each real number $x$ to the set $\backslash $ of all rationals less than or equal to $x$ (with the reals viewed as Dedekind cuts, this is nothing other than the inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iota( ...

in the set of sets of rationals). Because the rationals are dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

in $\backslash mathbb$, this map is injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...

, and because the rationals are countable, we have that $\backslash mathfrak\; c\; \backslash le\; 2^$.
#Let $\backslash ^$ be the set of infinite sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...

s with values in set $\backslash $. This set has cardinality $2^$ (the natural bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

between the set of binary sequences and $\backslash wp(\backslash mathbb\; N)$ is given by the indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...

). Now, associate to each such sequence $(a\_i)\_$ the unique real number in the interval $;\; href="/html/ALL/s/,1.html"\; ;"title=",1">,1$Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
Th ...

. It is not hard to see that this map is injective, for by avoiding points with the digit 1 in their ternary expansion, we avoid conflicts created by the fact that the ternary-expansion of a real number is not unique. We then have that $2^\; \backslash le\; \backslash mathfrak\; c$.
By the Cantor–Bernstein–Schroeder theorem we conclude that
The cardinal equality $\backslash mathfrak^2\; =\; \backslash mathfrak$ can be demonstrated using cardinal arithmetic:
By using the rules of cardinal arithmetic, one can also show that
where ''n'' is any finite cardinal ≥ 2, and
where $2\; ^$ is the cardinality of the power set of R, and $2\; ^\; >\; \backslash mathfrak\; c$.
Alternative explanation for 𝔠 = 2^{ℵ0}

decimal expansion
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
r = b_k b_\ldots b_0.a_1a_2\ldots
Here is the decimal separator, is ...

. For example,
(This is true even in the case the expansion repeats, as in the first two examples.)
In any given case, the number of digits is countable
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; ...

since they can be put into 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 ...

with the set of natural numbers $\backslash mathbb$. This makes it sensible to talk about, say, the first, the one-hundredth, or the millionth digit of π. Since the natural numbers have cardinality $\backslash aleph\_0,$ each real number has $\backslash aleph\_0$ digits in its expansion.
Since each real number can be broken into an integer part and a decimal fraction, we get:
where we used the fact that
On the other hand, if we map $2\; =\; \backslash $ to $\backslash $ and consider that decimal fractions containing only 3 or 7 are only a part of the real numbers, then we get
and thus
Beth numbers

The sequence of beth numbers is defined by setting $\backslash beth\_0\; =\; \backslash aleph\_0$ and $\backslash beth\_\; =\; 2^$. So $$ is the second beth number, beth-one: The third beth number, beth-two, is the cardinality of the power set of $\backslash mathbb$ (i.e. the set of all subsets of the real line):The continuum hypothesis

The famous continuum hypothesis asserts that $$ is also the secondaleph 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 ...

, $\backslash aleph\_1$. In other words, the continuum hypothesis states that there is no set $A$ whose cardinality lies strictly between $\backslash aleph\_0$ and $$
This statement is now known to be independent of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), as shown by Kurt Gödel
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an im ...

and Paul Cohen. That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero 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 ...

''n'', the equality $$ = $\backslash aleph\_n$ is independent of ZFC (case $n=1$ being the continuum hypothesis). The same is true for most other alephs, although in some cases, equality can be ruled out by König's theorem on the grounds of cofinality
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''.
This definition of cofinality relies on the axiom of choice, as it uses th ...

(e.g., $\backslash mathfrak\backslash neq\backslash aleph\_\backslash omega$). In particular, $\backslash mathfrak$ could be either $\backslash aleph\_1$ or $\backslash aleph\_$, where $\backslash omega\_1$ is the first uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by \omega_1 or sometimes by \Omega, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. W ...

, so it could be either a successor cardinal In set theory, one can define a successor operation on cardinal numbers in a similar way to the successor operation on the ordinal numbers. The cardinal successor coincides with the ordinal successor for finite cardinals, but in the infinite case th ...

or a limit cardinal
In mathematics, limit cardinals are certain cardinal numbers. A cardinal number ''λ'' is a weak limit cardinal if ''λ'' is neither a successor cardinal nor zero. This means that one cannot "reach" ''λ'' from another cardinal by repeated succes ...

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

or a singular cardinal.
Sets with cardinality of the continuum

A great many sets studied in mathematics have cardinality equal to $$. Some common examples are the following:Sets with greater cardinality

Sets with cardinality greater than $$ include: *the set of all subsets of $\backslash mathbb$ (i.e., power set $\backslash mathcal(\backslash mathbb)$) *the set 2indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...

s defined on subsets of the reals (the set $2^$ is isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

to $\backslash mathcal(\backslash mathbb)$ – the indicator function chooses elements of each subset to include)
*the set $\backslash mathbb^\backslash mathbb$ of all functions from $\backslash mathbb$ to $\backslash mathbb$
*the Lebesgue σ-algebra of $\backslash mathbb$, i.e., the set of all Lebesgue measurable sets in $\backslash mathbb$.
*the set of all Lebesgue-integrable functions from $\backslash mathbb$ to $\backslash mathbb$
*the set of all Lebesgue-measurable functions from $\backslash mathbb$ to $\backslash mathbb$
*the Stone–Čech compactification In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Ston ...

s of $\backslash mathbb$, $\backslash mathbb$ and $\backslash mathbb$
*the set of all automorphisms of the (discrete) field of complex numbers.
These all have cardinality $2^\backslash mathfrak\; c\; =\; \backslash beth\_2$ (beth 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 ...

).
References

Bibliography

*Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operato ...

, ''Naive set theory''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition).
* Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. .
* Kunen, Kenneth, 1980. '' Set Theory: An Introduction to Independence Proofs''. Elsevier. .
{{PlanetMath attribution, urlname=CardinalityOfTheContinuum, title=cardinality of the continuum
Cardinal numbers
Set theory
Infinity