In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). This cardinal number is often denoted by c, Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Alternative meaning: number of pitch classes in a set. ... This article is about sets in mathematics. ... Please refer to Real vs. ... In mathematics, the word continuum sometimes denotes the real line. ...

c = |R|

Properties

Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets. He famously showed that the set of real numbers is uncountably infinite; i.e. c is strictly greater than the cardinality of the natural numbers, ℵ₀ (aleph-null) Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. ... The cardinality of a set is a property that describes the size of the set by describing it using a cardinal number. ... In mathematics, an uncountable set is a set which is not countable. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. ...

In other words, there are strictly more real numbers than there are integers. Cantor proved this statement in a couple of different ways. See Cantor's first uncountability proof and Cantor's diagonal argument. Note: in order to fully understand this article you may want to refer to the set theory portion of the table of mathematical symbols. ...

A variation on 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 its power set, i.e. |A| < 2^|A|. One concludes that the power set P(N) of the natural numbers N is uncountable. It is then natural to ask whether the cardinality of P(N) is equal to c. It turns out that the answer is yes. One can prove this in two steps: Note: in order to fully understand this article you may want to refer to the set theory portion of the table of mathematical symbols. ... In mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

1. Define a map f : R → P(Q) from the reals to the power set of the rationals by sending each real number x to the set {q ∈ Q | q ≤ r} of all rationals less than or equal to x. This map is injective since the rationals are dense in R. Since the rationals are countable we have that .
2. Let {0,2}^N be the set of sequences with values in set {0,2}. This set clearly has cardinality (the natural bijection between the set of binary sequences and P(N) is given by the indicator function). Now associate to each such sequence (a_i) the unique real number in the inteval [0,1] with the ternary-expansion given by the digits (a_i), i.e. the i-th digit after the decimal point is a_i. The image of this map is called the Cantor set. 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 .

By the Cantor-Bernstein-Schroeder theorem we conclude that In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... This is a page about mathematics. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... In the mathematical subfield of set theory, the indicator function is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... Ternary is the base 3 numeral system. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In set theory, the Cantor-Bernstein-Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B. In terms of the...

The sequence of beth numbers is defined by setting and . So c is the first beth number, beth-one In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ...

The second beth number, , is the set of all subsets of the real line. In set theory and other branches of mathematics, ‭ב‬2 (pronounced beth two), or 2c (pronounced two to the power of c), is a certain cardinal number. ...

By using the rules of cardinal arithmetic one can show that

where n is any finite cardinal ≥ 2.

The continuum hypothesis

The famous continuum hypothesis asserts that c is also the first aleph number ℵ₁. In other words, the continuum hypothesis states that there is no set A whose cardinality lies strictly between ℵ₀ and c In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. ...

However, this statement is now known to be independent of the axioms of Zermelo-Fraenkel set theory. That is, both the hypothesis and its negation are consistent with these axioms. The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...

Sets with cardinality c

A great many sets studied in mathematics have cardinality equal to c. Some common examples are the following:

• the real numbers R
• any (nondegenerate) closed or open interval in R (such as the unit interval [0,1])
• the irrational numbers
• the transcendental numbers
• Euclidean space Rⁿ
• the complex numbers C
• the power set of the natural numbers (the set of all subsets of the natural numbers)
• the set of sequences of integers (i.e. all functions N → Z, often denoted Z^N)
• the set of sequences of real numbers, R^N
• the set of all continuous functions from R to R (the set of all functions R^R has cardinality 2^c)
• the Cantor set
• the Euclidean topology on Rⁿ (i.e. the set of all open sets in Rⁿ)

The text or formatting below is generated by a template which has been proposed for deletion. ... The term interval is used in the following contexts: cricket mathematics music time This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... In mathematics, a transcendental number is any irrational number that is not an algebraic number, i. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, given a set S, the power set of S, written P(S) or 2S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... This is a page about mathematics. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In set theory and other branches of mathematics, ‭ב‬2 (pronounced beth two), or 2c (pronounced two to the power of c), is a certain cardinal number. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

External links

• Cardinality of the Continuum (http://planetmath.org/?op=getobj&from=objects&id=5708) on PlanetMath.