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Encyclopedia > Cardinality of the continuum

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 mathfrak c, Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Please refer to Real vs. ... In mathematics, the word continuum sometimes denotes the real line. ...

mathfrak c = |mathbb R|

Contents

Properties

Uncountability

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, aleph_0 (aleph-null): Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... 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. ...

aleph_0 < mathfrak c

In other words, there are strictly more real numbers than there are integers. Cantor proved this statement in several different ways. See Cantor's first uncountability proof and Cantor's diagonal argument. Contrary to what most mathematicians believe, Georg Cantors first proof that the set of all real numbers is uncountable was not his famous diagonal argument, and did not mention decimal expansions or any other numeral system. ... Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ...


Cardinal equalities

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: In Zermelo-Fränkel set theory, Cantors theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantors theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

  1. Define a map f : RP(Q) from the reals to the power set of the rationals by sending each real number x to the set {q in mathbb{Q} | q le x} 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 the set of sets of rationals). This map is injective since the rationals are dense in R. Since the rationals are countable we have that mathfrak c le 2^{aleph_0}.
  2. Let {0,2}N be the set of infinite sequences with values in set {0,2}. This set clearly has cardinality 2^{aleph_0} (the natural bijection between the set of binary sequences and P(N) is given by the indicator function). Now associate to each such sequence (ai) the unique real number in the interval [0,1] with the ternary-expansion given by the digits (ai), i.e. the i-th digit after the decimal point is ai. 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 2^{aleph_0} le mathfrak c.

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, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards... In mathematics, inclusion is a partial order on sets. ... 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 mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... A bijective function. ... In the mathematical subfield of set theory, the indicator function, or characteristic 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 can mean: Ternary form, a form used for structuring music Ternary logic, a logic system with values true, false, and some other value Ternary numeral system, a base-3 counting system Ternary operation, an operation that takes three parameters Ternary plot or Ternary graph, a plot that shows the... 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...

mathfrak c = |mathcal{P}(mathbb{N})| = 2^{aleph_0}.

The sequence of beth numbers is defined by setting beth_0 = aleph_0 and beth_{k+1} = 2^{beth_k}. So c is the second beth number, beth-one In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ...

mathfrak c = beth_1

The third beth number, beth_2 = 2^mathfrak c, is the cardinality of the set of all subsets of the real line.


By using the rules of cardinal arithmetic one can show that Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ...

mathfrak c = n mathfrak c = aleph_0 mathfrak c = mathfrak c^n = n^{aleph_0} = {aleph_0}^{aleph_0} = mathfrak c^{aleph_0}

where n is any finite cardinal ≥ 2.


The continuum hypothesis

The famous continuum hypothesis asserts that c is also the first aleph number aleph_1. In other words, the continuum hypothesis states that there is no set A whose cardinality lies strictly between aleph_0 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, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ...

notexists A : aleph_0 < |A| < mathfrak c

However, this statement is now known to be independent of the axioms of Zermelo-Fraenkel set theory (ZFC). That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero natural number n, the equality c = aleph_n is independent of ZFC. (The case n = 1 is 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, e.g., mathfrak{c}neqaleph_omega. In particular, mathfrak{c} could be either aleph_1 or aleph_{omega_1}, where ω1 is the first uncountable ordinal, so it could be either a successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal. Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In set theory, Königs theorem (named after the Hungarian mathematician Julius König) colloquially states that if the axiom of choice holds and if I is a set and mi and ni are cardinal numbers for every i in I, and then The sum here is the cardinality... In mathematics, especially in order theory, a subset B of a partially ordered set A is cofinal if for every a in A there is a b in B such that a &#8804; b. ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In the theory of cardinal numbers, we can define a successor operation similar to that in the ordinal numbers. ... In mathematics, limit cardinals are a type of cardinal number. ... In set theory, a regular cardinal is an infinite well-orderable cardinal whose initial ordinal is regular, where a regular ordinal is an ordinal which is equal to its own cofinality. ... In set theory, a regular cardinal is an infinite well-orderable cardinal whose initial ordinal is regular, where a regular ordinal is an ordinal which is equal to its own cofinality. ...


Sets with cardinality c

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

In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... 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. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... This is a page about mathematics. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... 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...

References

  • Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
  • Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
  • Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9.

This article incorporates material from cardinality of the continuum on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...


  Results from FactBites:
 
PlanetMath: cardinality of the continuum (0 words)
The cardinality of the continuum, often denoted by
"cardinality of the continuum" is owned by yark.
This is version 12 of cardinality of the continuum, born on 2004-03-15, modified 2006-12-30.
Infinite Ink: The Continuum Hypothesis FAQ (0 words)
The continuum hypothesis was proposed by Georg Cantor in 1877 after he showed that the real numbers cannot be put into one-to-one correspondence with the natural numbers.
He used the Hebrew letter aleph to name the different levels of infinity: aleph_0 is the number of (or cardinality of) the natural numbers or any countably infinite set, and the next levels of infinity are aleph_1, aleph_2, aleph_3, et cetera.
The pursuit of the continuum problem has motivated a lot of work in set theory and in mathematics in general.
  More results at FactBites »

 
 

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