# continuum hypothesis

continuum hypothesis
a conjecture of set theory that the first infinite cardinal number greater than the cardinal number of the set of all positive integers is the cardinal number of the set of all real numbers.
[1935-40]

* * *

statement of set theory that the set of real numbers (the continuum) is in a sense as small as it can be. In 1873 the German mathematician Georg Cantor (Cantor, Georg) proved that the continuum is uncountable—that is, the real numbers are a larger infinity than the counting numbers—a key result in starting set theory as a mathematical subject. Furthermore, Cantor developed a way of classifying the size of infinite sets according to the number of its elements, or its cardinality. (See set theory: Cardinality and transfinite numbers (set theory).) In these terms, the continuum hypothesis can be stated as follows: The cardinality of the continuum is the smallest uncountable cardinal number.

In Cantor's notation, the continuum hypothesis can be stated by the simple equation 2ℵ0 = ℵ1, where ℵ0 is the cardinal number of an infinite countable set (such as the set of natural numbers), and the cardinal numbers of larger “well-orderable sets” are ℵ1, ℵ2, … , ℵα, … , indexed by the ordinal numbers. The cardinality of the continuum can be shown to equal 2ℵ0; thus, the continuum hypothesis rules out the existence of a set of size intermediate between the natural numbers and the continuum.

A stronger statement is the generalized continuum hypothesis (GCH): 2ℵα = ℵα + 1 for each ordinal number α. The Polish mathematician Wacław Sierpiński (Sierpiński, Wacław) proved that with GCH one can derive the axiom of choice.

As with the axiom of choice, the Austrian-born American mathematician Kurt Gödel (Gödel, Kurt) proved in 1939 that, if the other standard Zermelo-Fraenkel axioms (ZF; see the table—>) are consistent, then they do not disprove the continuum hypothesis or even GCH. That is, the result of adding GCH to the other axioms remains consistent. Then in 1963 the American mathematician Paul Cohen (Cohen, Paul Joseph) completed the picture by showing, again under the assumption that ZF is consistent, that ZF does not yield a proof of the continuum hypothesis.

Since ZF neither proves nor disproves the continuum hypothesis, there remains the question of whether to accept the continuum hypothesis based on an informal concept of what sets are. The general answer in the mathematical community has been negative: the continuum hypothesis is a limiting statement in a context where there is no known reason to impose a limit. In set theory, the power-set operation assigns to each set of cardinality ℵα its set of all subsets, which has cardinality 2ℵα. There seems to be no reason to impose a limit on the variety of subsets that an infinite set might have.

Herbert Enderton

* * *

Universalium. 2010.

### Look at other dictionaries:

• continuum hypothesis — The hypothesis proposed by Cantor that there is no set with a cardinality greater than that of the natural numbers but less than the cardinality of the set of all subsets of the set of natural numbers (the power set of that set). The generalized… …   Philosophy dictionary

• continuum hypothesis — noun The hypothesis which states that any infinite subset of must have the cardinality of either the set of natural numbers or of itself …   Wiktionary

• continuum hypothesis — Math. a conjecture of set theory that the first infinite cardinal number greater than the cardinal number of the set of all positive integers is the cardinal number of the set of all real numbers. [1935 40] …   Useful english dictionary

• The Continuum Hypothesis (album) — Infobox Album Name = The Contimuum Hypothesis Type = studio Artist = Epoch of Unlight Released = Start date|2005 Recorded = October 27 November 4, 2004 Genre = Melodic death/Black metal Length = 53:11 Label = The End Records Producer = Reviews =… …   Wikipedia

• Continuum — may refer to: Continuum (theory), anything that goes through a gradual transition from one condition, to a different condition, without any abrupt changes Contents 1 Linguistics 2 Mathematics …   Wikipedia

• Continuum mechanics — Continuum mechanics …   Wikipedia

• generalized continuum hypothesis — noun The hypothesis that, for each ordinal <math> alpha</math>, there is no cardinal number strictly between <math> aleph alpha</math> and <math style= vertical align: 0%; >2^ aleph alpha</math>, i.e. <math… …   Wiktionary

• Continuum (mathematics) — In mathematics, the word continuum has at least two distinct meanings, outlined in the sections below. For other uses see Continuum.Ordered setThe term the continuum sometimes denotes the real line. Somewhat more generally a continuum is a… …   Wikipedia

• Continuum (set theory) — In the mathematical field of set theory, the continuum means the real numbers, or the corresponding cardinal number, . The cardinality of the continuum is the size of the real numbers. The continuum hypothesis is sometimes stated by saying that… …   Wikipedia