1. Continuum Hypothesis - Wikipedia, The Free Encyclopedia In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor in 1877, about the possible sizes of infinite sets. http://en.wikipedia.org/wiki/Continuum_hypothesis

Continuum hypothesis From Wikipedia, the free encyclopedia Jump to: navigation search This article is about a hypothesis in set theory. For the assumption in fluid mechanics, see fluid mechanics In mathematics , the continuum hypothesis (abbreviated CH ) is a hypothesis , advanced by Georg Cantor in 1877, about the possible sizes of infinite sets . It states: There is no set whose cardinality is strictly between that of the integers and that of the real numbers. Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's twenty-three problems presented in the year 1900. The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo–Fraenkel set theory , the standard foundation of modern mathematics, provided ZF set theory is consistent The name of the hypothesis comes from the term the continuum for the real numbers. Contents Cardinality of infinite sets Impossibility of proof and disproof in ZFC Arguments for and against CH The generalized continuum hypothesis ... edit Cardinality of infinite sets Main article: Cardinal number Two sets are said to have the same cardinality or cardinal number if there exists a bijection (a one-to-one correspondence) between them. Intuitively, for two sets

2. Continuum Hypothesis - Encyclopedia Article - Citizendium Jul 16, 2009 In mathematics, the continuum hypothesis is the statement that any arbitrary infinite set of real numbers has either as many elements as http://en.citizendium.org/wiki/Continuum_hypothesis

Continuum hypothesis From Citizendium, the Citizens' Compendium Jump to: navigation search addthis_pub = 'citizendium'; addthis_logo = ''; addthis_logo_color = ''; addthis_logo_background = ''; addthis_brand = 'Citizendium'; addthis_options = ''; addthis_offset_top = ''; addthis_offset_left = ''; Main Article Talk Related Articles Bibliography External Links This is a draft article , under development and not meant to be cited; you can help to improve it. These unapproved articles are subject to edit intro In mathematics , the continuum hypothesis is the statement that any arbitrary infinite set of real numbers has either as many elements as there are real numbers or only as many elements as there are natural numbers (i.e., there is no intermediate size). This is equivalent to the statement that there are as many real numbers as there are elements in the smallest set which is larger than the set of natural numbers. Since the set of real numbers (or the real line) is also called the continuum this can be shortly expressed as: Any set of real numbers is either countable or equivalent to the continuum.

3. Continuum Hypothesis In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to http://www.fact-index.com/c/co/continuum_hypothesis.html

Main Page See live article Alphabetical index Continuum hypothesis In mathematics , the continuum hypothesis is a hypothesis about the possible sizes of infinite sets Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers (naively: whole numbers) is strictly smaller than the set of real numbers (naively: infinite decimals) The continuum hypothesis states the following: There is no set whose size is strictly between that of the integers and that of the real numbers. Or mathematically speaking, noting that the cardinality for the integers is (" aleph-null ") and the cardinality for the real numbers is , the continuum hypothesis says: The real numbers have also been called the continuum , hence the name. There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis , which is described at the end of this article. Table of contents 1 Investigating the continuum hypothesis 2 Impossibility of proof and disproof 3 The generalized continuum hypothesis 4 See also ... 5 References Investigating the continuum hypothesis Consider the set of all rational numbers . One might naively suppose that there are more rational numbers than integers, and fewer rational numbers than real numbers, thus disproving the continuum hypothesis. However, it turns out that the rational numbers can be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size as the set of integers.

4. Continuum Hypothesis - Simple English Wikipedia, The Free Encyclopedia The continuum hypothesis is a hypothesis that there is no set that is both bigger than that of the natural numbers and smaller than that of the real numbers http://simple.wikipedia.org/wiki/Continuum_hypothesis

Continuum hypothesis From Wikipedia, the free encyclopedia Jump to: navigation search The continuum hypothesis is a hypothesis that there is no set that is both bigger than that of the natural numbers and smaller than that of the real numbers Georg Cantor stated this hypothesis in 1877. There are infinitely many natural numbers, the cardinality of the set of natural numbers is infinite. This is also true for the set of real numbers, but there are more real numbers than natural numbers. We say that the natural numbers have infinite cardinality and the real numbers have infinite cardinality, but the cardinality of the real numbers is greater than the cardinality of the natural numbers. This hypothesis is the first problem on the list of 23 problems David Hilbert published in 1900. Kurt Gödel showed in 1939, that the hypothesis cannot be falsified using Zermelo–Fraenkel set theory . The Zermelo–Fraenkel set theory is the set theory commonly used in mathematics. Paul Cohen showed in the 1960s that the Zermelo-Fraenkel set theory cannot be use to prove the continuum hypothesis, either. For this, Cohen was awarded the Fields medal Retrieved from " http://simple.wikipedia.org/wiki/Continuum_hypothesis

5. Continuum Hypothesis Articles And Information In mathematics, the information continuum hypothesis is information hypothesis about the information possible sizes consistent with infinite sets. http://neohumanism.org/c/co/continuum_hypothesis.html

Home Archaeology Astronomy Biology ... Contact Us Advert: Buy Gold Current Article Continuum hypothesis In mathematics , the continuum hypothesis is a hypothesis about the possible sizes of infinite sets Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers (naively: whole numbers) is strictly smaller than the set of real numbers (naively: infinite decimals) The continuum hypothesis states the following: There is no set whose size is strictly between that of the integers and that of the real numbers. Or mathematically speaking, noting that the cardinality for the integers is (" aleph-null ") and the cardinality for the real numbers is , the continuum hypothesis says: The real numbers have also been called the continuum , hence the name. There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis , which is described at the end of this article. Table of contents showTocToggle("show","hide") 1 Investigating the continuum hypothesis 2 Impossibility of proof and disproof 3 The generalized continuum hypothesis 4 See also ... 5 References Investigating the continuum hypothesis Consider the set of all rational numbers . One might naively suppose that there are more rational numbers than integers, and fewer rational numbers than real numbers, thus disproving the continuum hypothesis. However, it turns out that the rational numbers can be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size as the set of integers.

6. Continuum Hypothesis The continuum hypothesis (CH), put forward by Cantor in 1877, says that the number of real numbers is the next level of infinity above countable infinity. http://www.daviddarling.info/encyclopedia/C/continuum_hypothesis.html

MATHEMATICS A B C ... CONTACT entire Web this site continuum hypothesis In 1874 Georg Cantor discovered that there is more than one level of infinity . The lowest level is called countable infinity ; higher levels are known as uncountable infinities . The natural numbers are an example of a countably infinite set and the real numbers are an example of an uncountably infinite set. The continuum hypothesis (CH), put forward by Cantor in 1877, says that the number of real numbers is the next level of infinity above countable infinity. It is called the continuum hypothesis because the real numbers are used to represent a linear continuum. Let c be the cardinality of (i.e., number of points in) a continuum, aleph -null, be the cardinality of any countably infinite set, and aleph-one be the next level of infinity above aleph-null. CH is equivalent to saying that there is no cardinal number between aleph-null and c , and that c = aleph-one. CH has been, and continues to be, one of the most hotly pursued problems in mathematics.

7. Continuum_hypothesis Encyclopedia Topics | Reference.com Copy paste this link to your blog or website to reference this page http://www.reference.com/browse/Continuum_hypothesis

8. Wolfram|Alpha: Continuum Hypothesis : Statement, Proof, Status, ... Complete information and computations for continuum hypothesis basic properties , history, http://www.wolframalpha.com/entities/famous_math_problems/continuum_hypothesis/g

document.write(''); Home Examples About FAQS ... Media Resources A Wolfram Web Resource Wolfram Research Wolfram Mathematica Wolfram Demonstrations ... Wolfram Tones Enter something to compute or figure out Calculate continuum hypothesis: statement, proof, status, awards, formulation, ... Input interpretation: Statement: Solution: Alternate names: History: More Current evidence: Associated prizes: Computed by Wolfram Mathematica Download as: PDF Live Mathematica A Wolfram Research Company Participate Entity Index continuum hypothesis: statement, proof, status, awards, formulation, ... Input: continuum hypothesis continuum hypothesis Statement There is no infinite set with a cardinal number between that of the (open curly double quote)small(close curly double quote) infinite set of integers and the (open curly double quote)large(close curly double quote) infinite set of real numbers. Solution undecidable Alternate names Cantor(close curly quote)s problem of the cardinal number of the continuum Hilbert(close curly quote)s first problem History Current evidence Proved by Go"del and Cohen to be undecidable within Zermelo-Frankel set theory with or without the axiom of choice, but there is no consensus on whether this is a solution to the problem.

9. The Continuum Hypothesis: A Mystery Of Mathematics? The continuum problem asks for a solution of the continuum hypothesis (CH), and is the first in Links http//en.wikipedia.org/wiki/continuum_hypothesis http://ercim-news.ercim.eu/the-continuum-hypothesis-a-mystery-of-mathematics

search... Subscribe/order back issues Back issues online About ERCIM News ... ERCIM web site ERCIM News 73 April 2008 Special theme: Maths for Everyday Life This issue in pdf (60 pages; 11.8 Mb) Next issue October 2010 Next special theme: Cloud Computing Platforms, Software and Applications Call for the next issue Get the latest issue to your desktop The Continuum Hypothesis: A Mystery of Mathematics? The concept of infinity has been a subject of speculation throughout history. In the nineteenth century, Georg Cantor produced an elegant theory in which the notion of infinity could be treated mathematically: set theory. The concepts investigated in set theory are the building blocks on which the whole of mathematics can be built. In set theory the size of an infinite set can be measured by its cardinality. Many problems regarding cardinal numbers that have a simple formulation are unsolved even after years of effort by outstanding mathematicians. The continuum problem asks for a solution of the continuum hypothesis (CH), and is the first in Hilbert's celebrated list of 23 problems. This problem has an ambivalent nature: it is comprehensible to a vast community of scientists, but the effort devoted to solving it shows that even in mathematics there are deceptively simple questions which may not have an answer. More precisely, the existence of 'solutions' depends on the philosophical attitude of the person seeking them: does every precisely stated mathematical question have a solution? Most Platonists believe so, and a large number of them believe that the latest research in set theory is leading to a satisfactory solution of the continuum problem in particular. A skeptical mathematician, on the other hand, is apt to believe that this problem will never be settled.

10. Continuum Hypothesis - Wiktionary Jul 15, 2010 continuum hypothesis. Definition from Wiktionary, the free Retrieved from http//en.wiktionary.org/wiki/continuum_hypothesis http://en.wiktionary.org/wiki/continuum_hypothesis

continuum hypothesis Definition from Wiktionary, the free dictionary Jump to: navigation search Contents English Noun Translations Abbreviations ... edit English Wikipedia has an article on: Continuum hypothesis Wikipedia edit Noun continuum hypothesis uncountable set theory The hypothesis which states that any infinite subset of must have the cardinality of either the set of natural numbers or of itself. edit Translations Hungarian: kontinuumhipotézis hu (hu) edit Abbreviations CH edit Derived terms generalized continuum hypothesis edit Translations Dutch: continuümhypothese f Hungarian: kontinuumhipotézis hu (hu) Retrieved from " http://en.wiktionary.org/wiki/continuum_hypothesis Categories English nouns Set theory Personal tools New features Log in / create account Namespaces Entry Discussion Variants Views Read Edit History Actions Search Navigation Main Page Community portal Preferences Requested entries ... Contact us Toolbox What links here Related changes Upload file Special pages ... Permanent link In other languages This page was last modified on 19 October 2010, at 04:08. Text is available under the Creative Commons Attribution/Share-Alike License ; additional terms may apply. See

11. Continuum Hypothesis: Facts, Discussion Forum, And Encyclopedia Article Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from http://www.absoluteastronomy.com/topics/Continuum_hypothesis

Home Discussion Topics Dictionary ... Login Continuum hypothesis Continuum hypothesis Topic Home Discussion Discussion Ask a question about ' Continuum hypothesis Start a new discussion about ' Continuum hypothesis Answer questions from other users Full Discussion Forum Encyclopedia In mathematics Mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.... , the continuum hypothesis (abbreviated CH ) is a hypothesis Hypothesis A hypothesis is a proposed explanation for an observable phenomenon. The term derives from the Greek, ὑποτιθέναι – hypotithenai meaning "to put under" or "to suppose." For a hypothesis to be put forward as a scientific hypothesis, the scientific method requires that one can test it... , advanced by Georg Cantor Georg Cantor Georg Ferdinand Ludwig Philipp Cantor was a mathematician, best known as the creator of set theory, which has become a fundamental theory in mathematics... in 1877, about the possible sizes of infinite sets. It states:

12. Kids.Net.Au - Encyclopedia > Continuum Hypothesis Kids.Net.Au is a search engine / portal for kids, children, parents, and teachers. The site offers a directory of child / kids safe websites, encyclopedia, dictionary, thesaurus http://encyclopedia.kids.net.au/page/co/Continuum_hypothesis

Search the Internet with Kids.Net.Au Encyclopedia > Continuum hypothesis Article Content Continuum hypothesis In mathematics , the continuum hypothesis is a hypothesis about the possible sizes of infinite sets Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers (naively: whole numbers) is strictly smaller than the set of real numbers (naively: infinite decimals) The continuum hypothesis states the following: There is no set whose size is strictly between that of the integers and that of the real numbers. Or mathematically speaking, noting that the cardinality The real numbers have also been called the continuum , hence the name. Consider the set of all rational numbers . One might naively suppose that there are more rational numbers than integers, and fewer rational numbers than real numbers, thus disproving the continuum hypothesis. However, it turns out that the rational numbers can be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size as the set of integers. If a set S was found that disproved the continuum hypothesis, it would be impossible to make a one-to-one correspondence between

13. Glossary: Continuum Hypothesis The Continuum hypothesis states that there is no no set whose cardinality lies between that of the Natural numbers and that of the Reals. http://www-history.mcs.st-and.ac.uk/Glossary/continuum_hypothesis.html

Continuum Hypothesis The Continuum hypothesis states that there is no no set whose cardinality lies between that of the Natural numbers and that of the Reals. The Generalised Continuum hypothesis states that if A is any set, there is no set whose cardinality lies between the cardinality of A and the cardinality of the set of all subsets of A The Continuum hypothesis has been shown to be independent of the other set-theory axioms.

14. Continuum Hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor in 1877, about the possible sizes of infinite sets. http://www.india-karnataka.info/Continuum_hypothesis

15. INDEPENDENCE OF THE CONTINUUM HYPOTHESIS File Format PDF/Adobe Acrobat Quick View http://www.math.mcmaster.ca/~luthermm/papers/Continuum_Hypothesis.pdf

16. Index Of /~gaillard/DIVERS/Continuum_Hypothesis Translate this page continuum-hypothesis.100424.dvi, 24-Apr-2010 1129, 20K. , continuum- hypothesis.100424.pdf, 24-Apr-2010 1129, 30K. , continuum-hypothesis.100424. ps http://www.iecn.u-nancy.fr/~gaillard/DIVERS/Continuum_Hypothesis/

Index of /~gaillard/DIVERS/Continuum_Hypothesis Name Last modified Size Description ... continuum-hypothesis.100616.pdf 16-Jun-2010 05:41 continuum-hypothesis.100616.tex 16-Jun-2010 05:41 Apache/2.2.9 (Debian) PHP/5.2.6-1+lenny4 with Suhosin-Patch mod_ssl/2.2.9 OpenSSL/0.9.8g Server at www.iecn.u-nancy.fr Port 80

17. The Continuum Hypothesis Solved: All Infinities Are The Same? Nope. : Good Math, Of all of the work in the history of mathematics, nothing seems to attract so much controversy, or even outright hatred as Cantor's diagonalization. The idea of comparing the http://scienceblogs.com/goodmath/2009/01/the_continuum_hypothesis_solve.php

18. Pierre-Yves Gaillard’s Weblog Is the Continuum Hypothesis true, false or undecidable in Bourbaki s set theory? http//www.iecn.unancy.fr/~gaillard/DIVERS/continuum_hypothesis/ http://gaillardpy.wordpress.com/

Mathematics October 15, 2010 by gaillardpy Posted in Uncategorized Leap of Faith August 6, 2010 by gaillardpy Posted in Uncategorized Peak July 28, 2010 by gaillardpy I think that mathematics reached its peak when Cantor stated the Continuum Hypothesis problem, and started regressing when Hilbert launched his foundation program.  Posted in Uncategorized Consistency July 27, 2010 by gaillardpy how likely a given formalization of mathematics is of being consistent So the question of knowing whether such consistency assumptions are realistic is inescapable. This question can be posed in a slightly more precise form: Let T L(T) R and (not R L(T) is infinite. For all practical purposes, the fact that L(T) is very large or is infinite makes no difference. But I think that, for any reasonable mathematical theory, L(T) is much more likely to be very large than to be infinite. It would be interesting to have a lower bound for L(T) Posted in Uncategorized The mathematical question whose answer I most ardently wish I knew June 3, 2010 by gaillardpy Posted in Uncategorized June 1, 2010 by gaillardpy

19. Continuum Hypothesis: Encyclopedia - Continuum Hypothesis In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of http://www.experiencefestival.com/a/Continuum_hypothesis/id/2009945

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