W. Hugh Woodin | Facebook He claims that these and related mathematical results lead (intuitively) to the conclusion that Continuum Hypothesis has a a http://ar-ar.facebook.com/pages/W-Hugh-Woodin/142883772390384
David Anderson The Continuum Hypothesis 2.1 years Notes on climbing 3.8 years About me 4.3 years Miscellanea 4.4 years Books 4.5 years Who's the best athlete? 4.5 years http://continuum-hypothesis.com/
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The Continuum Hypothesis: Information From Answers.com Artist Epoch of Unlight Rating Release Date March 08, 2005 Type Lyrics are included with the album Genre Rock Review Epoch of Unlight have never been the most prolific of http://www.answers.com/topic/the-continuum-hypothesis-album
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Solutions To The Continuum Hypothesis - MathOverflow The Continuum Hypothesis (CH) posed by Cantor in 1890 asserts that $ \aleph_1=2^ {\aleph_0}$. In other words, it asserts that every subset of the set of real http://mathoverflow.net/questions/23829/solutions-to-the-continuum-hypothesis
Extractions: login faq how to ask meta ... (A related MO question: http://mathoverflow.net/questions/14338/what-is-the-general-opinion-on-the-generalized-continuum-hypothesis The Continuum Hypothesis Cohen proved that the CH is independent from the axioms of set theory. (Earlier Goedel showed that a positive answer is consistent with the axioms). Several mathematicians proposed definite answers or approaches towards such answers regarding what the answer for the CH (and GCH) should be. I am aware of the existence of 2-3 approaches. One is by Woodin described in two 2001 Notices of the AMS papers ( part 1 part 2 Another by Shelah (perhaps in this paper ). See also the paper entitled " You can enter Cantor paradize " (Offered in Haim's answer.); There is a very nice presentation by Matt Foreman discussing Woodin's approach and some other avenues. Another description of Woodin's asnwer
Continuum Hypothesis The Continuum Hypothesis A Mystery of Mathematics? The proposal originally made by Georg Cantor that there is no infinite set with a cardinal number between that of the small http://www.continuumhypothesis.net/
Extractions: Home WebBoard AC_FL_RunContent( 'codebase','http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=9,0,28,0','width','700','height','178','src','/images_profiles/Misc30','quality','high','pluginspage','http://www.adobe.com/shockwave/download/download.cgi?P1_Prod_Version=ShockwaveFlash','movie','/images_profiles/Misc30','bgcolor','#FFFFFF','wmode','transparent' ); The Continuum Hypothesis: A Mystery of Mathematics? The proposal originally made by Georg Cantor that there is no infinite set with a cardinal number between that of the "small" infinite set of integers and the "large" infinite set of real numbers (the " continuum "). Symbolically, the continuum hypothesis is that Now,I have some idia that continuum hypothesis is Fasle. See my proof and please let me know what you think. Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined. Whereas mathematics began merely as a calculational tool for computation and tabulation of quantities, it has blossomed into an extremely rich and diverse set of tools, terminologies, and approaches which range from the purely abstract to the utilitarian. Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined. Whereas mathematics began merely as a calculational tool for computation and tabulation of quantities, it has blossomed into an extremely rich and diverse set of tools, terminologies, and approaches which range from the purely abstract to the utilitarian.
Extractions: document.write(''); Search Site: With all of these words With the exact phrase With any of these words Without these words Home My Britannica CREATE MY generalized ... NEW ARTICLE ... SAVE Table of Contents: generalized continuum hypothesis Article Article Related Articles Related Articles Citations LINKS Related Articles Aspects of the topic generalized continuum hypothesis are discussed in the following places at Britannica. axiomatic set theory in set theory (mathematics): Present status of axiomatic set theory Of far greater significance for the foundations of set theory is the status of AC relative to the other axioms of ZF. The status in ZF of the continuum hypothesis (CH) and its extension, the generalized continuum hypothesis (GCH), are also of profound importance. In the following discussion of these questions, ZF denotes Zermelo-Fraenkel set theory without AC. The first finding was obtained by... continuum hypothesis in continuum hypothesis (mathematics) A stronger statement is the generalized continuum hypothesis (GCH): 2
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The Continuum Hypothesis The Continuum Hypothesis The continuum hypothesis (CH) states that there are no sets bigger than the integers and smaller than the real numbers. http://continuum-hypothesis.com/ch.html
Extractions: The continuum hypothesis (CH) states that there are no sets bigger than the integers and smaller than the real numbers. (A set X is "bigger" than a set Y if there is no one-to-one function from X to Y). CH was formulated by Georg Cantor around 1880. This seems pretty straightforward, kind of like saying that there are no integers between and 1. It should be easy to prove or disprove, right? Wrong. No one made any progress on it, and in 1900 David Hilbert put it first on his famous list of 23 open problems. Remarkably, it turns out that CH can be neither proved nor disproved from the current axioms of mathematics. The two parts of this assertion were proved by Kurt Godel (in 1940) and Paul Cohen (in 1963). For his part of the proof, Cohen invented a general-purpose technique called "forcing". When I retire, I vow to learn about forcing, starting by reading Forcing for Dummies In other words, unless a new axiom comes along - some basic, obvious fact that has somehow eluded mathematicians to date - we will never know if CH is true or false . This is analogous to the Heisenberg Uncertainty Principle, which proves (from the axioms of quantum mechanics) that we can't measure both the position and momentum of a particle. It imposes a hard upper bound on what we puny humans can know.
Continuum Hypothesis: True, False, Or Neither? Is the Continuum Hypothesis True, False, or Neither? David J. Chalmers. Newsgroups sci.math From chalmers@bronze.ucs.indiana.edu (David Chalmers) http://consc.net/notes/continuum.html
Extractions: Date: Wed, 13 Mar 91 21:29:47 GMT Thanks to all the people who responded to my enquiry about the status of the Continuum Hypothesis. This is a really fascinating subject, which I could waste far too much time on. The following is a summary of some aspects of the feeling I got for the problems. This will be old-hat to set theorists, and no doubt there are a couple of embarrassing misunderstandings, but it might be of some interest to non-professionals. A basic reference is Gödel's "What is Cantor's Continuum Problem?", from 1947 with a 1963 supplement, reprinted in Benacerraf and Putnam's collection Philosophy of Mathematics . This outlines Gödel's generally anti-CH views, giving some "implausible" consequences of CH. "I believe that adding up all that has been said one has good reason to suspect that the role of the continuum problem in set theory will be to lead to the discovery of new axioms which will make it possible to disprove Cantor's conjecture." At one stage he believed he had a proof that C = aleph_2 from some new axioms, but this turned out to be fallacious. (See Ellentuck, "Gödel's Square Axioms for the Continuum", Mathematische Annalen 1975.)
Set Theory And The Continuum Hypothesis This exploration of a notorious mathematical problem is the work of the man who discovered the solution. The independence of the continuum hypothesis is the focus of this study http://store.doverpublications.com/0486469212.html
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Continuum Hypothesis | Define Continuum Hypothesis At Dictionary.com –noun Mathematics . 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 http://dictionary.reference.com/browse/continuum hypothesis
Continuum Hypothesis -- Math Fun Facts From the Fun Fact files, here is a Fun Fact at the Advanced level Continuum Hypothesis We have seen in the Fun Fact Cantor Diagonalization that the real numbers (the http://www.math.hmc.edu/funfacts/ffiles/30002.4-8.shtml
Extractions: From the Fun Fact files, here is a Fun Fact at the Advanced level: We have seen in the Fun Fact Cantor Diagonalization that the real numbers (the "continuum") cannot be placed in 1-1 correspondence with the rational numbers . So they form an infinite set of a different "size" than the rationals, which are countable. It is not hard to show that the set of all subsets (called the power set ) of the rationals has the same "size" as the reals. But is there a "size" of infinity between the rationals and the reals? Cantor conjectured that the answer is no. This came to be known as the Continuum Hypothesis Many people tried to answer this question in the early part of this century. But the question turns out to be PROVABLY undecidable ! In other words, the statement is indepedent of the usual axioms of set theory! It is possible to prove that adding the Continuum Hypothesis or its negation would not cause a contradiction.
Continuum Hypothesis Cantor's Continuum Hypothesis The Continuum Hypothesis Infinity has infinite ways to trouble our finite minds. http://users.forthnet.gr/ath/kimon/Continuum.htm
Extractions: The Continuum Hypothesis Infinity has ... infinite ways to trouble our finite minds. This was proved by Georg Cantor in 1874. The "smallest level" of infinity has to do with countable things that can be put in some order. . This seems strange: one set is a proper subset of another and still they have the same number of elements. This is exactly the definition of infinite sets. What about rational numbers? These are a superset of the natural numbers but still of class aleph . It turns out that there is a way to put rational numbers in order: 1, 2, 1/2, 1/3, 3, 4, 3/2, 2/3, 1/4 ... (the pattern is based on a diagram so it is not obvious as shown here). Things change when we examine the real numbers. There is no way to create a complete list of reals and this was shown by Cantor with a beautiful argument, the "diagonal" one: Suppose we had such a complete list of real numbers between and 1 : r1=0.a a a
Continuum Hypothesis@Everything2.com Y'know, if you log in, you can write something here, or contact authors directly on the site. Create a New User if you don't already have an account. http://www.everything2.com/title/continuum hypothesis
Buscalo.com - La Web Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert s twentythree problems presented in the year 1900. http://www.buscalo.com/index.php?page=search/web&search=continuum hypothesis
RedHotPawn.com : Posers And Puzzles : Infinity (and Beyond...) 9 posts 7 authors - Last post Mar 26Edit This is what I got after a quick search He also showed that the continuum hypothesis cannot be disproved from the accepted axioms of http://www.redhotpawn.com/board/showthread.php?threadid=89902&page=1
Constructible Universe | Facebook Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis . continuum hypothesis /a are true in the constructible universe . http://de-de.facebook.com/pages/Constructible-universe/133440370027811
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PlanetMath: Continuum Hypothesis The continuum hypothesis states that there is no cardinal number $\kappa$ such that $\aleph_0 \kappa 2^{\aleph_0}$ An equivalent statement is that $\aleph_1 =2^{\aleph_0}$ http://planetmath.org/encyclopedia/ContinuumHypothesis.html
Extractions: continuum hypothesis (Axiom) The continuum hypothesis states that there is no cardinal number such that An equivalent statement is that It is known to be independent of the axioms of ZFC The continuum hypothesis can also be stated as: there is no subset of the real numbers which has cardinality strictly between that of the reals and that of the integers . It is from this that the name comes, since the set of real numbers is also known as the continuum.
