21. Sci.math FAQ: The Continuum Hypothesis sci.math FAQ The continuum hypothesis 1994 Version 6.2 THE CONTINUUMHYPOTHESIS A basic reference is Godel s ``What is Cantor s Continuum Problem? http://www.faqs.org/faqs/sci-math-faq/AC/ContinuumHyp/

Usenet FAQs Search Web FAQs Documents ... RFC Index sci.math FAQ: The Continuum Hypothesis Newsgroups: sci.math alopez-o@neumann.uwaterloo.ca alopez-o@barrow.uwaterloo.ca Tue Apr 04 17:26:57 EDT 1995 Rate this FAQ N/A Worst Weak OK Good Great Related questions and answers Usenet FAQs Search Web FAQs ... RFC Index Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update June 15 2004 @ 00:31 AM

22. 2. Continuum Hypothesis 2. SOME FORMULATIONS OF continuum hypothesis Introduce the following notations. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

23. Mudd Math Fun Facts: Continuum Hypothesis This came to be known as the continuum hypothesis. It is possible to provethat adding the continuum hypothesis or its negation would not cause a http://www.math.hmc.edu/funfacts/ffiles/30002.4-8.shtml

hosted by the Harvey Mudd College Math Department Francis Su Any Easy Medium Advanced Search Tips List All Fun Facts Fun Facts Home About Math Fun Facts ... Other Fun Facts Features 181937235.18468 Fun Facts viewed since 20 July 1999. Francis Edward Su From the Fun Fact files, here is a Fun Fact at the Advanced level: Continuum Hypothesis We have seen in the Fun Fact How Many Reals 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.

24. An Intuitivistic Solution Of The Continuum Hypothesis For An intuitivistic solution of the continuum hypothesis for definable sets and resolution of the set theoretical paradoxes http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

25. PlanetMath: Continuum Hypothesis The continuum hypothesis states that there is no cardinal number The continuumhypothesis can also be stated as there is no subset of the real numbers http://planetmath.org/encyclopedia/ContinuumHypothesis.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About 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. "continuum hypothesis" is owned by Evandar view preamble View style: HTML with images page images TeX source See Also: axiom of choice Zermelo-Fraenkel axioms generalized continuum hypothesis Other names: CH Cross-references: integers strictly cardinality real numbers ... cardinal number There are 7 references to this object.

26. PlanetMath: Generalized Continuum Hypothesis The generalized continuum hypothesis states that for any infinite cardinal $ \lambda$there is no generalized continuum hypothesis is owned by yark. http://planetmath.org/encyclopedia/GeneralizedContinuumHypothesis.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About generalized continuum hypothesis (Axiom) The generalized continuum hypothesis states that for any infinite cardinal there is no cardinal such that An equivalent condition is that for every ordinal . Another equivalent condition is that for every ordinal Like the continuum hypothesis , the generalized continuum hypothesis is known to be independent of the axioms of ZFC "generalized continuum hypothesis" is owned by yark full author list owner history view preamble View style: HTML with images page images TeX source See Also: aleph numbers beth numbers continuum hypothesis cardinality ... cardinal exponentiation under GCH Other names: generalised continuum hypothesis, GCH Keywords: cardinality, cardinal

27. Continuum Hypothesis: True, False, Or Neither? Subject Re Status of the continuum hypothesis Subject continuum hypothesis.I just came across your posting about CH and found it quite interesting. http://consc.net/notes/continuum.html

Is the Continuum Hypothesis True, False, or Neither? David J. Chalmers Newsgroups: sci.math From: chalmers@bronze.ucs.indiana.edu (David Chalmers) Subject: Continuum Hypothesis - Summary 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.)

28. Continuum Hypothesis Cantor s continuum hypothesis. The continuum hypothesis states simply thataleph1= c, that is, there is no level of infinity between the one of http://users.forthnet.gr/ath/kimon/Continuum.htm

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 r2=0.a a a r3=0.a a a Where a ij take values in 0,1,...,9 and all numbers are written with infinite number of digits (e.g. 0.2=0.20000..., 1/7=0.142857142857...). Then we can create another real r'=0.b

29. Continuum Hypothesis - Definition Of Continuum Hypothesis In Encyclopedia In mathematics, the continuum hypothesis is a hypothesis about the possible sizesof infinite sets. Georg Cantor introduced the concept of cardinality to http://encyclopedia.laborlawtalk.com/Continuum_hypothesis

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law 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 is strictly smaller than the set of real numbers . 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 Contents showTocToggle("show","hide") 1 The size of a set 2 Investigating the continuum hypothesis 3 Impossibility of proof and disproof 4 The generalized continuum hypothesis ... 6 References The size of a set To state the hypothesis formally, we need a definition: we say that two sets

30. An Intuitivistic Solution Of The Continuum Hypothesis For Definable Sets And Res An intuitivistic solution of the continuum hypothesis for definable sets andresolution of the set theoretical paradoxes. http://www.farazgodrejjoshi.com/

An intuitivistic solution of the Continuum Hypothesis for definable sets and resolution of the set theoretic paradoxes. by Faraz Godrej Joshi faraz@farazgodrejjoshi.com View Paper A4 size in ( PDF format farazgodrejjoshi.pdf (74.6 KB) Download (.zip) Zipped PDF File. farazgodrejjoshi.zip (58.1 KB) Website by: Skindia Internet Pvt. Ltd.

31. An Intuitivistic Solution Of The Continuum Hypothesis For Definable Sets And Res An intuitivistic solution of the continuum hypothesis for definable sets andresolution of the set theoretical paradoxes. http://www.farazgodrejjoshi.com/about.htm

ABOUT THE PAPER The paper, titled "An intuitivistic solution of the Continuum Hypothesis for definable sets and resolution of the set theoretic paradoxes" , consists essentially of a solution of the Continuum Hypothesis for all sets which can intuitively be considered definable, with the section on paradoxes justifying its intuitive basis. The Continuum Hypothesis was for long regarded the most famous unsolved problem in mathematics. Subsequently in 1963, the works of Godel and Cohen proved the independence of the Continuum Hypothesis within the framework of an axiomatic set theory The paper, however, proves the Continuum Hypothesis for all definable sets within the framework of classical intuitive set theory which can arguably be considered the mathematical theory in which mathematicians originally sought a solution of the Continuum Hypothesis: A theory that is both all-encompassing in its generality and beautiful in its simplicity. Unfortunately, classical intuitive set theory was abandoned by mathematicians years ago in favour of several axiomatic set theories. This was primarily to side-step the apparent inconsistencies of classical intuitive set theory as manifested by the set theoretic paradoxes. Now since this paper basically reaffirms the stand of classical intuitive set theory, it was imperative to 'resolve' the set theoretic paradoxes to justify the intuitive approach in the first place. Also, since the solution of the Continuum Hypothesis entailed the use of concepts such as 'specifiable' or 'definable' real number and its counterpart 'unspecifiable' or 'undefinable' real number, it was necessary to resolve certain fundamental questions posed by Richard's paradox in particular and other set theoretic paradoxes in general.

32. Kids.net.au - Encyclopedia Continuum Hypothesis - If a set S was found that disproved the continuum hypothesis, Cantor believedthe continuum hypothesis to be true and tried for many years to prove it, http://www.kids.net.au/encyclopedia-wiki/co/Continuum_hypothesis

Web kids.net.au Thesaurus Dictionary Kids Categories Encyclopedia ... Contents 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 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

33. Continuum Hypothesis The continuum hypothesis (CH), put forward by Cantor in 1877, says that the It is called the continuum hypothesis because the real numbers are used to http://www.daviddarling.info/encyclopedia/C/continuum_hypothesis.html

Return to The Worlds of David Darling INDEX LATEST NEWS ARCHIVE ... Z 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.

34. Navier-Stokes Equations: Continuum Hypothesis These notes present the Navier Stokes equations, which are basis for the studyof fluid mechanics, aerodynamics, hydrodynamics, and gasdynamics. http://www.navier-stokes.net/nscont.htm

Foundations of Fluid Mechanics My Homepage Gallery of Fluid Mechanics Cambridge University Press Contact Index Introduction Web Stuff Our Sponsor What's New? Navier-Stokes Eqns Overview Notation Definition Continuum Hypoth. ... Summary Misc. Topics Constant Properties Inviscid Flows Incompressible Flows Conservative Forms ... Special Fluid Models Potential Flows Introduction Restrictions Aerodynamics Water Waves ... Acoustics Math Identities Vector Calculus Stokes' Theorem Gauss' Theorem Transport Theorems Great Books Introduction Continuum Mech. General Fluid Mech. Aero/Hydrodyn. ... Miscellaneous Navier-Stokes Equations Continuum Hypothesis In most treatments of fluid mechanics, the so-called continuum hypothesis is hurriedly stated during the first lecture or in the very first chapter of a text. While I think that the standard discussions are quite reasonable as far as they go, I have always felt that the additional concept of local thermodynamic equilibrium is essential in any preliminary discussion of fluid mechanics. Below I've provided a draft of my views on the subject. Standard Continuum Hypothesis The basis for much of classical mechanics is that the media under consideration is a continuum. Crudely speaking, matter is taken to occupy every point of the space of interest, regardless of how closely we examine the material. Such a view is perfectly reasonable from a modeling point of view as long as the resultant mathematical model generates results which agree with experiment. Among other things, such a model permits us to use the field representation, i.e., the view in which the velocities, pressures, and temperatures are taken to be piecewise continuous functions of space and time. Furthermore, it is well known that the standard macroscopic representation yields highly accurate predictions of the behavior of solids and fluids.

35. Continuum Hypothesis There is also a generalization of the continuum hypothesis called the The continuum hypothesis states that every subset of the continuum (= the real http://www.algebra.com/algebra/about/history/Continuum-hypothesis.wikipedia

Continuum hypothesis Regular View Dictionary View (all words explained) Algebra Help my dictionary with pronunciation , wikipedia etc 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 is strictly smaller than the set of real numbers . 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 of the real numbers is , the continuum hypothesis says: This implies: 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 saying: For all ordinals Contents The size of a set Investigating the continuum hypothesis Impossibility of proof and disproof The generalized continuum hypothesis ... References The size of a set Main article: Cardinal number To state the hypothesis formally, we need a definition: we say that two sets

36. Continuum Hypothesis -- Facts, Info, And Encyclopedia Article There is also a generalization of the continuum hypothesis called the generalized The continuum hypothesis states that every (A set whose members are http://www.absoluteastronomy.com/encyclopedia/c/co/continuum_hypothesis.htm

Continuum hypothesis [Categories: Set theory] In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics , the continuum hypothesis is a (A tentative theory about the natural world; a concept that is not yet verified but that if true would explain certain facts or phenomena) hypothesis about the possible sizes of (Click link for more info and facts about infinite) infinite (A group of things of the same kind that belong together and are so used) set s. (Click link for more info and facts about Georg Cantor) Georg Cantor introduced the concept of (Click link for more info and facts about cardinality) cardinality to compare the sizes of infinite sets, and he showed that the set of (Any of the natural numbers (positive or negative) or zero) integer s is strictly smaller than the set of (Any rational or irrational number) real number s. 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

37. Continuum Hypothesis@Everything2.com The continuum hypothesis is, loosely speaking, a postulate as to how Combining these facts, we see that the continuum hypothesis can be restated as http://www.everything2.com/index.pl?node_id=22495

38. Godel, K.: Consistency Of The Continuum Hypothesis. (AM-3). of the book Consistency of the continuum hypothesis. (AM3) by Godel,K., published by Princeton University Press....... http://www.pupress.princeton.edu/titles/1034.html

SEARCH: Keywords Author Title More Options Power Search Search Hints E-MAIL NOTICES NEW IN PRINT E-BOOKS ... One of Princeton University Press's Notable Centenary Titles. Consistency of the Continuum Hypothesis. (AM-3) Kurt Godel 69 pp. Shopping Cart Series: Annals of Mathematics Studies Phillip A. Griffiths, John N. Mather, and Elias M. Stein, Editors Subject Area: Mathematics VISIT OUR MATH WEBSITE Paper: Not for sale in Japan Shopping Cart: For customers in the U.S., Canada, Latin America, Asia, and Australia Paper: $27.95 ISBN: 0-691-07927-7 For customers in Europe, Africa, the Middle East, and India Prices subject to change without notice File created: 6/30/2005 Questions and comments to: webmaster@pupress.princeton.edu Princeton University Press

39. EPOCH OF UNLIGHT - The Continuum Hypothesis The End Records and Omega MailOrder offer a diverse selection of all genres ofmetal at great prices and excellent service. http://shopping.theendrecords.com/s.nl/sc.9/category.2/it.A/id.8900/.f

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40. Continuum Hypothesis The continuum hypothesis is that C = aleph 1, that there are no such Gödel showed in 1938 that the continuum hypothesis cannot be disproved using just http://www-users.cs.york.ac.uk/~susan/cyc/c/cont.htm

continuum hypothesis The smallest infinite cardinal number is (pronounced 'aleph null', or 'aleph naught'), the next is , then , and so on. There are integers. There are strictly more real numbers than integers (proof by 'diagonalisation'), in fact there are 2 reals; this is the cardinality of the continuum, or C . So C . We know that C cannot be less than , because the only infinite cardinal less than is . So, is C equal to, or greather than, If C , there would be sets with cardinality that would have strictly more elements than in the set of integers, but stricly fewer elements than in the set of reals. The continuum hypothesis is that C , that there are no such intermediate sized sets. disproved using just the axioms of set theory. Paul Cohen showed in 1963 that the continuum hypothesis cannot be proved using just the axioms of set theory. It is independent of those axioms. Shaughan Lavine. Understanding the Infinite Kevin Wald.

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