41. Continuum Hypothesis. Axiom Of Constructibility. Axiom Of Determinateness. Acker Trying to prove the continuum hypothesis, Cantor developed his theory of transfinite ordinal numbers. The Independence of the continuum hypothesis. Proc. Nat. http://www.ltn.lv/~podnieks/gt6_2.html

continuum hypothesis, axiom of constructibility, continuum problem, axiom of determinateness, constructibility, determinateness, axiom, set theory, Ackermann, continuum, Ackermann set theory Back to title page Left Adjust your browser window Right 2.4. Around the Continuum Problem 2.4.1. Counting Infinite Sets Trying to prove the continuum hypothesis, Cantor developed his theory of transfinite ordinal numbers . The origin of this concept was described in Section 2.1 . The idea behind is simple enough (to explain, but hard to discover). Counting a set means bringing of some very strong order among its members. After the counting of a finite set x is completed, its members are allocated in a linear order: x , x , ..., x n , where x is the first member, and x n is the last member of x (under this particular ordering). If we select any non-empty subset y of x, then y also contains both the first and the last members (under the same ordering of x). But infinite sets cannot be ordered in this way. How strong can be the orderings that can be introduced on infinite sets? For example, consider the "natural" ordering of the set w of all natural numbers. If you separate a non-empty subset y of w, then you can definitely find the first (i.e. the least) member of y, but for an infinite y you will not find the last element. Can each infinite set be ordered at least in this way? The precise framework is as follows. The relation R is called a

42. Continuum, Mu-Ency At MROB The continuum hypothesis states that there is no infinity between Aleph0 and the order of a continuum, which would mean that the order of the continuum is http://www.mrob.com/pub/muency/continuum.html

Continuum Robert P. Munafo, 2002 May 7. Roughly speaking, a continuum is a type of connected set that can be divided into smaller and smaller pieces infinitely many times and any such pieces, if they are obtained after a finite number of steps, have the same order as the original set. Examples of continuums are a straight line, a plane, a circle, a disc , the set of real numbers, and the set of complex numbers. It can be shown that all continuums have the same order The term "continuum" is also used to refer to an infinite quantity, equal to the order of any continuum. In other words, "continuum" can be used to mean "the number of points on a line" instead of meaning "a line". It was proven by Cantor in the late 1800's that the power set of the integers (or of any other set of order aleph ) has the same order as the set of reals or any other continuum. The Continuum Hypothesis states that there is no infinity between Aleph-0 and the order of a continuum, which would mean that the order of the continuum is Aleph-1 . Although it is called a "hypothesis", the truth or falsehood of the Continuum Hypothesis has been shown (by Godel and Paul Cohen) to be an axiomatic issue, like the parallel postulate in geometry, if one is working within Zermelo-Fraenkel set theory with the Axiom of Choice. Different systems of set theory and of transfinite quantities, each consistent within itself, can be constructed on the basis of whether or not the Continuum Hypothesis is taken to be true, false, or undetermined.

43. Erowid Experience Vaults: Aleph-2 - The Continuum Hypothesis Support Erowid by becoming a member and get an Erowid tshirt! The continuum hypothesis. Aleph-2. by 77k. DOSE T+ 000, 6 mg, oral, http://www.erowid.org/experiences/exp.php?ID=14854

44. 2. Continuum Hypothesis 2. SOME FORMULATIONS OF continuum hypothesis. Introduce the following notations. Now, there are two following main formulations of continuum hypothesis 8. http://www.mi.sanu.ac.yu/vismath/zen/zen2.htm

2. SOME FORMULATIONS OF CONTINUUM HYPOTHESIS X X X N N N D . Since D has, by the well-known Cantor's theorem, the power C D C 1) The classical Cantor Continuum Hypothesis formulation: C 2) The generalized Continuum Hypothesis formulation, by Cohen: P , where P ) is the power-set of any set A A ] by the following estimation of the Continuum Cardinality: "Thus, C is greater than n , where , and so on. " (p.282) [ ]. Therefore, we shall even not try to imagine visually a set of integers of a cardinality succeeding , and use the following most weak formulation of Continuum Hypothesis. 3) Whether there exists a set of integers, say M , such that a 1-1-correspondence between the set M and the set D of all real numbers (proper fractions, geometrical points) of the segment [0,1] can be realized? That is M M C ] ?, where M M , construct such the 1-1-corerespondence, and prove that the set M has the continual cardinality C NEXT VisMath HOME

45. Math Forum - Ask Dr. Math The continuum hypothesis. Date Wed, 24 May 1995 090405 +0800 From SheparD Subject Math Problem Although I am not a K12 type person my daughter is. http://mathforum.org/library/drmath/view/51437.html

Associated Topics Dr. Math Home Search Dr. Math The Continuum Hypothesis Date: Wed, 24 May 1995 09:04:05 +0800 From: SheparD Subject: Math Problem Although I am not a K12 type person my daughter is. She is the one with the math problem, but I am the one with the internet connection. But really it IS me with the problem... I volunteered to assist her with an essay assignment and I thought to retrieve some information from the net. But, alas, I can find no information on the net. I would only like to have you point me in the right direction, if you would. The problem: (or question as it may be) "The continuum theory, what is it and has it been resolved?" I would be grateful if you could provide any assistance to me. Thanks for your time, David Date: 9 Jun 1995 10:25:29 -0400 From: Dr. Ken Subject: Re: Math Problem Hello there! I'm sorry it's taken us so long to get back to you. If you're still interested, here's something I found in the Frequently-Asked-Questions for the sci.math newsgroup. If you want to look in the site yourself sometime, the site name is ftp.belnet.be (you can log in with the user name "anonymous") and this file's name is /pub/usenet-faqs/usenet-by-hierarchy/sci/math/ sci.math_FAQ:_The_Continuum_Hypothesis I found it by searching FAQs at the site http://mailserv.cc.kuleuven.ac.be/faq/faq.html

46. Re: Continuum Hypothesis Re continuum hypothesis. 18 Apr 2000 Re continuum hypothesis, by Antreas P. Hatzipolakis 19 Apr 2000 Re continuum hypothesis, by John Conway The Math Forum http://mathforum.org/epigone/math-history-list/twoudwolgix

a topic from math-history-list Re: Continuum Hypothesis post a message on this topic post a message on a new topic 18 Apr 2000 Re: Continuum Hypothesis , by Antreas P. Hatzipolakis 19 Apr 2000 Re: Continuum Hypothesis , by John Conway The Math Forum

47. Cogprints - Generalized Continuum Hypothesis And The Axiom Of Combinatorial Sets Generalized continuum hypothesis and the Axiom of Combinatorial Sets. Nambiar, Kannan (2002) Generalized continuum hypothesis and http://cogprints.ecs.soton.ac.uk/archive/00002169/

Cogprints Home About Browse Search ... Help Generalized Continuum Hypothesis and the Axiom of Combinatorial Sets Nambiar, Kannan Generalized Continuum Hypothesis and the Axiom of Combinatorial Sets Full text available as: PDF - Requires Adobe Acrobat Reader or other PDF viewer. Abstract Axiom of Combinatorial Sets is defined and used to derive Generalized Continuum Hypothesis. Keywords: Generalized continuum htpothesis; Axiom of combinatorial sets. Subjects: Philosophy Logic ID Code: Deposited By: Nambiar, Kannan Deposited On: 07 April 2002 Alternative Locations: http://www.rci.rutgers.edu/~kannan/science/combinatorial_axiom_screen.pdf Cogprints Editor: cogprints@ecs.soton.ac.uk Cogprints Technical Administrator: cogprints-admin@ecs.soton.ac.uk

48. Godel, K.: Consistency Of The Continuum Hypothesis. (AM-3). of the book Consistency of the continuum hypothesis. (AM Consistency of the continuum hypothesis. (AM3). Kurt Godel. Paper...... http://pup.princeton.edu/titles/1034.html

PRINCETON University Press SEARCH: Keywords Author Title More Options Power Search Search Hints E-MAIL NOTICES NEW IN PRINT E-BOOKS ... HOME PAGE 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: $24.95 ISBN: 0-691-07927-7 For customers in Europe, Africa, the Middle East, and India Prices subject to change without notice File created: 5/7/04 Questions and comments to: webmaster@pupress.princeton.edu Princeton University Press

49. Continuum Hypothesis continuum hypothesis. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. continuum hypothesis. http://www.fastload.org/co/Continuum_hypothesis.html

Continuum hypothesis Home Up About 'Continuum hypothesis' Advertisement 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

50. QuicklyFind : Continuum Hypothesis article. Investigating the continuum hypothesis. Consider the set of all rational numbers . The generalized continuum hypothesis. The http://www.quicklyfind.com/info/Continuum_hypothesis.htm

QUICKLY FIND Information on just about anything! Current topic : Continuum hypothesis - View Index - Search for : [[fr:Hypothèse du continu]] [[pl:Hipoteza continuum]] [[Category:Set theory]] Category:Conjectures 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 , which is described at the end of this article. 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: they are both

51. The Continuum Hypothesis The continuum hypothesis. http://www.jboden.demon.co.uk/SetTheory/ContinuumHypothesis.html

The Continuum Hypothesis

52. The Continuum Hypothesis The continuum hypothesis Notes for Math 446 M. Flashman Spring, 2002. I. Background Cantor 18451918 Investigation of discontinuities with Fourier series and http://www.humboldt.edu/~mef2/Courses/m446s02n2.html

The Continuum Hypothesis Notes for Math 446 M. Flashman Spring, 2002 I. Background: Cantor 1845-1918: Investigation of discontinuities with Fourier series and Set Theory Beginnings. Any infinite subset of the natural numbers or the integers is countable. The rational numbers are a countable set. "Godel counting" argument. The algebraic numbers are countable. [ Another first type of diagonal argument.] 1874 Cantor's proof that the number of points on a line segment are uncountable. (1874) A decimal based proof that there is an uncountable set of real numbers.(similar to 1891 proof) There is no onto function from R, the set of real numbers, to P(R), the set of all subsets of the real number. There are sets which are larger than the reals. The rational numbers between and 1 have "measure" zero. Any countable set of real numbers has "measure" zero. II. The XXth Century: An Age of Exploration and Discovery. Hilbert: (Finitistic Formalization of Arithmetic) The continuum hypothesis problem was the first of Hilbert's famous 23 problems delivered to the Second International Congress of Mathematicians in Paris in 1900. Hilbert's famous speech The Problems of Mathematics challenged (and still today challenge) mathematicians to solve these fundamental questions Brouwer: (1881-1966) (Rejection of the law of excluded middle for infinite sets) He rejected in mathematical proofs the Principle of the Excluded Middle, which states that any mathematical statement is either true or false. In 1918 he published a set theory, in 1919 a measure theory and in 1923 a theory of functions all developed without using the Principle of the Excluded Middle.

53. Continuum History San Diego 2002 The continuum hypothesis A Look at the History of the Real Numbers in The Second Millennium. Abstract After Cantor first demonstrated http://www.humboldt.edu/~mef2/Presentations/San Diego/Continuum History San Dieg

Monday January 7, 2002 2:20 p.m MAA Session on History of Mathematics in the Second Millennium, III Martin E Flashman flashman@humboldt.edu Department of Mathematics, Humboldt State University, Arcata, CA 955521 The Continuum Hypothesis: A Look at the History of the Real Numbers in The Second Millennium. Abstract: After Cantor first demonstrated that the real numbers (continuum) were uncountable, the hypothesis arose that the set of the real numbers was "the smallest" uncountable set. In 1900 David Hilbert made settling the continuum hypothesis the first problem on his now famous list of problems for this century. The author will discuss some of the historical, philosophical, and mathematical developments connected to this problem proceeding from issues of definition of the real numbers and proofs of uncountability to issues of consistency and models and proofs of the independence of this hypothesis and possibly some comments on its current status. (Received September 14, 2001) Outline of possible Discussion (depending on time allowed).

54. Math Lair - The Continuum Hypothesis The continuum hypothesis. Note that the notation a 0 and a 1 are used to represent aleph nought and aleph one respectively due to character set limitations. http://www.stormloader.com/ajy/continuum.html

The Continuum Hypothesis [Note that the notation a and a are used to represent aleph nought and aleph one respectively due to character set limitations]. How many real numbers are there? Cantor noted that there are Y X different numbers of X digits where Y is the base used (in this case, 10). Suppose there are precisely C real numbers that are specified by their decimal expansions 0.abcd . . . in which there are a digits each chosen from a set of 10 possibilities. Therefore there are 10 a possibilities. If we did the same thing in binary, we would get 2 a = C. Since C must be greater than a , we can see that almost all real numbers are transcendental The continuum hypothesis is the hypothesis that C = a . In other words, there is no set whose cardinal number lies between that of the natural numbers unprovable Last updated June 8, 2002. URL: http://www.stormloader.com/ajy/continuum.html For questions or comments email James Yolkowski Math Lair home page

55. Colloquium Announcement The Legacy of the continuum hypothesis. We will begin with a brief historical discussion of how the continuum hypothesis came to be formulated. http://math.dartmouth.edu/~colloq/f02/2002-December-05_1037657817.phtml

Homepage Contact Information Computing Help/Resources FAQ ... Make a comment Last reviewed 15 Aug 2003 Comments to webmaster Graphics by The Gimp Powered by PHP and Debian Linux Member and supporter of the NB: A PDF version of this announcement (suitable for posting) is also available. The Legacy of the Continuum Hypothesis Elizabeth Theta Brown Dartmouth College Thursday, December 5, 2002 102 Bradley Hall, 4 pm Tea 3:30 pm, Math Lounge Abstract: We will begin with a brief historical discussion of how the continuum hypothesis came to be formulated. This will be followed by an intuitive description of how the method of forcing works, and an heuristic account of how it can be used to address the continuum hypothesis as well as selected modern questions from analysis and set theory. Throughout, the talk will focus on intuition, motivation, and context. It will be accessible to graduate students. This talk will be accessible to graduate students.

56. Continuum Hypothesis TutorGig.com Encyclopedia continuum hypothesis. In mathematics, the continuum hypothesis is at the end of this article. Investigating the continuum hypothesis. http://www.tutorgig.com/encyclopedia/getdefn.jsp?keywords=Continuum_hypothesis

57. Infinite Ink: The Continuum Hypothesis, By Nancy McGough History, mathematics, metamathematics, and philosophy of Cantor s continuum hypothesis. Infinite Ink The continuum hypothesis, by Nancy McGough. http://www.spacetransportation.org/Detailed/70844.html

History, mathematics, metamathematics, and philosophy of Cantor's Continuum Hypothesis Home Math Logic and Foundations Set Theory : Infinite Ink: The Continuum Hypothesis, by Nancy McGough Infinite Ink: The Continuum Hypothesis, by Nancy McGough History, mathematics, metamathematics, and philosophy of Cantor's Continuum Hypothesis Visit this link Spacetransportation.org - Science Directory - Last Update: Sun May 23 2004

58. Axiom Of Choice And Continuum Hypothesis Home Math Logic and Foundations Set Theory Axiom of Choice and continuum hypothesis. Axiom of Choice and continuum hypothesis. http://www.spacetransportation.org/Detailed/70865.html

Part of the Frequently Asked Questions in Mathematics. Home Math Logic and Foundations Set Theory : Axiom of Choice and Continuum Hypothesis Axiom of Choice and Continuum Hypothesis Part of the Frequently Asked Questions in Mathematics. Visit this link Spacetransportation.org - Science Directory - Last Update: Sun May 23 2004

59. Numerical Analysis Of Nano-imprinting Process Based On Continuum Hypothesis The code is based on the continuum hypothesis with some modifycation to accommodate the size effect, such as slip along the boundary. Nanotech 2005. http://www.nanotech2004.com/2004program/showabstract.html?absno=184

60. HughWoodin Segal Theater The continuum hypothesis. Professor W Abstract. Can the problem of Cantor s continuum hypothesis be solved? Or does the formal http://nylogic.org/Conference/April2002/Friday/HughWoodin

nylogic.org Conference Friday / HughWoodin Calendar This Week Import Data Upcoming Talks ... Friday, April 5, 2002 5:00 pm GC Segal Theater The Continuum Hypothesis Professor W. Hugh Woodin University of California at Berkeley woodin@math.berkeley.edu Abstract. Can the problem of Cantor's Continuum Hypothesis be solved? Or does the formal independence of the problem from the axioms of Set Theory simply put an end to any reasonable discussion of whether the Continuum Hypothesis is true or false? New York Logic Colloquium Featured Speaker Login

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