61. The Continuum Hypothesis The continuum hypothesis. Posted by sol on September 13, 2002 at 173556 continuum hypothesis. What do we mean when we say continuum ? http://superstringtheory.com/forum/geomboard/messages2/117.html

String Theory Discussion Forum String Theory Home Forum Index The Continuum Hypothesis Follow Ups Post Followup Geometry II FAQ Posted by sol on September 13, 2002 at 17:35:56: Continuum Hypothesis What do we mean when we say "continuum"? Here's a description Albert Einstein gave on p. 83 of his Relativity: The Special and the General Theory: The surface of a marble table is spread out in front of me. I can get from any one point on this table to any other point by passing continuously from one point to a "neighboring" one, and repeating this process a (large) number of times, or, in other words, by going from point to point without executing "jumps." I am sure the reader will appreciate with sufficient clearness what I mean here by "neighbouring" and by "jumps" (if he is not too pedantic). We express this property of the surface by describing the latter as a continuum. So here we are describing something that is inherent in Superstring theory, and who is going to pave the way for me to understand this? By coming to the realization of the continuum, spoken by Einstein, it has made me realize, the value we could have assigned energy, and what did Kaluza do for Einstein that Einstein did for us?

62. Continuum Hypothesis - Information An online Encyclopedia with information and facts continuum hypothesis Information, and a wide range of other subjects. The generalized continuum hypothesis. http://www.book-spot.co.uk/index.php/Continuum_hypothesis

Continuum hypothesis - Information Home Mathematical and natural sciences Applied arts and sciences Social sciences and philosophy ... Interdisciplinary categories adsonar_pid=2712;adsonar_ps=1199;adsonar_zw=120;adsonar_zh=600;adsonar_jv='ads.adsonar.com'; de:Kontinuumshypothese fr:Hypothèse du continu pl:Hipoteza continuum 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 aleph-null 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")

63. Logic And Language Links - Continuum Hypothesis continuum hypothesis Gloss A hypothesis in set theory first proposed by Cantor. continuum hypothesis is a subtopic of set theory. http://staff.science.uva.nl/~caterina/LoLaLi/Pages/382.html

Siblings tell me more... under set theory iota operator descriptive set theory large cardinal determinacy ... TOP You have selected the concept continuum hypothesis Gloss: A hypothesis in set theory first proposed by Cantor. The set of all natural numbers N has a cardinal number Aleph_0. The power set of N will therefore have a cardinality of Aleph_0 to teh power of 2, which is denoted by c-the cardinal number of the set of real numbers (the continuum). Cantor's hypothesis is that no infinite cardinal lies between Aleph_0 and c. continuum hypothesis is a: subtopic of set theory continuum hypothesis has currently no subtopics. Long description: Not available yet. Search the hierarchy with v7 Caterina Caracciolo home page Home Search this site with Dowser Page generated on: 2004:3:15, 10:08 Information about LoLaLi.net Handbook Not available yet tell me more...

64. Logic And Language Links - Generalized Continuum Hypothesis generalized continuum hypothesis This concept has currently no gloss. generalized continuum hypothesis is a subtopic of set theory. http://staff.science.uva.nl/~caterina/LoLaLi/Pages/389.html

Siblings tell me more... under set theory iota operator descriptive set theory large cardinal determinacy ... TOP You have selected the concept generalized continuum hypothesis This concept has currently no gloss. generalized continuum hypothesis is a: subtopic of set theory generalized continuum hypothesis has currently no subtopics. Long description: Not available yet. Search the hierarchy with v7 Caterina Caracciolo home page Home Search this site with Dowser Page generated on: 2004:3:15, 10:08 Information about LoLaLi.net Handbook Not available yet tell me more...

65. Continuum Hypothesis continuum hypothesis. The proposal originally made Continuum ). Symbolically, the continuum hypothesis is that . Gödel showed that http://icl.pku.edu.cn/yujs/MathWorld/math/c/c640.htm

Continuum Hypothesis 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 showed that no contradiction would arise if the continuum hypothesis were added to conventional Zermelo-Fraenkel Set Theory . However, using a technique called Forcing , Paul Cohen (1963, 1964) proved that no contradiction would arise if the negation of the continuum hypothesis was added to Set Theory Set Theory being used, and is therefore Undecidable (assuming the Zermelo-Fraenkel Axioms together with the Axiom of Choice Conway and Guy (1996) give a generalized version of the Continuum Hypothesis which is also Undecidable : is for every See also Aleph-0 Aleph-1 Axiom of Choice Cardinal Number ... Zermelo-Fraenkel Set Theory References Cohen, P. J. ``The Independence of the Continuum Hypothesis.'' Proc. Nat. Acad. Sci. U. S. A. Cohen, P. J. ``The Independence of the Continuum Hypothesis. II.'' Proc. Nat. Acad. Sci. U. S. A.

66. Continuum Hypothesis A detailed presentation of the continuum hypothesis, which implies that there is an endless amount of real numbers in the world. http://www.hypography.com/info.cfm?id=17391

67. Encyclopedia: Continuum Hypothesis Updated Apr 23, 2004. Encyclopedia continuum hypothesis. Investigating the continuum hypothesis. Consider the set of all rational numbers. http://www.nationmaster.com/encyclopedia/Continuum-hypothesis

Supporter Benefits Signup Login Sources ... Pies Factoid #40 Women are flooding into the workforce in many Muslim countries. Interesting Facts Make your own graph: Hold down Control and click on several. Compare All Top 5 Top 10 Top 20 Top 100 Bottom 100 Bottom 20 Bottom 10 Bottom 5 All (desc) in category: Select Category Agriculture Crime Currency Democracy Economy Education Energy Environment Food Geography Government Health Identification Immigration Internet Labor Language Manufacturing Media Military Mortality People Religion Sports Taxation Transportation Welfare with statistic: view: Correlations Printable graph / table Pie chart Scatterplot with ... * Asterisk means graphable. Added May 21 Mortality stats Multi-users ½ price Catholic stats Top Graphs Richest Most Murderous Most Populous Most Militaristic ... More Stats Categories Agriculture Background Crime Currency ... Welfare Updated: Apr 23, 2004 Encyclopedia : Continuum hypothesis Sponsored links: fr:Hypoth se du continu 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

68. Path News.jmag.net!news.jmas.co.jp!nf9.iij.ad.jp!nr1.iij.ad.jp! From alopezo@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Newsgroups sci.math,news.answers,sci.answers Subject sci.math FAQ The continuum hypothesis Followup-To http://linas.org/mirrors/nntp/sci.math/faq.continuum.html

Path: news.jmag.net news.jmas.co.jp !nf9.iij.ad.jp!nr1.iij.ad.jp! news.iij.ad.jp news.qtnet.ad.jp !news1.mex.ad.jp!news0-mex-ad-jp!nr1.ctc.ne.jp! news.ctc.ne.jp !newsfeed.kddnet.ad.jp!newssvt07.tk!newsfeed.mesh.ad.jp!newsfeed.berkeley.edu!newsfeed.direct.ca!torn!watserv3.uwaterloo.ca!alopez-o From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Newsgroups: sci.math, news.answers daisy.uwaterloo.ca Summary: Part 25 of 31, New version Originator: alopez-o@neumann.uwaterloo.ca Originator: alopez-o@daisy.uwaterloo.ca Xref: news.jmag.net news.answers sci.answers:152 http://www.jazzie.com/ii/math/ch/ ... http://www.best.com/ ii/math/ch/ Alex Lopez-Ortiz alopez-o@unb.ca http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick Last updated: Sat Feb 19 00:01:06 2000

69. Normal Nonmetrizable Moore Space From Continuum Hypothesis Or Nonexistence Of In Sci US A. 1982 February; 79 (4) 1371 1372 Normal nonmetrizable Moore space from continuum hypothesis or nonexistence of inner models with measurable cardinals. http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=345971

70. THE INDEPENDENCE OF THE CONTINUUM HYPOTHESIS, II* Proc Natl Acad Sci US A. 1964 January; 51 (1) 105 110 THE INDEPENDENCE OF THE continuum hypothesis, II *. Paul J. Cohen. DEPARTMENT http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=300611

71. Peter Suber, Logical Systems, "Answers" À 1 is defined as the first cardinal greater than À 0 . Moreover, without the continuum hypothesis, we can prove that c = 2 À 0 (see Hunter s metatheorem http://www.earlham.edu/~peters/courses/logsys/answers.htm

Answers to Selected Exercises Peter Suber Philosophy Department Earlham College As in the exercise hand-out , page and theorem numbers refer to Geoffrey Hunter, Metalogic , University of California Press, 1971. To see what Day 1, Day 2, Day 3, etc. correspond to, see my syllabus Answer x.y corresponds to Day x , exercise y . When a question has sub-questions, then answer x.y.z corresponds to Day x , question y , sub-question z . Two systems S and S' may have the same theorems but different axioms and rules. This difference means they will differ in their proof theory. In S, some wff A might follow from another wff B, but this implication may not hold in S'. . Statement i is certainly true, in that every terminating, semantically bug-free program is obviously effective. Programming languages express what computers can do, and every step a computer takes is 'dumb' (even if putting many of these steps together is 'intelligent'). Statement ii may be true, but it is uncertain and unprovable. We'll never know whether a definite class of methods (those that are programmable) coincides with an indefinite class of methods (those that satisfy our intuition about 'dumbness'). The claim that statement B is true is called Church's Thesis, and will come up again on Day 27 (Hunter at 230ff). Statement i is false in this sense: many ineffective methods are programmable. Every method with an infinite loop is both ineffective and programmable.

72. Wikinfo | Continuum Hypothesis continuum hypothesis. from Wikinfo, an internet encyclopedia. Investigating the continuum hypothesis. Consider the set of all rational numbers. http://www.internet-encyclopedia.org/wiki.php?title=Continuum_hypothesis

73. No Title The continuum hypothesis. The logician K. Gödel (19061978) established that the continuum hypothesis is consistent with set theory. http://www.rpi.edu/~piperb/ugrad/phillip/

The Continuum Hypothesis. Phillip E. Folck Rensselaer Polytechnic Institute Undergraduate Mathematics Seminar March 23 rd AE 411 4:00pm Abstract The Continuum Hypothesis is a modern formalization of some of mankinds most philosophical questions concerning the nature of space and time. Zeno's paradox ( c . 490-435 BC), a.k.a. the stadium paradox, argues about the infinite divisiblity of time and space. The Continuum Hypothesis was conjectured by G. Cantor (1845-1918) at the end of the 19th century and has had a crucial role on the development of set theory and is at the foundations of modern mathematical analysis. The basic statement of the continuum hypothesis is: every infinite subset of is either countable or has the same cardinality as

74. The Continuum Hypothesis, Part I By W. Hugh Woodin Topology Atlas Document topd14 The continuum hypothesis, Part I. W. Hugh Woodin. From Volume 6, of TopCom. PDF file at www.ams.org; http://at.yorku.ca/t/o/p/d/14.htm

Topology Atlas Document # topd-14 The Continuum Hypothesis, Part I W. Hugh Woodin From Volume 6 , of TopCom PDF file at www.ams.org AtlasImage version Originally published in Notices of the AMS, June/July 2001 Volume 48, Number 6. Topology Atlas

75. AMCA: More On Countably Compact Spaces And The Continuum Hypothesis By Todd Eisw More on Countably Compact spaces and the continuum hypothesis presented by Todd Eisworth University of Kansas/Hebrew University of Jerusalem http://at.yorku.ca/c/a/a/o/49.htm

AMCA Document # caao-49 The 12th Summer Conference on General Topology and its Applications August 12-16, 1997 Nipissing University North Bay, ON, Canada Conference Organizers Ted Chase, Boguslaw Schreyer, Jodi Sutherland, Murat Tuncali and Stephen Watson View Abstracts Conference Homepage More on Countably Compact spaces and the Continuum Hypothesis presented by Todd Eisworth University of Kansas/Hebrew University of Jerusalem in the presence of the Continuum Hypothesis, and I would like to present a simple example or two of this in order to illustrate the technique. Date received: June 30, 1997 The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts

76. Consistency Of The Continuum Hypothesis. (AM-3) By Kurt Godel (Author) (Paperbac Buy Consistency of the continuum hypothesis. (AM3) by Kurt Godel (Author) (Paperback ) here at low prices. Consistency of the continuum hypothesis. (AM-3). http://www.rbookshop.com/mathematics/g/Kurt_Godel/Consistency_of_the_Continuum_H

AMAZON.COM Search All Products Electronics Books Classical Music DVD Kitchen Popular Music Software Toys VHS Video Video Games Enter keywords: Related Stores Math Videos Math Software Computers and Accessories Toys and Games NOTICE : All prices, availability, and specifications are subject to verification by their respective retailers. Privacy Policy info@rbookshop.com Last Modified : 6-5-2004 Consistency of the Continuum Hypothesis. (AM-3) Home Mathematics Books Kurt Godel Consistency of the Continuum Hypothesis. (AM-3) by Kurt Godel (Author) (Paperback ) Sales Rank: 452,291 At Amazon on 6-5-2004. Features Paperback: 72 pages ; Dimensions (in inches): 0.29 x 9.05 x 6.08 Publisher: Princeton Univ Pr; (September 1, 1940) Back To Top Consistency of the Continuum Hypothesis. (AM-3) Available from Amazon Price: $24.95 Updated on 6-5-2004. Home Mathematics Books Kurt Godel Search: All Products Books Magazines Popular Music Classical Music Video DVD Baby Electronics Software Outdoor Living Wireless Phones Keywords: NOTICE: All product prices, availability, and specifications

77. Sci.math FAQ: The Continuum Hypothesis Vorherige Nächste Index sci.math FAQ The continuum hypothesis. See also Nancy McGough s *continuum hypothesis article* or its *mirror*. http://www.uni-giessen.de/faq/archiv/sci-math-faq.continuum/msg00000.html

Index sci.math FAQ: The Continuum Hypothesis Subject : sci.math FAQ: The Continuum Hypothesis From alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Date : 17 Feb 2000 22:55:53 GMT Newsgroups sci.math news.answers sci.answers Summary : Part 25 of 31, New version http://www.jazzie.com/ii/math/ch/ http://www.best.com/ ii/math/ch/ Alex Lopez-Ortiz alopez-o@unb.ca http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick Index: Index sci-math-faq.continuum

78. Sci.math FAQ: The Continuum Hypothesis sci.math FAQ The continuum hypothesis. Subject sci.math FAQ The continuum hypothesis; From alopezo@neumann.uwaterloo.ca (Alex Lopez-Ortiz); http://www.uni-giessen.de/faq/archiv/sci-math-faq.ac.continuumhyp/msg00000.html

Index sci.math FAQ: The Continuum Hypothesis Subject : sci.math FAQ: The Continuum Hypothesis From alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Date : Fri, 17 Nov 1995 17:15:59 GMT Newsgroups sci.math sci.answers news.answers Sender news@undergrad.math.uwaterloo.ca (news spool owner) Summary : Part 28 of many, New version, Index: Index sci-math-faq.ac.continuumhyp

79. Transfinite Polynomials & The Continuum Hypothesis || Kuro5hin.org P. Transfinite polynomials the continuum hypothesis (Diaries) By The Writer Wed Jul 16th, 2003 at 102252 AM EST, The Writer s Diary. http://www.kuro5hin.org/story/2003/7/16/102252/834

create account help/FAQ contact links ... MLP We need your support: buy an ad premium membership k5 store Diaries By The Writer Wed Jul 16th, 2003 at 10:22:52 AM EST I'm probably missing something obvious, but surely there is a known order type which contains ordinals of the form: a + a + ... + a i i where a i are natural numbers? Because if there is, then we can map each of its elements to a real number between and 1, and hence, it must correspond to the cardinal c. This sounds too simple to be missed by mathematicians... so what's the flaw with my reasoning here? I'm sure there must be a flaw in my reasoning somewhere. Sponsors Managed Servers Managed Clusters Virtual Hosting Collocated Linux/FreeBSD Server ... As low as $45/month Root on your own FreeBSD or Linux server Very fast, triple-homed network NO hardware or setup fees, unlimited support Testimonials from K5 Users Login Make a new account Username: Password: Note: You must accept a cookie to log in. Poll Does such an order type exist? Yes, I know what it is too [please post comment] Yes, but it's impossible to describe in finite space

80. Continuum Hypothesis Article on continuum hypothesis from WorldHistory.com, licensed from Wikipedia, the free encyclopedia. Return Index continuum hypothesis. http://www.worldhistory.com/wiki/C/Continuum-hypothesis.htm

World History (home) Encyclopedia Index Localities Companies Surnames ... This Week in History Continuum hypothesis Continuum hypothesis in the news In mathematics , the continuum hypothesis is a hypothesis about the possible sizes of infinite set s. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integer s is strictly smaller than the set of 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 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 number s. 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 countable set s.

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