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
Extractions: 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?
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
Extractions: 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")
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
Extractions: 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:
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
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
Extractions: 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
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
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
Extractions: 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.
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
Extractions: 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
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
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
Extractions: 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.
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
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/
Extractions: 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
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
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
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
Extractions: Index 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
Extractions: 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
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
Extractions: World History (home) Encyclopedia Index Localities Companies Surnames ... This Week in History 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. 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.
