81. SS > Factoids > Continuum Hypothesis The smallest infinite cardinal number is 0 (pronounced 'aleph null', or 'aleph naught'), the next is (by definition) 1, then 2, and so on. http://www-users.cs.york.ac.uk/~susan/cyc/c/cont.htm

@import url(../../stepney.css); home factoids continuum hypothesis The smallest infinite cardinal number is (pronounced 'aleph null', or 'aleph naught'), the next is (by definition) , then , and so on. There are integers. There are strictly more real numbers than integers (proof by 'diagonalisation'), in fact there are 2 reals (proof by construction of a one-one mapping between the reals and the powerset of ); 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. Cantor's 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, and is undecidable.

82. The Continuum Hypothesis, Part I, Volume 48, Number 6 J UNE /J ULY 2001 N OTICES OF THE AMS 567 The Continuum Hypothesis, Part I W. Hugh Woodin Introduction Arguably the most famous formally unsolvable problem of mathematics is http://www.ams.org/notices/200106/fea-woodin.pdf

83. Comments And Homework In Chapter 1 File Format PDF/Adobe Acrobat Quick View http://www.math.uiowa.edu/~jsimon/COURSES/M132Fall07/M132Fall07_Handout_OnChapte

84. Axiom Of Choice And Continuum Hypothesis Alex LopezOrtiz Fri Feb 20 214530 EST 1998 http://www.cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node34.html

Next: The Axiom of Choice Up: Frequently Asked Questions in Mathematics Previous: Master Mind Axiom of Choice and Continuum Hypothesis The Axiom of Choice Cutting a sphere into pieces of larger volume The Continuum Hypothesis Alex Lopez-Ortiz Fri Feb 20 21:45:30 EST 1998

85. 연속체 가설 : Definition Of 연속체 가설 And Synonym Of 연속체 가� Translate this page Continuum hypothesis - Wikipedia, the free encyclopedia Establishing the truth or falsehood of the continuum hypothesis is the first of . http://others.sensagent.com/연속체 가설/ko-ko/

86. Continuum Hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor, about the possible sizes of infinite sets. Cantor introduced the concept of http://english.turkcebilgi.com/Continuum hypothesis

EnglishInfo Search Türkçe Home Information Translation ... continuum hypothesis Information about continuum hypothesis continuum hypothesis Information about continuum hypothesis Double click any English word, to find Turkish meaning fluid mechanics In mathematics , the continuum hypothesis (abbreviated CH ) is a hypothesis , advanced by Georg Cantor , about the possible sizes of infinite sets . Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he gave two proofs that cardinality of the set of integers is strictly smaller than that of the set of real numbers . His proofs, however, give no indication of the extent to which the cardinality of the natural numbers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question. It states: There is no set whose size is strictly between that of the integers and that of the real numbers. In light of Cantor's theorem that the sizes of these sets cannot be equal, this hypothesis states that the set of real numbers has minimal possible cardinality. The name of the hypothesis comes from the term the continuum for the real numbers.

87. Information 30 posts 9 authors - Last post Oct 18Hartman appears to be assuming the continuum hypothesis in the above passage. Even at the time of his death, wasn t it understood by all http://www.sciencechatforum.com/viewtopic.php?f=47&t=16412&start=0

88. Continuum Hypothesis | Facebook Welcome to the Facebook Community Page about Continuum hypothesis, a collection of shared knowledge concerning Continuum hypothesis. http://www.facebook.com/pages/Continuum-hypothesis/112429132106143

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89. The Continuum Hypothesis Problems Up Infinity cardinal numbers Previous The cardinality of The continuum hypothesis We showed in Proposition 1 that is the smallest infinite cardinal. http://mathcircle.berkeley.edu/BMC3/infinity/node12.html

Next: Problems Up: Infinity: cardinal numbers Previous: The cardinality of The continuum hypothesis We showed in Proposition that is the smallest infinite cardinal. It can be shown that there a next smallest cardinal called ; i.e., the only cardinals strictly smaller than are the finite ones and . Next come , .... Where does fit into this list, if anywhere? (A priori, it could be bigger than for every .) We know that , because we proved that . Cantor conjectured Continuum Hypothesis: In other words, he believed that there is no set whose cardinality is strictly between that of and that of The role of the continuum hypothesis in set theory is similar to the role of the parallel postulate in plane geometry. The parallel postulate (that given a line and a point not on , there exists a unique line through that does not intersect ) cannot be disproved from the other axioms of plane geometry, because it is actually true for the euclidean model of geometry. On the other hand, the parallel postulate cannot be proved either, since it is false in various noneuclidean models of geometry which do satisfy all the other axioms. Therefore the parallel postulate, or its negation, may be taken as a new axiom. Which one you choose will depend on your vision of what geometry is supposed to be. Similarly, whether you choose to accept the continuum hypothesis will depend on your idea of what a set is supposed to be.

90. Continuum Hypothesis - Definition Of Continuum Hypothesis By The Free Online Dic Still in typescript, this text contains the lecture notes for a course Cohen (19342007) taught at Harvard in spring 1965, shortly after his work on the continuum hypothesis. http://www.thefreedictionary.com/continuum hypothesis

91. CiteULike: Time And Classical And Quantum Mechanics: Indeterminacy Vs. Discontin by P Lynds Cited by 47 - Related articles http://www.citeulike.org/user/tictacgo/article/4836755

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92. Springer Online Reference Works In 1939 K. G del established the unprovability of the negation of the generalized continuum hypothesis (and hence the unprovability of the negation of the continuum hypothesis http://eom.springer.de/c/c025790.htm

93. JSTOR: An Error Occurred Setting Your User Cookie Your browser may not have a PDF reader available. Google recommends visiting our text version of this document. http://www.jstor.org/stable/2687835

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94. AUTHOR Mike TITLE Kissing Cousins DATE 1/30 - Meehawl - All Jan 3, 2005 /a For me, I see there s the a href= http//mathworld.wolfram.com/ ContinuumHypothesis.html Continuum /a a http://meehawl.com/Blogfiles-blogger/EXPORT-MV/2005_01_01_mtimport.txt

95. فرض متصله > Continuum Hypothesis (KudoZ) Farsi (Persian) to English translation of Continuum Hypothesis Mathematics Statistics (Science). http://www.proz.com/kudoz/farsi_persian_to_english/mathematics_statistics/204374

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