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Continuum Hypothesis:     more books (48)

Logic numbers and the continuum hypothesis (Transfigural mathematics series) by Lere Shakunle, 1991 The Consistency of the Continuum Hypothesis by Kurt Goedel, 1951 The Consistency of the Axiom of Choice and of the Continuum-Hypothesis by Kurt GODEL, 1951 The Consistency of the Axiom of Choice and of the Generalized Continuum- Hypothesis with the Axioms of Set Theory by Kurt Godel, 1961 On the consistency of the generalized continuum hypothesis (Polska Akademia Nauk. Instytut Matematyczny. Rozprawy matematyczne) by Ladislav Rieger, 1963 A Brief Tale of Infinity by H. Vic Dannon, 2007 Metacognitions in proneness towards hallucinations and delusions [An article from: Behaviour Research and Therapy] by F. Laroi, M. Van der Linden, Descriptive Set Theory and Definable Forcing (Memoirs of the American Mathematical Society) by Jindrich Zapletal, 2004-01 Greek Text of Jeremiah: A Revised Hypothesis by Sven Soderlund, 1987-03-01 Soul Hypothesis: Investigations into the Existence of the Soul by Mark C. Baker, Stewart Goetz, 2010-12-16 Real Numbers, Generalizations of the Reals, and Theories of Continua (Synthese Library) The Spirit of Science: From Experiment to Experience Language Instinct' Debate: Revised Edition by Geoffrey Sampson, 2005-04-01 Hunter vs. farmer Theory: Hypothesis, Thom Hartmann, Attention-deficit Hyperactivity Disorder, Adult Attention Deficit Hyperactivity Disorder, Adaptive ... Hard and Soft Science, Continuum Concept.

lists with details

41. AMCA: More On Countably Compact Spaces And The Continuum Hypothesis Presented By
    More on Countably Compact spaces and the continuum hypothesis by ToddEisworth University of Kansas/Hebrew University of Jerusalem  
    http://at.yorku.ca/cgi-bin/amca/caao-49

42. The Continuum Hypothesis
    one can show that card(P(N)) = card(R). In fact, this is a deep questioncalled the continuum hypothesis. This question results in  
    http://pirate.shu.edu/projects/reals/infinity/answers/conthyp.html
    
    Extractions: Example: Is there a cardinal number c with ? What is the most obvious candidate ? We need to find a set whose cardinality is bigger than N and less that that of R . The most obvious candidate would be the power set of N . However, one can show that In fact, this is a deep question called the continuum hypothesis . This question results in serious problems: Hence, it seems impossible to decide this question with our usual methods of proving theorems. Such undecidable questions do indeed exist for any reasonably complex logical system (such as set theory), and in fact one can even prove that such 'non-provable' statements must exist. To read more about this fascinating subject, look at the book Goedel's Proof or Goedel, Escher, Bach

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

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

45. Atlas: More On Countably Compact Spaces And The Continuum Hypothesis By Todd Eis
    More on Countably Compact spaces and the continuum hypothesis presentedby Todd Eisworth University of Kansas/Hebrew University of Jerusalem  
    http://atlas-conferences.com/c/a/a/o/49.htm

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

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

48. Continuum Hypothesis. Alternative Set Theories
    Trying to prove the continuum hypothesis, Cantor developed his theory of transfiniteordinal numbers. The Independence of the continuum hypothesis. Proc. Nat.  
    http://linas.org/mirrors/www.ltn.lv/2001.03.27/~podnieks/gt6_2.html
    
    Extractions: continuum hypothesis, axiom of constructibility, continuum problem, axiom of determinateness, constructibility, determinateness, axiom, set theory, descriptive, Ackermann, continuum Back to title page 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

49. The Continuum Hypothesis
    The continuum hypothesis. Georg Cantor. The continuum hypothesis arisesin the context of an inevitable evolutionary advance in mathematics.  
    http://www.math.rutgers.edu/courses/436/436-s00/Papers2000/brazza.html
    
    Extractions: Rutgers, Spring 2000 "Infinity is up on trial..." (Bob Dylan, Visions of Johanna , cited in In the Light of Logic ), pg. 28. These five words suffice to summarize the essence of Cantor's work. Cantor was tormented by opposition throughout his career. After conceiving and then proving his theorems on infinite sets, Cantor struggled against the negative reactions of his peers. It was not until the end of his lifetime that Cantor received the recognition he deserved. Cantor, a devout Christian, always held to his beliefs because to him, they came directly from G-d. "Where G-d was concerned, it was impossible to entertain hypotheses. There were no alternatives to be considered" (p. 238, Georg Cantor ). Georg Ferdinand Ludwig Philipp Cantor contributed greatly not only to discrete mathematics, but to every science based in mathematics. "Whatever the disappointments Cantor was to suffer, his transfinite set theory represented a revolution in the history of mathematics. Not a revolution in the sense of returning to ear lier starting points, but more a revolution in the sense of overthrowing older, established prejudices against the infinite in any actual, completed form." (Pg. 118, Georg Cantor). With his theory of sets and his introduction of the concept of infinite nu mbers, Cantor broke through the barriers of previous generations, and has allowed for the further exploration of areas that were previously unattainable.

50. Erowid Experience Vaults: Aleph-2 - The Continuum Hypothesis
    Make a donation today! The continuum hypothesis Aleph2 by 77k, DOSE T+ 000, 6 mg, oral, Phenethylamines - Other, (powder / crystals).  
    http://www.erowid.org/experiences/exp.php?ID=14854

51. Infinite Ink: Continuum Hypothesis References
    Chalmers, David. Is the continuum hypothesis True, False, or Neither? . Gibbs, Phil. Leary, Christopher C. The Measure Problem and the continuum hypothesis .  
    http://www.informatik.tu-darmstadt.de/RBG/service/FAQ2/Anleitungen/procmail/www.
    
    Extractions: The links give the book's description and price. Sometimes there are comments submitted by readers of the book or an interview with the author of the book. If you've read the book, how about submitting your thoughts about the book?! If you decide to buy some of these books, many of which are discounted, please buy them through these links. This will not increase the price you pay but it will help to support Infinite Ink. This service is provided by Infinite Ink in association with Amazon.com Books. Want to Help? Please let me know about: Burgess, John P. and Rosen, Gideon. A Subject With No Object: Strategies for Nominalistic Interpretation of Mathematics . Clarendon Pr, 1997.

52. Continuum Hypothesis
    continuum hypothesis. I dreamt I was a number, bristling with theinfinite nonrepeating pattern of decimal expansion. I felt the  
    http://www.visi.com/~shemus/poems/continuum.html

53. Continuum Hypothesis - Acapedia - Free Knowledge, For All
    continuum hypothesis. From Wikipedia, the free encyclopedia. In mathematics, thecontinuum hypothesis is a hypothesis about the possible sizes of infinite sets.  
    http://acapedia.org/aca/Continuum_hypothesis

54. Consistency Of The Continuum Hypothesis. (AM-3) By Kurt Godel (Paperback)
    Buy Consistency of the continuum hypothesis. (AM3) by Kurt Godel (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

55. Peter Suber, Logical Systems, "Answers"
    À 1 is defined as the first cardinal greater than À 0 . Moreover, without thecontinuum 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.

56. Peter Suber, Logical Systems, "Exercises"
    Answer. For a period in his career, Kurt Gödel believed that c= À 2 . Was this belief compatible with the continuum hypothesis?  
    http://www.earlham.edu/~peters/courses/logsys/exercise.htm
    
    Extractions: Exercises Peter Suber Philosophy Department Earlham College Day 1 ... Day 29 Selected exercises are answered in a separate 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 Day 1 If we specify a formal system by giving only its language, axioms, and rules, then we are omitting its theorems. Why isn't this inadequate? [Answer] What's wrong with this statement: "One-to-one correspondence makes no sense applied to infinite sets, for the method of pairing-off can only be effective for a finite number of steps"? Explain how it is that an infinite set can be put into 1-1 correspondence with one of its own proper subsets. An example is not enough. Why is this kind of 1-1 correspondence not a contradiction? Prove that a set is infinite iff it can be put into 1-1 correspondence with at least one of its proper subsets.

57. Sci.math FAQ: The Continuum Hypothesis
    sci.math FAQ The continuum hypothesis. Subject sci.math FAQ The ContinuumHypothesis; 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

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

59. Continuum Hypothesis And Chaos
    continuum hypothesis and Chaos continuum hypothesis insists thatthere isn't any cardinal number between À 0 and À. Cohen has  
    http://www.geocities.com/tontokohirorin/mathematics/continuum/continuum.htm
    
    Extractions: Continuum hypothesis insists that there isn't any cardinal number between and . Cohen has proved the continuum hypothesis is independent from ZF axioms. Therefore as long as considering cardinal number of a certain set on ZF axioms, it may happen that cardinal number doesn't satisfy the continuum hypothesis. I thought there might be found the set which has the cardinal number between and in a chaotic phenomenon.

60. Mathematics Encyclopedia -- Platonic Realms
    continuum n. In mathematics, the real numbers or real number line. continuum hypothesisn. The claim that there is no set of intermediate cardinality between  
    http://www.mathacademy.com/cgi-bin/ref_main.cgi?continuum hypothesis

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