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

AMCA Document # caao-49 The 12th Summer Conference on General Topology and its Applications August 12-16, 1997 Nipissing University North Bay, ON, Canada Organizers Ted Chase, Boguslaw Schreyer, Jodi Sutherland, Murat Tuncali, Stephen Watson View Abstracts Conference Homepage More on Countably Compact spaces and the Continuum Hypothesis 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 Atlas Mathematical Conference Abstracts

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

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 card( P N )) = card( R In fact, this is a deep question called the continuum hypothesis . This question results in serious problems: In the 1940's the German mathematician Goedel showed that if one denies the existence of an uncountable set whose cardinalities is less than the cardinality of the continuum, no logical contradictions to the axioms of set theory would arise. One the other hand, it was shown recently that the existence of an uncountable set with cardinality less than that of the continuum would also be consistent with the axioms of set theory. 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

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.

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

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).

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

Atlas 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 Conferences Inc.

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

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

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

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

continuum hypothesis, axiom of constructibility, continuum problem, axiom of determinateness, constructibility, determinateness, axiom, set theory, descriptive, Ackermann, continuum Back to title page 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

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

The Continuum Hypothesis Cesare Brazza History of Mathematics 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.

Trapped in a frame? Break free now! continuum hypothesis T HE C ONTINUUM H YPOTHESIS R EFERENCES Each book that can be purchased through Amazon.com is preceded by one or both of the following links: links to the hardcover edition (mnemonic: r e d =ha rd cover) links to the paperback edition (mnemonic: p u r ple= p ape r back) 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: links to any of the authors listed below. links to any info about the books, e.g., table of contents, sample chapters, reviews, etc. 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

Continuum Hypothesis I dreamt I was a number, bristling with the infinite nonrepeating pattern of decimal expansion. I felt the darkness burn and I was one with it. Out of a harsh continuum a sharp and final point defined my very self against a field of else. I felt the dark fathom's pressure, murky and malignant, control my passage slow along a narrow pass - and through a shattered lens recursive thoughts I sense: there's no where else to go. There's nothing else that is. All contents and design

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

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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

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 More Stores Ray Ban Store Sporting Goods Store Electronics Store Electronics Store ... Toy Store NOTICE : All prices, availability, and specifications are subject to verification by their respective retailers. Privacy Policy info@rbookshop.com Last Modified : 3-16-2003 Consistency of the Continuum Hypothesis. (AM-3) Home Mathematics Books Kurt Godel Consistency of the Continuum Hypothesis. (AM-3) by Kurt Godel (Paperback) Sales Rank: 408,474 List Price: $20.00 At Amazon on 3-16-2003. Features Paperback: 72 pages ; Dimensions (in inches): 0.29 x 9.05 x 6.08 Publisher: Princeton University Press; ; (September 1, 1940) ISBN: 0691079277 Consistency of the Continuum Hypothesis. (AM-3) Available from Amazon Price: $20.00 Updated on 3-16-2003. 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

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

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

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] Explain why two formal systems with the same set of theorems might not be identical. [Answer] Day 2 Which of these statements is true? "If a method is programmable (written in a programming language and executable on a machine), then it is effective." "If a method is effective, then it is programmable." [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

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

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

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

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

Continuum Hypothesis and Chaos 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. When n is arbitrary natural number, the following equation is satisfied, n On the other hand, satisfies the following equation. = m (m: arbitrary natural number greater than 1) Therefore if there is a cardinal number a between and , it would satisfy the following inequality. n a Now we consider the following series of set A j (j = 1, 2, ... ). n, e j s.t. j n j e j (j, n: natural number, e : real number) ... (A) There are various series satisfying the above condition, e.g. A j j j (j = 1, 2, ... ) Let b the cardinal number of A j (j ). Actually it depends on individual series. We can not say whether there exists a series of set A j having the cardinal number b a or not on ZF. However if it is supposed that there is a series A

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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