1. Infinite Ink: The Continuum Hypothesis By Nancy McGough History, mathematics, metamathematics, and philosophy of Cantor's continuum hypothesisCategory Science Math Logic and Foundations Set Theory......History, mathematics, metamathematics, and philosophy of Cantor's continuum hypothesis. Morethanks coming Search the Net for continuum hypothesis . http://www.ii.com/math/ch/

mathematics T HE C ONTINUUM H YPOTHESIS By Nancy McGough nm noadsplease.ii.com Overview 1.1 What is the Continuum Hypothesis? 1.2 Current Status of CH Alternate Overview Assumptions, Style, and Terminology 2.1 Assumptions 2.1.1 Audience Assumptions 2.1.2 Mathematical Assumptions 2.2 Style 2.3 Terminology 2.3.1 The Word "continuum" 2.3.2 Ordered Sets 2.3.3 More Terms and Notation Mathematics of the Continuum and CH 3.1 Sizes of Sets: Cardinal Numbers aleph c aleph 3.1.2 CH and GCH 3.1.3 Sample Cardinalities 3.2 Ordering Sets: Ordinal Numbers 3.3 Analysis of the Continuum 3.3.1 Decomposing the Reals 3.3.2 Characterizing the Reals 3.3.3 Characterizing Continuity 3.4 What ZFC Does and Does Not Tell Us About c Metamathematics and CH 4.1 Consistency, Completeness, and Compactness of ... 4.1.1 a Logical System 4.1.2 an Axiomatic Theory 4.2 Models of ... 4.2.1 Real Numbers 4.2.2 Set Theory 4.2.2.1 Inner Models 4.2.2.2 Forcing and Outer Models 4.3 Adding Axioms to Zermelo Fraenkel Set Theory 4.3.1 Axioms that Imply CH or GCH 4.3.1.1 Explicitly Adding CH or GCH 4.3.1.2 V=L: Shrinking the Set Theoretic Universe

2. The Continuum Hypothesis A workshop featuring a number of lectures surveying the current insights into the continuum problem and its variations. MSRI, Berkeley, CA, USA; 29 May 1 June 2001. http://zeta.msri.org/calendar/workshops/WorkshopInfo/94/show_workshop

Calendar The Continuum Hypothesis May 29, 2001 to June 1, 2001 Organized by: Hugh Woodin and John Steel The workshop will feature a number of lectures surveying the current insights into the continuum problem and its variations. Group Photo Group Photo (larger version) Continuum Hypothesis Schedule Lectures on Streaming Video: This event is now over. Resources from this event, including streaming video , may be available. MSRI Home Page Search the MSRI Website Subject and Title Index webmaster@msri.org

3. Axiom Of Choice And Continuum Hypothesis Part of the Frequently Asked Questions in Mathematics. http://db.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

4. Infinite Ink: The Continuum Hypothesis FAQ The continuum hypothesis was proposed by Georg Cantor in 1877 after he showedthat the real numbers cannot be put into oneto-one correspondence with the http://www.ii.com/math/ch/faq/

Trapped in a frame? Break free now! mathematics faq T HE C ONTINUUM H YPOTHESIS FAQ By Nancy McGough nm noadsplease.ii.com This is a draft of an article that will become part of the sci.math FAQ , which is regularly posted to the sci.math news group. The continuum hypothesis was proposed by Georg Cantor in 1877 after he showed that the real numbers cannot be put into one-to-one correspondence with the natural numbers. Cantor hypothesized that the number of real numbers is the next level of infinity above the number of natural numbers. He used the Hebrew letter aleph to name the different levels of infinity: aleph_0 is the number of (or cardinality of) the natural numbers or any countably infinite set, and the next levels of infinity are aleph_1, aleph_2, aleph_3, et cetera. Since the reals form the quintessential continuum, Cantor named the cardinality of the reals c , for continuum. Cantor's original formulation of the continuum hypothesis, or CH, can be stated as either: card( R c where `card( R )' means `the cardinality of the reals.' An amazing fact that Cantor also proved is that the cardinality of the set of all subsets of the natural numbers the power set of N or P( N is equal to the cardinality of the reals. So, another way to state CH is:

5. Continuum Hypothesis: True, False, Or Neither? Is the continuum hypothesis True, False, or Neither? Thanks to all the peoplewho responded to my enquiry about the status of the continuum hypothesis. http://www.u.arizona.edu/~chalmers/notes/continuum.html

Is the Continuum Hypothesis True, False, or Neither? David J. Chalmers Newsgroups: sci.math From: chalmers@bronze.ucs.indiana.edu (David Chalmers) Subject: Continuum Hypothesis - Summary Date: Wed, 13 Mar 91 21:29:47 GMT Thanks to all the people who responded to my enquiry about the status of the Continuum Hypothesis. This is a really fascinating subject, which I could waste far too much time on. The following is a summary of some aspects of the feeling I got for the problems. This will be old-hat to set theorists, and no doubt there are a couple of embarrassing misunderstandings, but it might be of some interest to non-professionals. A basic reference is Gödel's "What is Cantor's Continuum Problem?", from 1947 with a 1963 supplement, reprinted in Benacerraf and Putnam's collection Philosophy of Mathematics . This outlines Gödel's generally anti-CH views, giving some "implausible" consequences of CH. "I believe that adding up all that has been said one has good reason to suspect that the role of the continuum problem in set theory will be to lead to the discovery of new axioms which will make it possible to disprove Cantor's conjecture." At one stage he believed he had a proof that C = aleph_2 from some new axioms, but this turned out to be fallacious. (See Ellentuck, "Gödel's Square Axioms for the Continuum", Mathematische Annalen 1975.)

6. An Intuitivistic Solution Of The Continuum Hypothesis For Definable Sets And Res An intuitivistic solution of the continuum hypothesis for definable sets and resolution of the set theoretical paradoxes http://www.farazgodrejjoshi.com/

An intuitivistic solution of the Continuum Hypothesis for definable sets and resolution of the set theoretic paradoxes. by Faraz Godrej Joshi faraz@farazgodrejjoshi.com View Paper A4 size in ( PDF format farazgodrejjoshi.pdf (74.6 KB) Download (.zip) Zipped PDF File. farazgodrejjoshi.zip (58.1 KB) Website by: Skindia Internet Pvt. Ltd.

7. Continuum Hypothesis -- From MathWorld that no contradiction would arise if the continuum hypothesis were added to conventional ZermeloFraenkel set theory. http://www.treasure-troves.com/math/ContinuumHypothesis.html

Foundations of Mathematics Set Theory Cardinal Numbers Math Contributors ... Szudzik Continuum Hypothesis Portions of this entry contributed by Matthew Szudzik 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 C (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, p. 282) recount a generalized version of the continuum hypothesis originally due to Hausdorff in 1908 which is also undecidable : is for every ? The continuum hypothesis follows from generalized continuum hypothesis, so

8. The Continuum Hypothesis The continuum hypothesis. See also. Nancy McGough's *continuum hypothesisarticle* or its *mirror*. http//www.jazzie.com/ii/math/ch/. http://db.uwaterloo.ca/~alopez-o/math-faq/mathtext/node37.html

Next: Formulas of General Interest Up: Axiom of Choice and Previous: Cutting a sphere into The Continuum Hypothesis A basic reference is Godel's ``What is Cantor's Continuum Problem?", from 1947 with a 1963 supplement, reprinted in Benacerraf and Putnam's collection Philosophy of Mathematics. This outlines Godel's generally anti-CH views, giving some ``implausible" consequences of CH. "I believe that adding up all that has been said one has good reason to suspect that the role of the continuum problem in set theory will be to lead to the discovery of new axioms which will make it possible to disprove Cantor's conjecture." At one stage he believed he had a proof that C = aleph_2 from some new axioms, but this turned out to be fallacious. (See Ellentuck, ``Godel's Square Axioms for the Continuum", Mathematische Annalen 1975.) Maddy's ``Believing the Axioms", Journal of Symbolic Logic 1988 (in 2 parts) is an extremely interesting paper and a lot of fun to read. A bonus is that it gives a non-set-theorist who knows the basics a good feeling for a lot of issues in contemporary set theory. Most of the first part is devoted to ``plausible arguments" for or against CH: how it stands relative to both other possible axioms and to various set-theoretic ``rules of thumb". One gets the feeling that the weight of the arguments is against CH, although Maddy says that many ``younger members" of the set-theoretic community are becoming more sympathetic to CH than their elders. There's far too much here for me to be able to go into it in much detail.

9. Conference In Honor Of D. A. Martin's 60th Birthday Held in coordination with the Mathematical Sciences Research Institute workshop on The continuum hypothesis. University of California, Berkeley, CA, USA; 2728 May 2001. http://www.math.berkeley.edu/~steel/martin.html

Conference in Honor of D. A. Martin's 60th Birthday May 27 - 28, 2001 The University of California, Berkeley Organizers: Stephen Jackson , University of North Texas, Denton, jackson@jove.acs.unt.edu John R. Steel , University of California, Berkeley, steel@math.berkeley.edu W. Hugh Woodin , University of California, Berkeley, woodin@math.berkeley.edu Presented under the auspices of the The University of California and in coordination with the Mathematical Sciences Research Institure workshop The Continuum Hypothesis The conference focused on topics close to Martin's work. Here is the meeting schedule, with copies of the speakers' presentations, as available. May 27, morning 8:45-9:30 : Coffee, etc. in 1015 Evans 9:30-10:30 : Theodore Slaman, University of California, Berkeley, ``High'' is definable in the partial order of the Turing degrees of the recursively enumerable sets, abstract and slides of talk 10:30-11:00 : Coffee, etc. in 1015 11:00-12:00 : Stephen Jackson, University of North Texas, A survey of the inductive analysis of L(R) assuming determinacy slides of talk 12:00-2:00 : Lunch May 27, afternoon

10. How Much Lager Is The Mightiness Of The Continuum Than The Countable Infinity? The continuum hypothesis. One assumption by Cantor, called the continuum hypothesis, CH, is that c= 1, but it has been http://hemsidor.torget.se/users/m/mauritz/math/num/pow.htm

Created 980404. Last change 981106. Previous : The mightiness of the Continuum, Cantors Diagonal Proof Up : Contents . Next : Imaginary and Complex numbers, C How much lager is the mightiness of the continuum than the countable infinity? If we have a mapping of the reals to the naturals, and all reals are mapped to one integer, then we will have at least one integer mapped to by an infinite number of reals. If it would not be so, then we could 'split' the finite number of mappings on the number n to n, n+1, n+2,n+3, and so on , and 'push' the other mappings up a desired number of steps. As an end result we would have a mapping of the reals on the integers, but Cantors diagonal proof proved this impossible, and the above follows. In the most used set theory can we define an infinite numbers of infinities where each infinity is defined as the power set set of any non empty set is defined to exist, and can be shown to always have a larger cardinality than the set it is a power set of.

11. Event Schedule Calendar. continuum hypothesis Schedule. Date, Time, Speaker, Title. Tuesday, May29, 915 am to 1015 am, Tomek Bartoszynski, Continuous images of strongly meagersets. http://zeta.msri.org/calendar/workshops/WorkshopInfo/94/schedule?WKS_ID=94

12. Continuum Hypothesis - Wikipedia continuum hypothesis. From Wikipedia, the free encyclopedia. In mathematics, thecontinuum hypothesis is a hypothesis about the possible sizes of infinite sets. http://www.wikipedia.org/wiki/Continuum_hypothesis

13. Sci.math FAQ: The Continuum Hypothesis uwaterloo.ca (Alex LopezOrtiz) Subject sci.math FAQ The continuum hypothesis Summary Part 28 of many, New version, http://www.faqs.org/faqs/sci-math-faq/AC/ContinuumHyp

sci.math FAQ: The Continuum Hypothesis Newsgroups: sci.math sci.answers news.answers From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject: sci.math DI76Mo.8s1@undergrad.math.uwaterloo.ca alopez-o@neumann.uwaterloo.ca Organization: University of Waterloo Followup-To: sci.math alopez-o@barrow.uwaterloo.ca Tue Apr 04 17:26:57 EDT 1995 By Archive-name By Author By Category By Newsgroup ... Help Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update March 05 2003 @ 01:20 AM

14. Continuum Hypothesis - Wikipedia continuum hypothesis. (Redirected from Generalized continuum hypothesis). ReferencesNancy McGough. The continuum hypothesis, http//www.ii.com/math/ch/; http://www.wikipedia.org/wiki/Generalized_continuum_hypothesis

15. Continuum Hypothesis -- From MathWorld continuum hypothesis, Portions of this entry contributed by Matthew Szudzik.The Symbolically, the continuum hypothesis is that . Gödel http://mathworld.wolfram.com/ContinuumHypothesis.html

Foundations of Mathematics Set Theory Cardinal Numbers Math Contributors ... Szudzik Continuum Hypothesis Portions of this entry contributed by Matthew Szudzik 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 C (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, p. 282) recount a generalized version of the continuum hypothesis originally due to Hausdorff in 1908 which is also undecidable : is for every ? The continuum hypothesis follows from generalized continuum hypothesis, so

16. Continuum -- From MathWorld The continuum hypothesis, first proposed by Georg Cantor, holds that the cardinalnumber of the continuum is the same as that of Aleph1. The surprising truth http://mathworld.wolfram.com/Continuum.html

Foundations of Mathematics Set Theory Cardinal Numbers Continuum The nondenumerable set of real numbers , denoted C . It satisfies and where is Aleph-0 . It is also true that However, is a set larger than the continuum. Paradoxically, there are exactly as many points C on a line (or line segment ) as in a plane , a three-dimensional space , or finite hyperspace , since all these sets can be put into a one-to-one correspondence with each other. The continuum hypothesis , first proposed by Georg Cantor holds that the cardinal number of the continuum is the same as that of Aleph-1 . The surprising truth is that this proposition is undecidable , since neither it nor its converse contradicts the tenets of set theory Aleph-0 Aleph-1 Continuum Hypothesis ... Denumerable Set Author: Eric W. Weisstein Wolfram Research, Inc.

17. Navier-Stokes Equations: Continuum Hypothesis NavierStokes Equations continuum hypothesis. In most treatments offluid mechanics, the so-called continuum hypothesis is hurriedly http://www.eng.vt.edu/fluids/msc/ns/nscont.htm

Navier-Stokes Equations Continuum Hypothesis In most treatments of fluid mechanics, the so-called continuum hypothesis is hurriedly stated during the first lecture or in the very first chapter of a text. While I think that the standard discussions are quite reasonable as far as they go, I have always felt that the additional concept of local thermodynamic equilibrium is essential in any preliminary discussion of fluid mechanics. Below I've provided a draft of my views on the subject. The basis for much of classical mechanics is that the media under consideration is a continuum. Crudely speaking, matter is taken to occupy every point of the space of interest, regardless of how closely we examine the material. Such a view is perfectly reasonable from a modeling point of view as long as the resultant mathematical model generates results which agree with experiment. Among other things, such a model permits us to use the field representation, i.e., the view in which the velocities, pressures and temperatures are taken to be piecewise continuous functions of space and time. Furthermore, it is well known that the standard macroscopic representation yields highly accurate predictions of the behavior of solids and fluids. However, most treatments of the continuum hypothesis are concerned with the widely accepted assumption of the molecular nature of matter. Clearly, as the length scales of a particular problem become smaller and smaller, the molecular structure will eventually become evident. We must certainly recognize that our continuum models cannot be accurate at length scales approaching those of the molecular world. This observation appears to be the basis for most discussions of the continuum hypothesis found in texts on fluid mechanics.

18. Continuum Hypothesis Susan Stepney's Home Page Indexcontinuum hypothesis. The continuum hypothesisis that C = 1 , that there are no such intermediate sized sets. http://www-users.cs.york.ac.uk/~susan/cyc/c/cont.htm

continuum hypothesis The smallest infinite cardinal number is (pronounced 'aleph null', or 'aleph naught'), the next is , then , and so on. There are integers. There are strictly more real numbers than integers (proof by 'diagonalisation'), in fact there are 2 reals; 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. The 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. Shaughan Lavine. Understanding the Infinite Kevin Wald.

19. Solution To Continuum Hypothesis Discovery of Dependent Sets leads to key counterexample, which leadsto solution of the continuum hypothesis. Discovery of Dependent http://home.sprintmail.com/~websterkehr/

Discovery of Dependent Sets (a.k.a. Hinged Sets) leads to key counterexample, which leads to solution of Continuum Hypothesis. Adobe Acrobat Reader is Required to Read Papers (to Install Click the Link Below) This paper contains the official solution to the Continuum Hypothesis A Study of Four Definitions and the Solution to the Continuum Hypothesis (37 Pages) This is a reference paper that contains more detail on many key Continuum Hypothesis issues Hinged Sets and the Answer to the Continuum Hypothesis (119 Pages) This paper solves Hilbert's Second Problem Extension on Turing Machines (17 Pages) De Witte's experiment disproves Einstein's theory that there is no Universal Reference Frame Understanding the De Witte Experiment (Physics) Click Here to download Adobe Acrobat Reader Email author at: webster.r.kehr@mail.sprint.com See a Picture of the Author In case you can't get Adobe to work, here is the Abstract and Overview of my solution to the Continuum Hypothesis paper: Abstract: When a set is defined, one of the key things that must be done is determine the "cardinal number" of the set. The "cardinal number" represents its "size," meaning "how many" elements the set has. While many axioms and definitions have been developed to attempt to set up a consistent method of determining the "size" of transfinite sets, in fact there is still much work to be done in studying different ways to determine the actual "size" of some transfinite sets.

20. 2. Continuum Hypothesis 2. SOME FORMULATIONS OF continuum hypothesis. Introduce the following notations. Now,there are two following main formulations of continuum hypothesis 8. http://members.tripod.com/vismath1/zen/zen2.htm

2. SOME FORMULATIONS OF CONTINUUM HYPOTHESIS X X X N N N D . Since D has, by the well-known Cantor's theorem, the power C D C 1) The classical Cantor Continuum Hypothesis formulation: C 2) The generalized Continuum Hypothesis formulation, by Cohen: P , where P ) is the power-set of any set A A ] by the following estimation of the Continuum Cardinality: "Thus, C is greater than n , where , and so on. " (p.282) [ ]. Therefore, we shall even not try to imagine visually a set of integers of a cardinality succeeding , and use the following most weak formulation of Continuum Hypothesis. 3) Whether there exists a set of integers, say M , such that a 1-1-correspondence between the set M and the set D of all real numbers (proper fractions, geometrical points) of the segment [0,1] can be realized? That is M M C ] ?, where M M , construct such the 1-1-corerespondence, and prove that the set M has the continual cardinality C NEXT VisMath HOME

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