1. Infinite Ink: The Continuum Hypothesis By Nancy McGough History, mathematics, metamathematics, and philosophy of Cantor s continuum hypothesis. More thanks 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. 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

3. Infinite Ink: The Continuum Hypothesis FAQ The continuum hypothesis was proposed by Georg Cantor in 1877 after he showed that 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:

4. 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. [Organizer Summary] MSRI Home Page Search the MSRI Website Subject and Title Index ... webmaster@msri.org

5. Continuum Hypothesis The continuum hypothesis. Infinity has infinite ways to trouble our finite minds level of infinity after aleph0. The continuum hypothesis states simply that aleph1= c, that is http://users.forthnet.gr/ath/kimon/Continuum.htm

The Continuum Hypothesis Infinity has ... infinite ways to trouble our finite minds. This was proved by Georg Cantor in 1874. The "smallest level" of infinity has to do with countable things that can be put in some order. . This seems strange: one set is a proper subset of another and still they have the same number of elements. This is exactly the definition of infinite sets. What about rational numbers? These are a superset of the natural numbers but still of class aleph . It turns out that there is a way to put rational numbers in order: 1, 2, 1/2, 1/3, 3, 4, 3/2, 2/3, 1/4 ... (the pattern is based on a diagram so it is not obvious as shown here). Things change when we examine the real numbers. There is no way to create a complete list of reals and this was shown by Cantor with a beautiful argument, the "diagonal" one: Suppose we had such a complete list of real numbers between and 1 : r1=0.a a a r2=0.a a a r3=0.a a a Where a ij take values in 0,1,...,9 and all numbers are written with infinite number of digits (e.g. 0.2=0.20000..., 1/7=0.142857142857...). Then we can create another real r'=0.b

6. Continuum Hypothesis - Wikipedia, The Free Encyclopedia In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinitesets smaller than the set of real numbers. The continuum hypothesis states the following http://www.wikipedia.org/wiki/Continuum_hypothesis

Continuum hypothesis From Wikipedia, the free encyclopedia. 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 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. Table of contents 1 Investigating the continuum hypothesis 2 Impossibility of proof and disproof 3 The generalized continuum hypothesis 4 See also ... edit Investigating the continuum hypothesis Consider the set of all rational numbers . 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

7. The Continuum Hypothesis The continuum hypothesis. May 29, 2001 to June 1, 2001. The workshop will feature a number of lectures surveying the current insights into the continuum problem and its variations. Group Photo. Group http://www.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. [Organizer Summary] MSRI Home Page Search the MSRI Website Subject and Title Index ... webmaster@msri.org

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

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Set Theory Cardinal Numbers ... 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

9. Aleph-1 -- From MathWorld The continuum hypothesis asserts that , where c is the cardinality of the large infinite set of real numbers (called the continuum in set theory). http://mathworld.wolfram.com/Aleph-1.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Set Theory Cardinal Numbers Aleph-1 The set theory symbol for the smallest infinite set larger than Aleph-0 , and equal to the cardinality of the set of countable ordinal numbers The continuum hypothesis asserts that where c is the cardinality of the "large" infinite set of real numbers (called the continuum in set theory ). However, the truth of the continuum hypothesis depends on the version of set theory you are using and so is undecidable Curiously enough, n -dimensional space has the same number of points ( c ) as one-dimensional space , or any finite interval of one-dimensional space (a line segment ), as was first recognized by Georg Cantor Aleph-0 Cardinality Continuum ... search Eric W. Weisstein. "Aleph-1." From MathWorld A Wolfram Web Resource.

10. 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

11. Continuum Hypothesis From MathWorld continuum hypothesis from MathWorld 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 \aleph http://rdre1.inktomi.com/click?u=http://mathworld.wolfram.com/ContinuumHypothesi

12. Continuum Hypothesis - Wikipedia, The Free Encyclopedia continuum hypothesis. From Wikipedia, the free encyclopedia. Investigating the continuum hypothesis. Consider the set of all rational numbers. http://en.wikipedia.org/wiki/Continuum_hypothesis

Continuum hypothesis From Wikipedia, the free encyclopedia. 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 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. Table of contents 1 Investigating the continuum hypothesis 2 Impossibility of proof and disproof 3 The generalized continuum hypothesis 4 See also ... edit Investigating the continuum hypothesis Consider the set of all rational numbers . 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

13. Continuum Hypothesis: True, False, Or Neither? Is the continuum hypothesis True, False, or Neither? who responded to my enquiry about the status of the continuum hypothesis. This is a really fascinating subject, which I 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.)

14. Continuum Hypothesis - Wikipedia, The Free Encyclopedia continuum hypothesis. (Redirected from Generalized continuum hypothesis). Investigating the continuum hypothesis. Consider the set of all rational numbers. http://en.wikipedia.org/wiki/Generalized_continuum_hypothesis

Continuum hypothesis From Wikipedia, the free encyclopedia. (Redirected from Generalized continuum hypothesis 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 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. Table of contents 1 Investigating the continuum hypothesis 2 Impossibility of proof and disproof 3 The generalized continuum hypothesis 4 See also ... edit Investigating the continuum hypothesis Consider the set of all rational numbers . 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

15. The Continuum Hypothesis The continuum hypothesis. See also. Nancy McGough s *continuum hypothesis article* 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.

16. The Continuum Hypothesis, Part I, Volume 48, Number 6 Could the continuum hypothesis be similarly. solved? These questions are the subject of this ar For the problem of the continuum hypothesis, I. shall focus on one specific approach http://www.ams.org/notices/200106/fea-woodin.pdf

17. Continuum Hypothesis: True, False, Or Neither? Is the continuum hypothesis True, False, or Neither? Thanks to all the people who responded to my enquiry about the status of the continuum hypothesis. http://jamaica.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.)

18. The Continuum Hypothesis, Part II, Volume 48, Number 7 AUGUST2001NOTICES OF THEAMS681The ContinuumHypothesis, Part IIW dinals and if the continuum hypothesis is to be http://www.ams.org/notices/200107/fea-woodin.pdf

19. Sci.math FAQ: The Continuum Hypothesis sci.math FAQ The continuum hypothesis. Newsgroups sci.math From alopezo@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Subject sci.math http://www.faqs.org/faqs/sci-math-faq/AC/ContinuumHyp/

Usenet FAQs Search Web FAQs Documents ... RFC Index sci.math FAQ: The Continuum Hypothesis Newsgroups: sci.math alopez-o@neumann.uwaterloo.ca alopez-o@barrow.uwaterloo.ca Tue Apr 04 17:26:57 EDT 1995 Rate this FAQ N/A Worst Weak OK Good Great Current Top-Rated FAQs Are you an expert in some area? Share your knowledge and earn expert points by giving answers or rating people's questions and answers! This section of FAQS.ORG is not sanctioned in any way by FAQ authors or maintainers. Questions awaiting answers: 17142 questions related to FAQs 2985 general questions 8012 answered questions Usenet FAQs Search ... RFC Index Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update June 05 2004 @ 00:23 AM

20. Navier-Stokes Equations: Continuum Hypothesis NavierStokes Equations continuum hypothesis. Click here to see my latest discussion of the continuum hypothesis as it applies to the Navier-Stokes equations. 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. I now have a new web site on the Navier-Stokes equations. Click here to see my latest discussion of the continuum hypothesis as it applies to the Navier-Stokes equations. 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.

Page 1     1-20 of 84    1  | 2  | 3  | 4  | 5  | Next 20