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

4. Infinite Ink The Continuum Hypothesis By Nancy McGough History, mathematics, metamathematics, and philosophy of Cantor's continuum hypothesis http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

5. Infinite Ink: The Continuum Hypothesis FAQ The continuum hypothesis was proposed by Georg Cantor in 1877 after he showedthat the Cantor s original formulation of the continuum hypothesis, or CH, 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:

6. Infinite Ink The Continuum Hypothesis FAQ THE continuum hypothesis FAQ By Nancy McGough (nm@noadsplease.ii.com) http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

7. Continuum Hypothesis -- From MathWorld 282) recount a generalized version of the continuum hypothesis The continuumhypothesis follows from generalized continuum hypothesis, so ZF+GCH CH . http://mathworld.wolfram.com/ContinuumHypothesis.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Set Theory Cardinal Numbers ... Szudzik Continuum Hypothesis 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 (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

8. Ernst Friedrich Ferdinand Zermelo Describes educational background and motivation towards working on set theory and the continuum hypothesis. Page lists other major contributions made by this person. http://www.stetson.edu/~efriedma/periodictable/html/Zr.html

Ernst Friedrich Ferdinand Zermelo Ernst Zermelo's father was a college professor, so Zermelo was brought up in a family where academic pursuits were encouraged. He graduated from gymnasium in 1889. At this time it was the custom for students in Germany to study at a number of different universities, and that is what Zermelo did. He studied at Berlin, Halle and Freiburg, and the subjects he studied were quite wide ranging and included mathematics, physics and philosophy. At these universities he attended courses by Frobenius, Lazarus, Fuchs, Planck, Schmidt, and Schwarz. Zermelo began to undertake research in mathematics after completing his first degree. His doctorate was completed in 1894 when the University of Berlin awarded him the degree for a dissertation on the calculus of variations. In this thesis he extended Weierstrass's method for the extrema of integrals over a class of curves to the case of integrands depending on derivatives of arbitrarily high order, at the same time giving a careful definition of the notion of neighbourhood in the space of curves. After the award of his doctorate, Zermelo remained at the University of Berlin where he was appointed assistant to Planck who held the chair of theoretical physics there. At this stage Zermelo's work was turning more towards areas of applied mathematics and, under Planck's guidance, he began to work for his habilitation thesis studying hydrodynamics.

9. 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 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

10. Aleph-1 -- From MathWorld The continuum hypothesis asserts that aleph_1c , where c is the However,the truth of the continuum hypothesis depends on the version of set theory http://mathworld.wolfram.com/Aleph-1.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Set Theory Cardinal Numbers Aleph-1 Aleph-1 is the set theory symbol for the smallest infinite set larger than Aleph-0 ), which in turn is equal to the cardinality of the set of countable ordinal numbers The continuum hypothesis asserts that , where 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, -dimensional space has the same number of points ( ) as one-dimensional space , or any finite interval of one-dimensional space (a line segment ), as was first recognized by Georg Cantor SEE ALSO: Aleph-0 Cardinality Continuum Continuum Hypothesis ... [Pages Linking Here] CITE THIS AS: Eric W. Weisstein. "Aleph-1." From

11. Continuum Hypothesis True, False, Or Neither? Is the continuum hypothesis True, False, or Neither? David J. Chalmers http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

12. Continuum Hypothesis - Wikipedia, The Free Encyclopedia Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be Hence, CH is independent of the ZermeloFränkel axiom system and of the 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 of the real numbers is , the continuum hypothesis says: This implies: 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 saying: For all ordinals Contents The size of a set Investigating the continuum hypothesis Impossibility of proof and disproof The generalized continuum hypothesis ... edit The size of a set Main article: Cardinal number To state the hypothesis formally, we need a definition: we say that two sets

13. 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

14. Continuum Hypothesis - Wikipedia, The Free Encyclopedia continuum hypothesis http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

15. Continuum Hypothesis - Wikipedia, The Free Encyclopedia There is also a generalization of the continuum hypothesis called the The generalized continuum hypothesis (GCH) states that if an infinite set s 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 of the real numbers is , the continuum hypothesis says: This implies: 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 saying: For all ordinals Contents The size of a set Investigating the continuum hypothesis Impossibility of proof and disproof The generalized continuum hypothesis ... edit The size of a set Main article: Cardinal number To state the hypothesis formally, we need a definition: we say that two sets

16. Cantor's Donut Paradox Reassessing Uncountability and the continuum hypothesis with Reference to a Novel Geometrical Analogue of the Aleph Series. http://www.cantorsdonutparadox.co.uk/

cantorsdonutparadox.co.uk UNDER CONSTRUCTION For more information contact webmaster@cantorsdonutparadox.co.uk Is your domain name available? .biz .br.com .cn.com .co.uk .com .de.com .eu.com .gb.com .gb.net .hu.com .info .jpn.com .me.uk .name .net .no.com .org .org.uk .qc.com .ru.com .sa.com .se.com .se.net .uk.com .uk.net .us .us.com .uy.com .web.com .za.com

17. Sci.math FAQ The Continuum Hypothesis sci.math FAQ The continuum hypothesis http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

18. Streaming Video - Spring 2001 continuum hypothesis May 29 June 1, 2001 Tomek Bartoszynski Continuous Images of Strongly Meager Sets http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

19. The Continuum Hypothesis The continuum hypothesis. A basic reference is Godel s ``What is Cantor s Continuum Nancy McGough s *continuum hypothesis article* or its *mirror*. 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.

20. Navier-Stokes Equations Continuum Hypothesis General Fluid Mech. Aero/Hydrodyn. Viscous Flow Compressible Flow Miscellaneous. NavierStokes Equations continuum hypothesis http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

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