61. Identity Of Particles And Continuum Hypothesis S13.001 Identity of Particles and continuum hypothesis. AlexanderA. Berezin (McMaster University). Why all electrons are the same? http://www.eps.org/aps/meet/APR01/baps/abs/S2380001.html

Previous abstract Graphical version Next abstract Session S13 - General Physics. ORAL session, Monday afternoon, April 30 Room 12-13, Renaissance Hotel Identity of Particles and Continuum Hypothesis Alexander A. Berezin (McMaster University) Part S of program listing

62. Identity Of Particles And Continuum Hypothesis Previous abstract Graphical version Text version Next abstractSession S13 General Physics. ORAL session, Monday afternoon http://www.eps.org/aps/meet/APR01/baps/abs/G2380001.html

Previous abstract Text version Next abstract Session S13 - General Physics. ORAL session, Monday afternoon, April 30 Room 12-13, Renaissance Hotel Identity of Particles and Continuum Hypothesis Alexander A. Berezin (McMaster University) Part S of program listing

63. PlanetMath: Generalized Continuum Hypothesis generalized continuum hypothesis, (Axiom). The generalized continuum hypothesisstates that for any infinite cardinal there is no cardinal such that . http://planetmath.org/encyclopedia/GeneralizedContinuumHypothesis.html

Math for the people, by the people. Encyclopedia Books Papers Expositions ... Random Login create new user name: pass: forget your password? Main Menu the math Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports information Docs Classification News Legalese ... TODO List generalized continuum hypothesis (Axiom) The generalized continuum hypothesis states that for any infinite cardinal there is no cardinal such that Equivalently, for every ordinal Like the continuum hypothesis , the generalized continuum hypothesis is known to be independent of the axioms of ZFC "generalized continuum hypothesis" is owned by Evandar view preamble View style: HTML with images page images TeX source See Also: continuum hypothesis cardinality axiom of choice Zermelo-Fraenkel axioms Other names: generalised continuum hypothesis GCH Keywords: cardinality, cardinal Cross-references: ZFC axioms independent continuum hypothesis ... infinite This is version 6 of generalized continuum hypothesis, born on 2002-01-03, modified 2002-08-01. Object id is 1184, canonical name is GeneralizedContinuumHypothesis.

64. PlanetMath: \Diamond_S$ ( ) owned by Henry. diamond is equivalent to club and continuumhypothesis ( is equivalent to and continuum hypothesis) owned by Henry. http://planetmath.org/encyclopedia/D/

Math for the people, by the people. Encyclopedia Books Papers Expositions ... Random Login create new user name: pass: forget your password? Main Menu the math Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports information Docs Classification News Legalese ... TODO List PlanetMath Encyclopedia (browse by subject) - 1 entry - 1 entry - 1 entry - 1 entry A - 215 entries B - 164 entries C - 534 entries D - 223 entries DAG (acyclic graph) owned by Logan Darboux's theorem (analysis) owned by mathwizard proof of Darboux's theorem owned by ariels Darboux's Theorem (symplectic geometry) owned by bwebste DCC (descending chain condition) owned by antizeus DCT (discrete cosine transform) owned by akrowne de Bruijn digraph owned by vampyr decide (decision problem) owned by Henry decision problem owned by Henry deck transformation owned by Dr_Absentius deck transformation (deck transformation) owned by Dr_Absentius decomposition (complementary subspace) owned by rmilson decomposition group owned by djao Dedekind cuts owned by rmilson Dedekind domain owned by saforres Dedekind-Hasse norm (Dedekind-Hasse valuation) owned by Henry Dedekind-Hasse valuation (Dedekind-Hasse valuation) owned by Henry Dedekind-Hasse valuation owned by Henry Dedekind infinite owned by Evandar Dedekind zeta function owned by bwebste deduction (deductions are ) owned by Henry deductions are delta 1 (deductions are ) owned by Henry deductions are owned by Henry definable owned by Timmy

65. Www.math.niu.edu/~rusin/known-math/99/not_CH From rupert4050@mydeja.com Subject Re continuum hypothesis Date Thu, 16 Dec1999 011410 GMT Newsgroups sci.math Keywords Natural assumptions which http://www.math.niu.edu/~rusin/known-math/99/not_CH

From: rupert4050@my-deja.com Subject: Re: Continuum Hypothesis Date: Thu, 16 Dec 1999 01:14:10 GMT Newsgroups: sci.math Keywords: Natural assumptions which imply negation of Continuum Hypothesis In article such that <=x <=1 and <=y <=1, and for each x the set of y such that is in S is countable. That set is pretty thin in the unit square, right? Now, consider its reflection in the diagonal y=x. That's the set of points such that such that :beta

66. Www.math.niu.edu/~rusin/known-math/99/CH jeremy@jboden.demon.co.uk Subject Re Question Date Tue, 23 Feb 1999 233916+0000 Newsgroups sci.math Keywords What is the continuum hypothesis ? http://www.math.niu.edu/~rusin/known-math/99/CH

From: Jeremy Boden Subject: Re: Question Date: Tue, 23 Feb 1999 23:39:16 +0000 Newsgroups: sci.math Keywords: What is the "Continuum Hypothesis"? In article , Pepero , "Nick Tsocanos"

67. 1. Introduction THEORETICAL ABSTRACTIONS. ABSTRACT As is known, G.Cantor formulatedhis famous continuum hypothesis in the end of the XIX Century. http://www.mi.sanu.ac.yu/vismath/zen/zen1.htm

68. How Many Real Numbers Are There? The proposal that 2 (aleph0) = aleph-1 became known as Cantor'scontinuum hypothesis. It turned out to be intimately connected http://www.maa.org/devlin/devlin_6_01.html

Devlin's Angle June 2001 How many real numbers are there? How many real numbers are there? One answer is, "Infinitely many." A more sophisticated answer is "Uncountably many," since Georg Cantor proved that the real line the continuum cannot be put into one-one correspondence with the natural numbers. But can we be more precise? Cantor introduced a system of numbers for measuring the size of infinite sets: the alephs. The name comes from the symbol Cantor used to denote his infinite numbers, the Hebrew letter aleph a symbol not universally available for web pages. He defined an entire infinite hierarchy of these infinite numbers (or cardinals), aleph-0 (the first infinite cardinal, the size of the set of natural numbers), aleph-1 (the first uncountable cardinal), aleph-2, etc. The infinite cardinals can be added and multiplied, just as the finite natural numbers can, only it's much easier to learn the answers. The sum or product of any two infinite cardinals is simply the larger of the two. You can also raise any finite or infinite cardinal to any finite or infinite cardinal power. And this is where things rapidly become tricky. To pick the simplest tricky case, if K is an infinite cardinal, what is the value of 2

69. 2. Epistemic Modalities We may for example say of a child that it fails to judge whether the continuum hypothesisis true simply because the continuum hypothesis is beyond the grasp of http://www.hf.uio.no/filosofi/njpl/vol1no1/beliefs/node2.html

Next: 3. Naming in Belief Up: On Beliefs Previous: 1. Introduction 2. Epistemic Modalities Nathan Salmon has, I think, in his book Frege's Puzzle , made a strong case for holding that the principle of substitutivity of coreferential terms holds in belief contexts, and that there are such cases as I have pointed out where a person believes a proposition p and its negation because the person is disposed to assent to one sentence s which expresses p and at the same time to assent to a sentence s ' which expresses the negation of p . Salmon also considers a situation where S is disposed to assent to one sentence s which expresses the proposition p and at the same time expressly withholding judgment with respect to the same sentence s , and his analysis of the situation gives, I think, a quite plausible account of what is going on. The reader is referred to his discussion. In the following I briefly sketch Salmon's analysis. I then make some refinements of his analysis which make it somewhat more transparent why many, in fact most, people have had the intuition that we for example cannot infer that S believes that Tully is an author from the fact that Cicero is Tully and S believes that Cicero is an author. I also want to suggest that my refinements make it possible to provide a new solution to the problem concerning when we can quantify into belief contexts, and to account for Donnellan's distinction between a referential and an attributive use of definite description. Before suggesting these refinements I give some examples which should make it clear, I hope, that some analysis along the lines suggested by Salmon must be the appropriate kind of analysis. The examples should also provide ample evidence for my principle DP.

70. Orðasafn: G GCH , = general continuum hypothesis. general. 1 víðtækur general position.= 5 generalized, general continuum hypothesis, - general mean. http://www.hi.is/~mmh/ord/safn/safnG.html

G g. c. d. greatest common divisor Galois extension Galois-útvíkkun, = normal separable extension Galois field endanlegt svið, Galois-svið, = finite field Galois group Galois-grúpa. Galois theory Galois-kenning. gamble game of chance game , spil, leikur, keppni. game of chance líkindaleikur (), áhættuleikur (), = gamble game of hazard game of hazard game of chance game rule leikregla. game theory spilfræði, spilafræði, leikjafræði, keppnisfræði, = theory of games game tree leiktré. gamma function gammafall, -> factorial function gap , gloppa, eyða, -> lacunary value gap series gloppuröð, eyðuröð, = lacunar series lacunary series gap theorem gloppusetning, eyðusetning. gauge , kvarði. gauge group kvarðagrúpa, kvörðunargrúpa. gauge transformation umkvörðun, kvarðafærsla, kvarðabreyting. Gauss curvature Gauss-krappi, Gauss-sveigja, heildarkrappi, heildarsveigja, = total curvature Gauss distribution Gauss-dreifing, normleg dreifing, = normal distribution Gauss elimination method reiknirit Gauss, útrýmingaraðferð Gauss, = Gaussian algorithm Gauss number plane Gaussian plane Gaussian algorithm Gauss elimination method Gaussian complex integer Gaussian integer Gaussian curve Gauss-ferill, skekkjuferill, normlegur dreififerill, =

71. FOM: Cantor'sTheorem & Paradoxes & Continuum Hypothesis FOM Cantor'sTheorem Paradoxes continuum hypothesis. Kanovei kanovei@wmwap1.math.uniwuppertal.deMon, 12 Feb 01 202045 +0100 http://www.cs.nyu.edu/pipermail/fom/2001-February/004767.html

Kanovei kanovei@wmwap1.math.uni-wuppertal.de Mon, 12 Feb 01 20:20:45 +0100 Previous message: Next message: Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... neilt@mercutio.cohums.ohio-state.edu Cantor's theorem does not depend on the assumption that the power set of the continuum exists. Once again, the proof has two important issues, 1) technically - the diagonal method (which in fact was used, perhaps, earlier by Du Bois Reymond in "scaling" sequences, near 1872) 2) foundationally - the postulate that all elements of P(N) (or P(X),for "arbitrary" X - which was not much meaningful for a XIX century mathematician) are "already given" and no new ones can appear in the course of the proof. The rest is a couple of lines of ordinary transformations. Something similar to 2) appears in many paradoxes, say those of Liar and Barber in the form of a mess between "has been" and "has been and will always be". V.Kanovei Previous message: Next message: Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]

72. FOM: Cantor'sTheorem & Paradoxes & Continuum Hypothesis FOM Cantor'sTheorem Paradoxes continuum hypothesis. Andrej BauerAndrej.Bauer@andrej.com 11 Feb 2001 223137 +0100 Previous http://www.cs.nyu.edu/pipermail/fom/2001-February/004754.html

Andrej Bauer Andrej.Bauer@andrej.com 11 Feb 2001 22:31:37 +0100 Previous message: Next message: Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] "Robert Tragesser" < rtragesser@hotmail.com Isn't it fair to say that that proof of Cantor's Theorem is a capital example of the sort of purely logical, nonconstructive proof which motivated Brouwer's churlish observations about the logical? Is it right to say that a constructive proof of Cantor's Theorem would provide an answer to the truth of the Generalized Continuum Hypothesis? (I mean of course a constructive proof within Cantor-Zermelo set theories.) A "yes" here would be good for making dramatically proofs. http://andrej.com Previous message: Next message: Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]

73. Gödel, Kurt (1906-1978) -- From Eric Weisstein's World Of Scientific Biography Gödel showed that no contradiction would arise if the continuum hypothesiswere added to conventional ZermeloFraenkel set theory. http://scienceworld.wolfram.com/biography/Goedel.html

Branch of Science Mathematicians Nationality American ... Austrian Austrian-American mathematician who proved that, if you begin with any sufficiently strong consistent system of axioms there will always be statements within the system governed by those axioms that can neither be proved or disproved on the basis of those axioms Hence, it in undecidable on the basis of those axioms whether the system contains paradoxes The formal statement of this fact is known as which states that if T is a set of axioms in a first-order language, and a statement p holds for any structure M satisfying T , then p can be formally deduced from T in some appropriately defined fashion. 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 Additional biographies: MacTutor (St. Andrews)

74. Zermelo Biography from the MacTutor History of Mathematics archive.Category Science Math History People Zermelo, Ernst...... Cantor had put forward the continuum hypothesis in 1878, conjecturing that everyinfinite subset of the continuum is either countable (ie can be put in 11 http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Zermelo.html

75. Untitled The continuum hypothesis. This *continuum hypothesis*, as it is called,was number 1 on Hilbert's famous list of problems in 1900. http://www.maths.monash.edu.au/research/seminars/lunch/310701.html

LUNCHTIME MATHS These talks will be self-contained presentations of mathematical concepts and theorems. They are intended to be accessible to all mathematics students. Coffee, tea and drinks will be provided. 1 pm, Tuesday 31st July 2001 The Continuum Hypothesis Associate Professor John Stillwell Monash University Most mathematicians use the continuum of real numbers (the "number line"), but no one really understands it. Ever since 1874, when Cantor discovered that there are more real numbers than integers, the central question of set theory has been: how many real numbers are there? Cantor believed the answer to be: the smallest infinity beyond the infinity of integers. This *continuum hypothesis*, as it is called, was number 1 on Hilbert's famous list of problems in 1900. It is still open, and perhaps may never be settled. In this talk I'll explain the basic theory of real numbers and sets which, if not actually making the continuum clear, at least shows why it is mysterious. I'll also mention some recent developments, aimed at *disproving* the continuum hypothesis.

76. Kurt Seifried - Informaiton Security / Articles / Interview With Elias Levy The question of whether the cardinality of the real numbers is alef one is calledthe continuum hypothesis (since the real numbers are used to represent a http://www.seifried.org/security/articles/20011015-elias-levy-interview.html

Interview with Elias Levy (Bugtraq) Kurt Seifried, kurt@seifried.org Bugtraq is probably the best security mailing list around. However while the quasi-founder (technically Aleph1 didn't start Bugtraq as I was surprised to find out) is quite prominent online I wasn't able to find any detailed information about him or Bugtraq (except for one old interview). So here for you to enjoy is an interview with Aleph1. Kurt: Where does the name Aleph1 come from? Elias: Its comes from transfinite mathematics. There exists many "infinite" numbers or sets. The first infinite number is small omega or alef null. It is also called countable infinity. Many infinite sets can be mapped one-to-one with each other. For example, the set of all natural numbers can be mapped one-to-one with the set of odd natural numbers. Yet one is a subset of the other. Both these sets are said to have a cardinality of alef null. Alef One is the first cardinal number after alef null (i.e. the first set that cannot be mapped one-to-one to a set of cardinality alef null). Alef one is a funny number. One of the reasons its difficult to grasp is because it is a regular number. A number 'n' is regular if it cannot be represented by the sum of less than 'n' ordinals less than 'n'. Of all the ordinals between and alef one only 0, 1, 2 and alef null are regular. You can't write as the sum of less than terms less than (doesn't make sense). Nor can you write 1 as the sum of less than 1 terms less than 1. Nor can you write 2 as the sum of less than 2 terms less than 2. If you live in an universe were all you known is the number 1 how can you grasp the number 2? Alef One is also regular. Its difficult to reach alef one from bellow.

77. The Continuum Hypothesis next up previous contents Next Formulas of General Interest Up Axiom ofChoice and Previous Cutting a sphere into. The continuum hypothesis. http://cage.rug.ac.be/~hvernaev/FAQ/node38.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 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.

78. Transfinite Numbers And Set Theory The continuum hypothesis. It's natural to ask real numbers. The continuumhypothesis states that such is not the case. Whether this http://www.math.utah.edu/~alfeld/math/sets.html

Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah Transfinite Numbers and Set Theory Note: A much more thorough and precise discussion of the topics illustrated here is the article Set Theory in the Macropedia of the Encyclopedia Britannica (1992 edition). Basic Concepts and Notation How could one generalize the concept of a natural number beyond infinity? It turns out that there is a natural way that leads to surprising discoveries. It's based on the concept of a set . According to George Cantor (1845-1918), the founder of set theory), The individual objects of the set are its elements. A set may have no elements, in which case it is called the empty set and denoted by There is only one empty set. Sets can be defined in words, or by listing the elements between curly braces separated by commas, or between curly braces containing some other defining symbols. A set is finite if it's empty or it contains a finite number of elements. It is

79. 5.2 Continuum Hypothesis DYNAMICS, © Howard D. Curtis. 5.2 continuum hypothesis. At thispoint let us invoke the continuum hypothesis. That is, the number http://faculty.erau.edu/curtish/Dynamics_Modules/chapter_5/5_2_continuum_hypothe

DYNAMICS 5.2 Continuum Hypothesis At this point let us invoke the continuum hypothesis. That is, the number of particles in our system is allowed to increase without bound, while the mass m remains finite. The mass becomes distributed smoothly throughout the interior of the system boundary, like the air within a room, and the mass of each particle becomes infinitesimal dm . All of the above summations over the n mass points of a discrete system become integrals over the continuously distributed mass, Thus, Equation (5.1.2) becomes and Equation (5.1.30) takes on the form DYNAMICS

80. Hippias: Limited Area Search Of Philosophy On The Internet 3. continuum hypothesis True, False, or Neither? This web site hasmoved. You 3. continuum hypothesis True, False, or Neither? Is http://hippias.evansville.edu/search.cgi?cantor

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