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Continuum Hypothesis: more books (21)

Set Theory and the Continuum Problem (Oxford Logic Guides) by Raymond M. Smullyan, Melvin Fitting, 1996-11-14

Metacognitions in proneness towards hallucinations and delusions [An article from: Behaviour Research and Therapy] by F. Laroi, M. Van der Linden,

A Brief Tale of Infinity by H. Vic Dannon, 2007

Descriptive Set Theory and Definable Forcing (Memoirs of the American Mathematical Society) by Jindrich Zapletal, 2004-01

Real Numbers, Generalizations of the Reals, and Theories of Continua (Synthese Library)

The Spirit of Science: From Experiment to Experience

The 'Language Instinct' Debate by Geoffrey Sampson, 2005-05-15

lists with details

Sci.math FAQ: The Continuum Hypothesis
sci.math FAQ The continuum hypothesis 1994 Version 6.2 THE CONTINUUMHYPOTHESIS A basic reference is Godel s ``What is Cantor s Continuum Problem?
http://www.faqs.org/faqs/sci-math-faq/AC/ContinuumHyp/
2. Continuum Hypothesis
2. SOME FORMULATIONS OF continuum hypothesis Introduce the following notations.
http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
Mudd Math Fun Facts: Continuum Hypothesis
This came to be known as the continuum hypothesis. It is possible to provethat adding the continuum hypothesis or its negation would not cause a
http://www.math.hmc.edu/funfacts/ffiles/30002.4-8.shtml

Extractions: Francis Edward Su From the Fun Fact files, here is a Fun Fact at the Advanced level: We have seen in the Fun Fact How Many Reals that the real numbers (the "continuum") cannot be placed in 1-1 correspondence with the rational numbers. So they form an infinite set of a different "size" than the rationals, which are countable. It is not hard to show that the set of all subsets (called the power set ) of the rationals has the same "size" as the reals. But is there a "size" of infinity between the rationals and the reals? Cantor conjectured that the answer is no. This came to be known as the Continuum Hypothesis Many people tried to answer this question in the early part of this century. But the question turns out to be PROVABLY undecidable ! In other words, the statement is indepedent of the usual axioms of set theory! It is possible to prove that adding the Continuum Hypothesis or its negation would not cause a contradiction.
An Intuitivistic Solution Of The Continuum Hypothesis For
An intuitivistic solution of the continuum hypothesis for definable sets and resolution of the set theoretical paradoxes
http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126
PlanetMath: Continuum Hypothesis
The continuum hypothesis states that there is no cardinal number The continuumhypothesis can also be stated as there is no subset of the real numbers
http://planetmath.org/encyclopedia/ContinuumHypothesis.html

Extractions: continuum hypothesis (Axiom) The Continuum Hypothesis states that there is no cardinal number such that An equivalent statement is that It is known to be independent of the axioms of ZFC The continuum hypothesis can also be stated as: there is no subset of the real numbers which has cardinality strictly between that of the reals and that of the integers . It is from this that the name comes, since the set of real numbers is also known as the continuum.
PlanetMath: Generalized Continuum Hypothesis
The generalized continuum hypothesis states that for any infinite cardinal $ \lambda$there is no generalized continuum hypothesis is owned by yark.
http://planetmath.org/encyclopedia/GeneralizedContinuumHypothesis.html

Extractions: generalized continuum hypothesis (Axiom) The generalized continuum hypothesis states that for any infinite cardinal there is no cardinal such that An equivalent condition is that for every ordinal . Another equivalent condition is that for every ordinal Like the continuum hypothesis , the generalized continuum hypothesis is known to be independent of the axioms of ZFC
Continuum Hypothesis: True, False, Or Neither?
Subject Re Status of the continuum hypothesis Subject continuum hypothesis.I just came across your posting about CH and found it quite interesting.
http://consc.net/notes/continuum.html

Extractions: 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.)
Continuum Hypothesis
Cantor s continuum hypothesis. The continuum hypothesis states simply thataleph1= c, that is, there is no level of infinity between the one of
http://users.forthnet.gr/ath/kimon/Continuum.htm

Extractions: 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
Continuum Hypothesis - Definition Of Continuum Hypothesis In Encyclopedia
In mathematics, the continuum hypothesis is a hypothesis about the possible sizesof infinite sets. Georg Cantor introduced the concept of cardinality to
http://encyclopedia.laborlawtalk.com/Continuum_hypothesis

Extractions: 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 Contents showTocToggle("show","hide") 1 The size of a set
An Intuitivistic Solution Of The Continuum Hypothesis For Definable Sets And Res
An intuitivistic solution of the continuum hypothesis for definable sets andresolution of the set theoretical paradoxes.
http://www.farazgodrejjoshi.com/
An Intuitivistic Solution Of The Continuum Hypothesis For Definable Sets And Res
An intuitivistic solution of the continuum hypothesis for definable sets andresolution of the set theoretical paradoxes.
http://www.farazgodrejjoshi.com/about.htm

Extractions: The paper, titled "An intuitivistic solution of the Continuum Hypothesis for definable sets and resolution of the set theoretic paradoxes" , consists essentially of a solution of the Continuum Hypothesis for all sets which can intuitively be considered definable, with the section on paradoxes justifying its intuitive basis. The Continuum Hypothesis was for long regarded the most famous unsolved problem in mathematics. Subsequently in 1963, the works of Godel and Cohen proved the independence of the Continuum Hypothesis within the framework of an axiomatic set theory The paper, however, proves the Continuum Hypothesis for all definable sets within the framework of classical intuitive set theory which can arguably be considered the mathematical theory in which mathematicians originally sought a solution of the Continuum Hypothesis: A theory that is both all-encompassing in its generality and beautiful in its simplicity. Unfortunately, classical intuitive set theory was abandoned by mathematicians years ago in favour of several axiomatic set theories. This was primarily to side-step the apparent inconsistencies of classical intuitive set theory as manifested by the set theoretic paradoxes. Now since this paper basically reaffirms the stand of classical intuitive set theory, it was imperative to 'resolve' the set theoretic paradoxes to justify the intuitive approach in the first place. Also, since the solution of the Continuum Hypothesis entailed the use of concepts such as 'specifiable' or 'definable' real number and its counterpart 'unspecifiable' or 'undefinable' real number, it was necessary to resolve certain fundamental questions posed by Richard's paradox in particular and other set theoretic paradoxes in general.
Kids.net.au - Encyclopedia Continuum Hypothesis -
If a set S was found that disproved the continuum hypothesis, Cantor believedthe continuum hypothesis to be true and tried for many years to prove it,
http://www.kids.net.au/encyclopedia-wiki/co/Continuum_hypothesis

Extractions: Web kids.net.au Thesaurus Dictionary Kids Categories Encyclopedia ... Contents 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 (naively: whole numbers) is strictly smaller than the set of real numbers (naively: infinite decimals) The continuum hypothesis states the following: Or mathematically speaking, noting that the cardinality The real numbers have also been called the continuum , hence the name. 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. If a set S was found that disproved the continuum hypothesis, it would be impossible to make a one-to-one correspondence between
Continuum Hypothesis
The continuum hypothesis (CH), put forward by Cantor in 1877, says that the It is called the continuum hypothesis because the real numbers are used to
http://www.daviddarling.info/encyclopedia/C/continuum_hypothesis.html

Extractions: In 1874 Georg Cantor discovered that there is more than one level of infinity . The lowest level is called countable infinity ; higher levels are known as uncountable infinities . The natural numbers are an example of a countably infinite set and the real numbers are an example of an uncountably infinite set. The continuum hypothesis (CH), put forward by Cantor in 1877, says that the number of real numbers is the next level of infinity above countable infinity. It is called the continuum hypothesis because the real numbers are used to represent a linear continuum. Let c be the cardinality of (i.e., number of points in) a continuum, aleph -null, be the cardinality of any countably infinite set, and aleph-one be the next level of infinity above aleph-null. CH is equivalent to saying that there is no cardinal number between aleph-null and c , and that c = aleph-one. CH has been, and continues to be, one of the most hotly pursued problems in mathematics.
Navier-Stokes Equations: Continuum Hypothesis
These notes present the Navier Stokes equations, which are basis for the studyof fluid mechanics, aerodynamics, hydrodynamics, and gasdynamics.
http://www.navier-stokes.net/nscont.htm

Extractions: 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. Standard Continuum Hypothesis 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.
Continuum Hypothesis
There is also a generalization of the continuum hypothesis called the The continuum hypothesis states that every subset of the continuum (= the real
http://www.algebra.com/algebra/about/history/Continuum-hypothesis.wikipedia

Extractions: Regular View Dictionary View (all words explained) Algebra Help my dictionary with pronunciation , wikipedia etc 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: 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: To state the hypothesis formally, we need a definition: we say that two sets
Continuum Hypothesis -- Facts, Info, And Encyclopedia Article
There is also a generalization of the continuum hypothesis called the generalized The continuum hypothesis states that every (A set whose members are
http://www.absoluteastronomy.com/encyclopedia/c/co/continuum_hypothesis.htm

Extractions: In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics , the continuum hypothesis is a (A tentative theory about the natural world; a concept that is not yet verified but that if true would explain certain facts or phenomena) hypothesis about the possible sizes of (Click link for more info and facts about infinite) infinite (A group of things of the same kind that belong together and are so used) set s. (Click link for more info and facts about Georg Cantor) Georg Cantor introduced the concept of (Click link for more info and facts about cardinality) cardinality to compare the sizes of infinite sets, and he showed that the set of (Any of the natural numbers (positive or negative) or zero) integer s is strictly smaller than the set of (Any rational or irrational number) real number s. The continuum hypothesis states the following:
Continuum Hypothesis@Everything2.com
The continuum hypothesis is, loosely speaking, a postulate as to how Combining these facts, we see that the continuum hypothesis can be restated as
http://www.everything2.com/index.pl?node_id=22495
Godel, K.: Consistency Of The Continuum Hypothesis. (AM-3).
of the book Consistency of the continuum hypothesis. (AM3) by Godel,K., published by Princeton University Press.
http://www.pupress.princeton.edu/titles/1034.html

Extractions: Shopping Cart: For customers in the U.S., Canada, Latin America, Asia, and Australia Paper: $27.95 ISBN: 0-691-07927-7 For customers in Europe, Africa, the Middle East, and India Prices subject to change without notice File created: 6/30/2005 Questions and comments to: webmaster@pupress.princeton.edu
EPOCH OF UNLIGHT - The Continuum Hypothesis
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Continuum Hypothesis
The continuum hypothesis is that C = aleph 1, that there are no such Gödel showed in 1938 that the continuum hypothesis cannot be disproved using just
http://www-users.cs.york.ac.uk/~susan/cyc/c/cont.htm

Extractions: 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.

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