21. Continuum Hypothesis continuum hypothesis. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Investigating the continuum hypothesis. http://www.fact-index.com/c/co/continuum_hypothesis.html

Main Page See live article Alphabetical index 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 (naively: whole numbers) is strictly smaller than the set of real numbers (naively: infinite decimals) 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 ... 5 References 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.

22. 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.

23. 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. Part I The continuum hypothesis. http://www.farazgodrejjoshi.com/page1.htm

An intuitivistic solution of the Continuum Hypothesis for definable sets and resolution of the set theoretic paradoxes Part I : The Continuum Hypothesis Introduction This paper has two parts, Part I and Part II. Part I consists of an intuitive proof of the Continuum Hypothesis (CH) for all definable sets. Part II is a systematic treatment of the set theoretic paradoxes and is complementary to Part I: It justifies the intuitive approach adopted in Part I notwithstanding the problems posed by the paradoxes and it reveals that the scope of Part I extends to include all 'definable' sets - even the 'paradoxical' ones! The arguments involving as they do sets of real numbers, it is necessary now to dwell on certain fundamental concepts associated with real numbers and their sets. A fundamental concept associated with real numbers is that of 'characteristic property' (CP), which is any finitely describable property that is 'meaningful' (true or false) for each real number. Thus for example, the property 'is a rational number' being finitely describable and meaningful for each real number, is a CP of real numbers. Another fundamental concept is that of 'general negation' of a CP, which as its name suggests is that CP obtained by inserting the word 'not' at the appropriate place in a given CP so as to get the 'opposite' of that CP. For example, 'is

24. Mudd Math Fun Facts: Continuum Hypothesis From the Fun Fact files, here is a Fun Fact at the Advanced level continuum hypothesis. This came to be known as the continuum hypothesis. http://www.math.hmc.edu/funfacts/ffiles/30002.4-8.shtml

hosted by the Harvey Mudd College Math Department Francis Su Any Easy Medium Advanced Search Tips List All Fun Facts Fun Facts Home About Math Fun Facts ... Other Fun Facts Features 1553353 Fun Facts viewed since 20 July 1999. Francis Edward Su From the Fun Fact files, here is a Fun Fact at the Advanced level: Continuum Hypothesis 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.

25. Continuum Hypothesis - Encyclopedia Article About Continuum Hypothesis. Free Acc encyclopedia article about continuum hypothesis. continuum hypothesis in Free online English dictionary, thesaurus and encyclopedia. continuum hypothesis. http://encyclopedia.thefreedictionary.com/Continuum hypothesis

Dictionaries: General Computing Medical Legal Encyclopedia Continuum hypothesis Word: Word Starts with Ends with Definition In mathematics Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'. In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation; other views are described in Philosophy of mathematics. Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships. Click the link for more information. , the continuum hypothesis is a hypothesis A hypothesis is a proposed explanation for a phenomenon. In the hypothetico-deductive method, a hypothesis should be falsifiable, meaning that it is possible that it be shown to be false, usually by observation. As an example, a reader who comes upon a high-quality article in Encyclopedia might form a hypothesis that Encyclopedia articles can only be edited by highly qualified professors with multiple Ph.Ds. Click the link for more information.

26. Continuum Hypothesis. - Encyclopedia Article About Continuum Hypothesis.. Free A encyclopedia article about continuum hypothesis.. continuum hypothesis. in Free online English dictionary, thesaurus and encyclopedia. continuum hypothesis. http://encyclopedia.thefreedictionary.com/Continuum Hypothesis.

Dictionaries: General Computing Medical Legal Encyclopedia Continuum Hypothesis. Word: Word Starts with Ends with Definition In mathematics Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'. In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation; other views are described in Philosophy of mathematics. Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships. Click the link for more information. , the continuum hypothesis is a hypothesis A hypothesis is a proposed explanation for a phenomenon. In the hypothetico-deductive method, a hypothesis should be falsifiable, meaning that it is possible that it be shown to be false, usually by observation. As an example, a reader who comes upon a high-quality article in Encyclopedia might form a hypothesis that Encyclopedia articles can only be edited by highly qualified professors with multiple Ph.Ds. Click the link for more information.

27. Continuum Hypothesis :: Online Encyclopedia :: Information Genius Investigating the continuum hypothesis. Consider the set of all rational numbers. One might continuum. The generalized continuum hypothesis. The http://www.informationgenius.com/encyclopedia/c/co/continuum_hypothesis.html

Quantum Physics Pampered Chef Paintball Guns Cell Phone Reviews ... Science Articles Continuum hypothesis Online 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 (naively: whole numbers) is strictly smaller than the set of real numbers (naively: infinite decimals) 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 ... 5 References 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.

28. What Is CH? What is the continuum hypothesis? The continuum hypothesis is the first problem on David Hilbert s famous list of 23 unsolved mathematics problems. http://www.wall.org/~aron/whatisch.html

What is the continuum hypothesis? The continuum hypothesis is the first problem on David Hilbert's famous list of 23 unsolved mathematics problems. The history of this problem stretches back to the brilliant man who created (or discovered, whichever you prefer) set theory. Georg Cantor 1, b 2). So far this is quite intuitive. But Cantor applied this principle to infinite sets as well. So the set of all even natural numbers has the same cardinality as the set of all natural numbers, even though the latter has all the elements of the former and more! The mapping looks like this: In the same way you can map any infinite set of natural numbers to any other infinite set of natural numbers. They are the same size. Cantor gave this infinity the name aleph_0, after the first letter of the Hebrew alphabet. Now, infinity would be kind of boring if aleph_0 was all there was. But Cantor proved that some sets are larger than aleph_0. The set in question was the set of real numbers, or the set of points on a line. Cantor proved that the cardinality of the reals cannot be mapped on to the natural numbers. This is the proof: Take the real numbers between and 1. Now suppose there was a mapping of the real numbers between and 1 to the natural numbers. It would look something like this:

29. CH Directory Enter at your own risk. A cool page on the continuum hypothesis Toward any dissenters, I quote the first Peano Postualate. What is the continuum hypothesis? http://www.wall.org/~aron/chdir.html

Warning: Severely complicated mathematics within. Enter at your own risk. A cool page on the continuum hypothesis Note: The reader should be warned that throughout these pages I include as a natural number. Toward any dissenters, I quote the first Peano Postualate. What is the continuum hypothesis? My Philosophy of Mathematics Reference This stuff is nessesary to comprehend the very last section. It has more complex set theoretical notions. An advanced set theorist should already know this stuff. And finally... My discoveries on the matter Math not your thing? Back to my homepage

30. Example 2.2.9: The Continuum Hypothesis Example 2.2.9 The continuum hypothesis. Is card(P(N)) = card(R). In fact, this is a deep question called the continuum hypothesis. This http://www.shu.edu/projects/reals/infinity/answers/conthyp.html

Example 2.2.9: The Continuum Hypothesis Is there a cardinal number c with ? What is the most obvious candidate ? Back We need to find a set whose cardinality is bigger than N and less that that of R . The most obvious candidate would be the power set of N . However, one can show that card( P N )) = card( R In fact, this is a deep question called the continuum hypothesis . This question results in serious problems: In the 1940's the German mathematician Goedel showed that if one denies the existence of an uncountable set whose cardinalities is less than the cardinality of the continuum, no logical contradictions to the axioms of set theory would arise. One the other hand, it was shown recently that the existence of an uncountable set with cardinality less than that of the continuum would also be consistent with the axioms of set theory. Hence, it seems impossible to decide this question with our usual methods of proving theorems. Such undecidable questions do indeed exist for any reasonably complex logical system (such as set theory), and in fact one can even prove that such 'non-provable' statements must exist. To read more about this fascinating subject, look at the book Goedel's Proof or Goedel, Escher, Bach

31. 2.2. Uncountable Infinity What is card(N) + card(N) ? What is card(N) card(N) ? What is card(R) + card(N) ? What is card(R) + card(R) ? Example 2.2.9 The continuum hypothesis. http://www.shu.edu/projects/reals/infinity/uncntble.html

2.2. Uncountable Infinity IRA The last section raises the question whether it is at all possible to have sets that contain more than countably many elements. After all, the examples of infinite sets we encountered so far were all countable. It was Georg Cantor who answered that question: not all infinite sets are countable. Proposition 2.2.1: An Uncountable Set The open interval (0, 1) is uncountable. Proof Note that this proposition assumes the existence of the real numbers. At this stage, however, we have only defined the integers and rationals. We are not supposed to know anything about the real numbers. Therefore, this proposition should - from a strictly logical point of view - be rephrased: if there are 'real numbers' in the interval (0, 1) then there must be uncountably many. However, the real numbers, of course, do exist, and are thus uncountable. As for a more elementary uncountable set, one could consider the following: the set of all infinite sequences of 0's and 1's is uncountable The proof of this statement is similar to the above proposition, and is left as an exercise. What about other familiar sets that are uncountable ?

32. PlanetMath: Continuum Hypothesis continuum hypothesis, (Axiom). The continuum hypothesis states that there is no cardinal number such that . continuum hypothesis is owned by Evandar. http://planetmath.org/encyclopedia/ContinuumHypothesis.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information Docs Classification News Legalese ... TODO List 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. "continuum hypothesis" is owned by Evandar view preamble View style: HTML with images page images TeX source See Also: axiom of choice Zermelo-Fraenkel axioms generalized continuum hypothesis Other names: CH Cross-references: integers cardinality real numbers subset ... cardinal number There are 6 references to this object.

33. PlanetMath: $\Diamond$ Is Equivalent To $\clubsuit$ And Continuum Hypothesis is equivalent to and continuum hypothesis, (Theorem). See Also $\Diamond$, $\clubsuit$ Other names diamond is equivalent to club and continuum hypothesis. http://planetmath.org/encyclopedia/DiamondIsEquivalentToClubsuitAndContinuumHypo

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information Docs Classification News Legalese ... TODO List is equivalent to and continuum hypothesis (Theorem) If is a stationary subset of and implies then Moreover, this is best possible: is consistent with is equivalent to and continuum hypothesis" is owned by Henry view preamble View style: HTML with images page images TeX source See Also: Other names: diamond is equivalent to club and continuum hypothesis Attachments: proof of is equivalent to and continuum hypothesis (Proof) by Henry Cross-references: consistent subset stationary This is version 3 of is equivalent to and continuum hypothesis , born on 2002-07-31, modified 2002-08-02. Object id is 3247, canonical name is DiamondIsEquivalentToClubsuitAndContinuumHypothesis. Accessed 875 times total.

34. Continuum Hypothesis Definition Meaning Information Explanation continuum hypothesis definition, meaning and explanation and more about continuum hypothesis. Free continuum hypothesis. definition http://www.free-definition.com/Continuum-hypothesis.html

A B C D ... Contact Beta 0.71 powered by: akademie.de PHP PostgreSQL Google News about your search term Continuum hypothesis In mathematics , the continuum hypothesis is a hypothesis about the possible sizes of infinite set s. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integer s is strictly smaller than the set of real number s. 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. aleph-null 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. Inhaltsverzeichnis 1 Investigating the continuum hypothesis 2 Impossibility of proof and disproof 3 The generalized continuum hypothesis 4 See also ... 5 References Investigating the continuum hypothesis Consider the set of all rational number s. 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 countable set s.

35. Navier-Stokes Equations: Continuum Hypothesis NavierStokes Equations continuum hypothesis. In most treatments of fluid mechanics, the so-called continuum hypothesis is hurriedly http://www.navier-stokes.net/nscont.htm

Foundations of Fluid Mechanics My Homepage Gallery of Fluid Mechanics Cambridge University Press Contact Index Introduction Web Stuff Our Sponsor What's New? Navier-Stokes Eqns Overview Notation Definition Continuum Hypoth. ... Summary Misc. Topics Inviscid Flows Incompressible Flows Conservative Forms Bernoulli Equations ... Special Fluid Models Potential Flows Introduction Restrictions Aerodynamics Water Waves ... Acoustics Math Identities Vector Calculus Stokes' Theorem Gauss' Theorem Transport Theorems Great Books Introduction Continuum Mech. General Fluid Mech. Aero/Hydrodyn. ... Miscellaneous 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. 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.

36. Continuum Hypothesis Contents continuum hypothesis. But this. It was one of the great unsolved problems of mathematics, called ``The continuum hypothesis . Finally http://cs.wwc.edu/~aabyan/CII/BOOK/book/node43.html

Next: The Indescribable Up: Countability Previous: Arithmetic of the infinite Contents Continuum Hypothesis But are there any transfinite cardinal numbers between and C? The name has been given to the smallest transfinite cardinal number larger than . Then . The above question can be rephrased as, ``Is ?" Cantor guessed that this is so, but was unable to prove this. It was one of the great unsolved problems of mathematics, called ``The Continuum Hypothesis". Finally it was shown that this was not provable from the usual axioms of set theory! It is usually assumed as an additional axiom. Anthony A. Aaby 2003-11-11

37. Continuum Hypothesis Susan Stepney s Home Page Indexcontinuum hypothesis. The continuum hypothesis is 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.

38. Learn More About Continuum Hypothesis In The Online Encyclopedia. Visit the Online Encyclopedia and learn more and get your questions answered about continuum hypothesis. see previous page. continuum hypothesis. http://www.onlineencyclopedia.org/c/co/continuum_hypothesis.html

You are here: Online Encyclopedia Enter a phrase or search word in the box below. You can enter multiple phrases at a time by putting a comma between each word.(e.g. cat ,dog ,lion ) Press the search button to start your search. Hint: Play with putting spaces before and after your words to see the different results you get. see previous page 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 (naively: whole numbers) is strictly smaller than the set of real numbers (naively: infinite decimals) 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

39. Continuum Hypothesis From FOLDOC continuum hypothesis. Proving or disproving the continuum hypothesis was the first problem on Hilbert s famous list of problems in 1900. http://www.swif.uniba.it/lei/foldop/foldoc.cgi?continuum hypothesis

40. Continuum Hypothesis - Wikipedia, The Free Encyclopedia PhatNav s Encyclopedia A Wikipedia . continuum hypothesis. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. http://www.phatnav.com/wiki/wiki.phtml?title=Continuum_hypothesis

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