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Topic: Generalized continuum hypothesis

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	Continuum hypothesis - Encyclopedia, History, Geography and Biography
		The generalized continuum hypothesis (GCH) states that if an infinite set's cardinality lies between that of an infinite set S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as the power set of S: there are no in-betweens.
		This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power set of the integers.
		GCH is also independent of the Zermelo-Fränkel set theory axioms, and also of the axiom of choice.
	www.arikah.com /encyclopedia/Continuum_hypothesis (1002 words)

	Continuum hypothesis -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		The continuum hypothesis states that every (A set whose members are members of another set; a set contained within another set) subset of the continuum (= the (Any rational or irrational number) real numbers) which contains the integers either has the same cardinality as the integers or the same cardinality as the continuum.
		The continuum hypothesis is closely related to many statements in (The abstract separation of a whole into its constituent parts in order to study the parts and their relations) analysis, point set (The configuration of a communication network) topology and (Click link for more info and facts about measure theory) measure theory.
		This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the (Click link for more info and facts about power set) power set of the integers.
	www.absoluteastronomy.com /encyclopedia/c/co/continuum_hypothesis.htm (1032 words)

	Continuum hypothesis: Definition and Links by Encyclopedian.com - All about Continuum hypothesis (Site not responding. Last check: 2007-10-08)
		Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standard Zermelo-Fraenkel set theory axiom system, even if the axiom of choice is adopted.
		The generalized continuum hypothesis (GCH) states that if a set's cardinality lies between that of an infinite set S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as the power set of S: there are no in-betweens.
		GCH is also independent of the Zermel-Fraenkel set theory axioms and it implies the axiom of choice.
	www.encyclopedian.com /co/Continuum-hypothesis.html (931 words)

	Continuum hypothesis - Wikipedia
		The continuum hypothesis is the hypothesis that there is no set whose cardinality is strictly between that of the integers and that of the real numbers.
		First conjectured by Cantor, the hypothesis became the first on David Hilbert's list of important open questions that was presented at the International Mathematical Congress in the year 1900 in Paris.
		The Continuum hypothesis states that every subset of the continuum which contains the integers either has the same cardinality as the integers or the same cardinality as the continuum.
	nostalgia.wikipedia.org /wiki/Continuum_hypothesis (613 words)

	Kurt Gödel - Wikipedia, the free encyclopedia
		He also produced celebrated work on the continuum hypothesis, showing that it cannot be disproven from the accepted set theory axioms, assuming that those axioms are consistent.
		He returned to teaching in 1937 and during this time he worked on the proof of consistency of the continuum hypothesis; he would go on to show that this hypothesis cannot be disproved from the common system of axioms of set theory.
		Gödel showed that both the axiom of choice and the generalized continuum hypothesis are true in the constructible universe, and therefore must be consistent.
	www.wikipedia.com /wiki/Kurt_Godel (1959 words)

	new_site (Site not responding. Last check: 2007-10-08)
		The continuum hypothesis reflects Cantor’s inability to construct a set with cardinality between that of the natural numbers and that of the real numbers.
		According to [2,p.189], "Mathematicians do not tend to assume the Continuum Hypothesis as an additional axiom of set theory mostly since they cannot convince themselves that this statement is "true" as many of them have done for the axioms of ZFC including the axiom of choice.
		To reconcile his quantum hypothesis with his conception of wave radiation, he avoided the conclusion that radiation energy must be made of particles, and postulated that radiation is a transition between the energy levels of an oscillator.
	www.gauge-institute.org (690 words)

	Continuum hypothesis
		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:
		There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis, which is described at the end of this article.
		To state the hypothesis formally, we need a definition: we say that two sets S and T have the same cardinality or cardinal number if there exists a bijection.
	www.brainyencyclopedia.com /encyclopedia/c/co/continuum_hypothesis.html (897 words)

	PlanetMath: generalized continuum hypothesis
		The generalized continuum hypothesis states that for any infinite cardinal
		Like the continuum hypothesis, the generalized continuum hypothesis is known to be independent of the axioms of ZFC.
		This is version 11 of generalized continuum hypothesis, born on 2002-01-03, modified 2004-04-02.
	planetmath.org /encyclopedia/GeneralizedContinuumHypothesis.html (83 words)

	PlanetMath: continuum hypothesis
		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.
		This is version 8 of continuum hypothesis, born on 2002-01-03, modified 2003-12-31.
	planetmath.org /encyclopedia/ContinuumHypothesis.html (127 words)

	Cardinal number - Wikipedia
		This is easily visualized using Cantor's diagonal argument; classic questions of cardinality (for instance the continuum hypothesis) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals.
		The continuum hypothesis states that there are no cardinals strictly between aleph-0 and 2
		The continuum hypothesis is independent from the usual axioms of set theory, the Zermelo-Fraenkel axioms together with the axiom of choice (ZFC).
	nostalgia.wikipedia.org /wiki/Cardinality (1132 words)

	Continuum, Mu-Ency at MROB
		Examples of continuums are a straight line, a plane, a circle, a disc, the set of real numbers, and the set of complex numbers.
		The Continuum Hypothesis states that there is no infinity between Aleph-0 and the order of a continuum, which would mean that the order of the continuum is Aleph-1.
		The Generalized Continuum Hypothesis states that if N is the order of set S and M is the order of the power set of S, there exist no sets that have more elements than N and fewer elements than M. This would mean that the order of a power set of a continuum is Aleph-2.
	www.mrob.com /pub/muency/continuum.html (491 words)

	Read about Continuum hypothesis at WorldVillage Encyclopedia. Research Continuum hypothesis and learn about Continuum ... (Site not responding. Last check: 2007-10-08)
		subset of the continuum (= the real numbers) which contains the integers either has the same cardinality as the integers or the same cardinality as the continuum.
		Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standard
		This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the
	encyclopedia.worldvillage.com /s/b/Continuum_hypothesis (873 words)

	Continuum Hypothesis and the Axiom of Choice (Site not responding. Last check: 2007-10-08)
		Definition 0.6 The continuum hypothesis (CH) says that there is no cardinality strictly between countable and continuum.
		In the 1930s Gödel showed that no contradiction arises if the GCH is added to ZFC, where ZF stands for the Zermelo-Fraenkel axioms of set theory and C stands for the Axiom of Choice.
		Therefore the Continuum Hypothesis is independent of ZFC.
	people.cs.uchicago.edu /~laci/reu03/n2_7/node2.html (131 words)

	Real closed field - Wikipedia, the free encyclopedia
		The weight of F, which is the minimum size of a dense subset of F. These three cardinal numbers tell us much about the order properties of any real closed field, though it may be difficult to discover what they are, especially if we are not willing to invoke generalized continuum hypothesis.
		The characteristics of real closed fields become much simpler if we are willing to assume the generalized continuum hypothesis.
		This is the most commonly used hyperreal number field in nonstandard analysis, and its uniqueness is equivalent to the continuum hypothesis.
	en.wikipedia.org /wiki/Real_closed_field (866 words)

	set theory from FOLDOC (Site not responding. Last check: 2007-10-08)
		Cantorian set theory Set theory in which either the generalized continuum hypothesis or the axiom of choice is an axiom.
		In 1938 Goedel proved that the axiom of choice, continuum hypothesis, and generalized continuum hypothesis are theorems (even if not axioms) of constructible set theory.
		Non-Cantorian set theory Set theory in which either the negation of the generalized continuum hypothesis (GCH) or the negation of the axiom of choice (AC) is an axiom.
	lgxserver.uniba.it /lei/foldop/foldoc.cgi?set+theory (300 words)

	Station Information - Suslin hypothesis
		The Suslin hypothesis, also called the Souslin hypothesis, is the assertion that every countable-chain-condition dense complete linear order without endpoints is isomorphic to the real line.
		The Generalized Suslin Hypothesis asserts that for every infinite regular cardinal κ every tree of height κ either has a branch of length κ or an antichain of cardinality κ.
		The Suslin hypothesis is independent of ZFC, and is independent of both the Generalized Continuum Hypothesis and of the negation of the Continuum Hypothesis.
	www.stationinformation.com /encyclopedia/s/su/suslin_hypothesis.html (154 words)

	KURT GODEL
		He proved the incompleteness of axioms for arithmetic (his most famous result), as well as the relative consistency of the axiom of choice and continuum hypothesis with the other axioms of set theory.
		Later that year he proved the consistency of the generalized continuum hypothesis with the axioms of set theory, and he lectured on his set-theoretic results at the IAS in 1938-39.
		In 1942 Gödel attempted to prove that the axiom of choice and continuum hypothesis are independent of (not implied by) the axioms of set theory.
	www.usna.edu /Users/math/meh/godel.html (900 words)

	Aleph number - Encyclopedia, History, Geography and Biography (Site not responding. Last check: 2007-10-08)
		This is harder than most explicit descriptions of "generation" in algebra (for example vector spaces, groups, etc.) because in those cases we only have to close with respect to finite operations — sums, products, and the like.
		In Zermelo-Fraenkel set theory with the axiom of choice, the celebrated continuum hypothesis is equivalent to the identity
		This proposition is independent of "ZFC", i.e., of Zermelo-Fraenkel set theory conjoined with the axiom of choice: it can be neither proved nor disproved within the context of that axiom system.
	www.arikah.net /encyclopedia/Aleph-null (746 words)

	Set Theory: Foundations of Mathematics
		Continuum Hypothesis is derived from an axiom called Axiom of Monotonicity.
		Generalized continuum hypothesis is derived from the first axiom, and the infinitesimal is visualized using the latter axiom.
		Generalized Continuum Hypothesis is derived from a simple axiom called Axiom of Combinatorial Sets.
	www.ece.rutgers.edu /~knambiar/intuitive_set_theory.html (390 words)

	Science of Value - Wikipedia, the free encyclopedia
		Starting from the claim that a person can eventually think of a countable infinity of things, Hartman concludes the intension of man is a denumerably infinite set of predicates; which means that man, according to this first definition, is appropriately to be measured by a denumerable infinity.
		However he quickly passes to the conclusion that we also have a countable infinity of levels of thought, and that therefore we can think of a countable infinity of things using a countable infinity of thought levels, giving us the cardinality of the continuum of thoughts.
		Hartman believes the generalized continuum hypothesis is true, and therefore claims the intension of man consists of
	www.wikipedia.org /wiki/Science_of_Value (1112 words)

	Aleph number - Wikipedia, the free encyclopedia
		An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the sigma-algebra generated by an arbitrary collection of subsets.
		The process involves defining, for each countable ordinal, via transfinite induction, a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of Ω.
		It follows from ZFC (Zermelo-Fraenkel set theory with the axiom of choice) that the celebrated continuum hypothesis, CH, is equivalent to the identity
	en.wikipedia.org /wiki/Aleph-null (737 words)

	Logic and Language Links - continuum hypothesis
		Gloss: A hypothesis in set theory first proposed by Cantor.
		The power set of N will therefore have a cardinality of Aleph_0 to teh power of 2, which is denoted by c-the cardinal number of the set of real numbers (the continuum).
		Cantor's hypothesis is that no infinite cardinal lies between Aleph_0 and c.
	www.astro.uva.nl /~caterina/LoLaLi/Pages/382.html (66 words)

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