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Topic: Continuum hypothesis

Related Topics

Generalized continuum hypothesis

Power set

Axiomatic set theory

Aleph number

Beth number

Cardinal number

Axiom of constructibility

Ordinal

Continuum

Axiom of choice

Set

Real number

Constructivism (math)

Paul Cohen

Rational number

	Continuum hypothesis - Wikipedia, the free encyclopedia
		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.
		This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power set of the integers.
		The Independence of the Continuum Hypothesis, II Paul J. Cohen Proceedings of the National Academy of Sciences of the United States of America, Vol.
	en.wikipedia.org /wiki/Continuum_hypothesis (1064 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 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.
	nostalgia.wikipedia.org /wiki/Continuum_hypothesis (614 words)

	Continuum - Wikipedia, the free encyclopedia
		Continuum (mathematics), mathematical sets that can be contrasted with the properties of discrete spaces
		Continuum hypothesis - "There is no set whose size is strictly between that of the integers and that of the real numbers."
		Continuum Convention, a roleplaying convention dedicated to the worlds of Moorcock, Greg Stafford, H.P.Lovecraft and many others.
	en.wikipedia.org /wiki/Continuum (196 words)

	Encyclopedia: Continuum hypothesis
		There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis, which is described at the end of this article.
		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 says that the cardinality of the power set of each infinite cardinal is the next infinite cardinal.
	www.nationmaster.com /encyclopedia/Continuum_hypothesis (2412 words)

	AREA33 : Metaphysical Dome : the condensation of transcendent thought
		The continuum hypothesis, symbolically represented as 2?0 = ?1 roughly translates to mean that the infinite realm of rational numbers is "one level of infinity" smaller than the infinite realm of real numbers which includes both rational and irrational numbers.
		A further claim of the continuum hypothesis is that the infinite realm of real numbers can be paralleled to the infinity of the infinitely small, because both are infinites by division.
		The continuum hypothesis is the claim that the answer to Cantor's question is that the set of real numbers is the next size of infinity up from the set of natural numbers.
	www.area33.com /continuum.html (1560 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)

	Continuum hypothesis - Encyclopedia, History, Geography and Biography
		Or mathematically speaking, noting that the cardinality for the integers $\mathbb{Z}$ is $\aleph_0$ ("aleph-null") and the cardinality of the real numbers $\mathbb{R}$ is $2^{\aleph_0}$ , the continuum hypothesis says:
		Cantor's diagonal argument shows thatthe integers and the continuum do not have the same cardinality.
		Continuum hypothesis, The size of a set, Impossibility of proof and disproof, Arguments pro and con, The generalized continuum hypothesis, References, See also, External links, PlanetMath sourced articles, Set theory, Model theory and Hilbert's problems.
	www.arikah.com /encyclopedia/Continuum_hypothesis (1052 words)

	continuum hypothesis
		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.
	www.daviddarling.info /encyclopedia/C/continuum_hypothesis.html (242 words)

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

	[No title] (Site not responding. Last check: 2007-11-05)
		This conjecture, the Continuum Hypothesis is stated formally as \begin{displaymath} c = \aleph_1 \end{displaymath} or equivalently: there is no set whose size is greater than that of the natural numbers but less than that of the real numbers.
		The Continuum Hypothesis would be the first problem enumerated by Hilbert at the turn of the 20th century in his list of 23 unsolved problems.
		Yet, it is unlikely that Hilbert himself fully recognized the profundity of the Continuum Hypothesis and its implications on the fundamental basis of mathematics.
	mars.drw.net /acw83/paper2/ee.txt (3460 words)

	Navier-Stokes Equations: Continuum Hypothesis (Site not responding. Last check: 2007-11-05)
		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.
		This observation appears to be the basis for most discussions of the continuum hypothesis found in texts on fluid mechanics.
		While the above continuum hypothesis is sufficient to ensure that the field representation holds, the Navier-Stokes equations require more stringent contraints on their validity.
	www.navier-stokes.net /nscont.htm (785 words)

	contiuum hypothesis: Philosophy Forums (Site not responding. Last check: 2007-11-05)
		Moreover, to adopt the negation of the continuum hypothesis as an axiom does not require that set theory have sentences that we take to be descriptions of a set whose cardinality is between the aleph-0 and the cardinality of the continuum.
		But for the continuum hypothesis, first, the objects in question aren't lined up for us, and, second, even when we do have an object for inspection, we still don't have a mechanical test to see whether it upholds the continuum hypothesis (that is, to see what the cardinality of the object is).
		One point about the continuum hypothesis: Rather than mention that a model is not required to mention sets as we unsually understand them, it would have been even more pertinent for me to mention that a model does not have to use the standard membership relation.
	forums.philosophyforums.com /thread/16808 (4120 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)

	The continuum hypothesis (Site not responding. Last check: 2007-11-05)
		In 1940 Gödel proved that the continuum hypothesis cannot be disproved from the other axioms of set theory.
		The role of the continuum hypothesis in set theory is similar to the role of the parallel postulate in plane geometry.
		Similarly, whether you choose to accept the continuum hypothesis will depend on your idea of what a set is supposed to be.
	mathcircle.berkeley.edu /BMC3/infinity/node12.html (192 words)

	1. Introduction (Site not responding. Last check: 2007-11-05)
		So, that Continuum Hypothesis can not "genetically" be a problem of and have an attitude to either modern meta-mathematics or modern mathematical logic.
		Continuum Hypothesis is a rather dramatic example of what can be called (from our today's point of view) an absolutely undecidable assertion,..." (p.13).
		The complete absence of any progress in the Continuum Hypothesis proof (or dispoof) on the way of modern meta-mathematics during last decades confirms the validity of Cohen's pessimism.
	www.mi.sanu.ac.yu /vismath/zen/zen1.htm (540 words)

	Continuum hypothesis - RecipeFacts (Site not responding. Last check: 2007-11-05)
		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 $S \leftrightarrow T$ .
		If a set S were found that disproved the continuum hypothesis, it would be impossible to make a one-to-one correspondence between S and the set of integers, because there would always be elements of set S that were "left over".
		Like CH, GCH is also independent of the Zermelo-Fränkel set theory axioms, and the axiom of choice.
	www.recipeland.com /encyclopaedia/index.php/Continuum_hypothesis (1145 words)

	Wikinfo \| Continuum hypothesis (Site not responding. Last check: 2007-11-05)
		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.
		Chris Freiling in 1986 presented an argument against CH: he showed that the negation of CH is equivalent to a statement about probabilities which he calls "intuitively true", but others have disagreed.
		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.
	www.wikinfo.org /wiki.php?title=Continuum_hypothesis (990 words)

	Continuum Hypothesis
		Let's call it c from "continuum" to emphasize the fact that the set of real numbers is mapped onto something continuous in space, namely the line.
		Kurt Gödel proved in 1940 that the Continuum Hypothesis (CH) is consistent with the accepted axioms of set theory (the Zermelo-Fraenkel-Skolem system).
		But in 1963 Paul Cohen proved that the reverse also holds, that is the refutation of the CH is also consistent with the same axioms.
	users.forthnet.gr /ath/kimon/Continuum.htm (712 words)

	Science News
		While Cantor was struggling with the continuum hypothesis, other mathematicians were exploring the implications of Cantor's dramatically broad vision of sets.
		They hold that, if the continuum hypothesis can't be resolved within the standard framework of mathematics, then the hypothesis must be inherently vague.
		To them, if the standard axioms can't settle the continuum hypothesis, it's not that the hypothesis is a meaningless question, but rather that the axioms are insufficient.
	www.phschool.com /science/science_news/articles/infinite_wisdom.html (2238 words)

	PlanetMath: a shorter proof: Martin's axiom and the continuum hypothesis
		PlanetMath: a shorter proof: Martin's axiom and the continuum hypothesis
		"a shorter proof: Martin's axiom and the continuum hypothesis" is owned by x_bas.
		This is version 8 of a shorter proof: Martin's axiom and the continuum hypothesis, born on 2003-08-24, modified 2004-03-15.
	planetmath.org /encyclopedia/Dense3.html (125 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.
	staff.science.uva.nl /~caterina/LoLaLi/Pages/382.html (66 words)

	Continuum hypothesis
		Or mathematically speaking, noting that the cardinality for the integers $\mathbf{Z}$ is $\aleph_0$ and the cardinality for the real numbers $\mathbf{R}$ is $2^{\aleph_0}$ , the continuum hypothesis says:
		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 S → T.
		GCH is also independent of the Zermel-Fraenkel set theory axioms and it implies the axiom of choice.
	www.termsdefined.net /co/continuum-hypothesis.html (1104 words)

	sci.math FAQ: The Continuum Hypothesis
		Most of those who disbelieve CH think that the continuum is likely to have very large cardinality, rather than aleph_2 (as Godel seems to have suggested).
		most were wary even of suggesting that the Riemann Hypothesis necessarily has a determinate truth-value.) On the other hand, Maddy's contemporaries discussed in her paper seemed quite happy to speculate about the ``truth" or ``falsity" of CH.
		The integers are not only a bedrock, but also any finite number of power sets seem to be quite natural Intuitively are also natural which would point towards the fact that CH may be determinate one way or the other.
	www.faqs.org /faqs/sci-math-faq/AC/ContinuumHyp (1193 words)

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