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Topic: Cantors diagonal argument

	Cantors Diagonal argument - Wikipedia
		A logical argument devised by Georg Cantor to demonstrate that the real numbers are not countably infinite.
		The diagonal argument is an example of reductio ad absurdum because it proves a certain proposition (the interval (0,1) is not countably infinite) by showing that the assumption of its negation leads to a contradiction.
		A generalized form of the diagonal argument was used by Cantor to show that for every set S the power set of S, i.e., the set of all subsets of S (here written as P(S)), is larger than S itself.
	nostalgia.wikipedia.org /wiki/Cantors_Diagonal_argument (681 words)

	Kids.Net.Au - Encyclopedia > Cantors Diagonal argument (Site not responding. Last check: )
		Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite.
		The diagonal argument is an example of reductio ad absurdum because it proves a certain proposition (the interval (0,1) is not countably infinite) by showing that the assumption of its negation leads to a contradiction.
		Analogues of the diagonal argument are widely used in mathematics to prove the existence or nonexistence of certain objects.
	www.kids.net.au /encyclopedia-wiki/ca/Cantors_Diagonal_argument (797 words)

	Cantors - Hutchinson encyclopedia article about Cantors (Site not responding. Last check: )
		In Protestant churches, the music director is known as the cantor.
		The Jewish cantor, or chazan, who leads the singing in synagogue has had a training not only in music and voice work, but in chanting the special prayers for different occasions.
		Cantors World, a leading producer of cantorial programs, is coordinating the event.
	encyclopedia.farlex.com /Cantors (265 words)

	PlanetMath: Cantor's diagonal argument
		One of the starting points in Cantor's development of set theory was his discovery that there are different degrees of infinity.
		In essence, Cantor discovered two theorems: first, that the set of real numbers has the same cardinality as the power set of the naturals; and second, that a set and its power set have a different cardinality (see Cantor's theorem).
		This is version 6 of Cantor's diagonal argument, born on 2002-02-18, modified 2003-01-09.
	planetmath.org /encyclopedia/CantorsDiagonalArgument.html (358 words)

	"Uncountable": finite intuitions and Cantor's diagonal (beyond Peano)
		Cantor's motivation was about Fourier and the question of infinite frequency spectrum, that is the idea of "continuïty" --> what we now call the Reals.
		It's cardinality N is countable, and you may imagine that the whole lattice (of infinite "width and hight") 2^N is "infinitely larger" (Cantor's diagonal argument).
		Cantor, with his countable w x w square table, and its diagonal (;-) came pretty close: the difference between linear disctrete [ Peano's {+1}* ] and the continuum of reals [ {a,b}* ] is one of sequential (generating) dimension, countable: seqdim=1, uncountable (reals): seqdim >1, e.g.
	home.vianetworks.nl /users/benschop/finite.htm (6597 words)

	Cantor’s Diagonal Argument
		Because of its importance to the IFF metastack, we discuss, quote and paraphrase here in some detail Cantor’s diagonal argument as presented in the book Sets for Mathematics (2003) by Lawvere and Rosebrugh.
		As discussed in that book, in the nineteenth century George Cantor proved an important theorem using a diagonal argument, which implies the result that any set X is smaller than its power set
		Cantor, however, used his method to prove positive results, namely inequalities between cardinalities.
	grouper.ieee.org /groups/suo/IFF/notes/cantors-diagonal-argument.html (721 words)

	Cantors Diagonal argument (Site not responding. Last check: )
		Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countablyinfinite.
		A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem : for every set S the power set of S, i.e., the set of all subsets of S (here written as P(S)), is larger than S itself.
		Analogues of the diagonal argument are widely used in mathematics to prove the existence or nonexistence of certain objects.For example, the conventional proof of the unsolvability of the haltingproblem is essentially a diagonal argument.
	www.therfcc.org /cantors-diagonal-argument-341472.html (742 words)

	Cantor's Diagonal Argument
		The classic form of the argument goes as follows: given a mapping (N f :{mapping (N:{0,1})}) there is a mapping g defined by; g(i) = not(f(i,i)) where ({0,1}: not :{0,1}) swaps 0 <-> 1; clearly, g isn't f(i) for any i in N, since, by construction, it is distinguishable from each.
		That set a revolution in motion which came to its resolution, via the heroics of Russel, Whitehead and their peers, in the work of Kurt Gödel. Listen closely, and you may be able to hear the first pre-echoes of Gödel's fork somewhere in the background of the saga's first breakthrough.
		The crucial property of the diagonal argument is that f(N) is not a member of {f(n) : n in N}.
	www.chaos.org.uk /~eddy/math/diagonal.html (834 words)

	Cantor's diagonal argument - RecipeFacts (Site not responding. Last check: )
		Cantor's diagonal argument, also called the diagonalization argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.
		The diagonal argument was not Cantor's first proof of the uncountability of the real numbers; it was actually published three years after his first proof.
		The uncountability of the real numbers was already established by Cantor's first uncountability proof, but it also follows from this result.
	www.recipeland.com /facts/Cantor%27s_diagonal_argument (1098 words)

	BBC - h2g2 - What I don't like about Cantor’s Diagonal Argument (Site not responding. Last check: )
		Cantor's Diagonal Argument is a powerful but subtle technique for proving certain theorems about countability.
		Cantor's mechanism is supposed to generate a number we don't have in the sequence by confounding us at every turn - by making every digit it chooses one it didn't find.
		So I don't have a problem with the diagonal argument as such, but I believe it's considerably more subtle than most people give it credit for, precisely because of the deep understanding of infinities that is required to make effective use of it.
	www.bbc.co.uk /dna/h2g2/A386787 (1414 words)

	Cardinal number
		While in the finite world these two concepts coincide, when dealing with infinite sets one has to distinguish between the two.
		The position aspect leads to ordinal numbers, which were also discovered by Cantor, while the size aspect is generalized by the cardinal numbers described here.
		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.
	www.ebroadcast.com.au /lookup/encyclopedia/ca/Cardinality.html (1135 words)

	Three beliefs that lend illusory legitimacy to Cantor’s diagonal argument
		In the second case, Gödel’s use of the diagonal argument, in his seminal 1931 paper on formally undecidable propositions [Go31a], is purely illustrative.
		Although Turing appears to argue in his paper [Tu36] that Cantor’s argument can be taken to establish the Platonic existence of an uncomputable Turing real number, he seems to have been ambivalent about using the argument unrestrictedly whilst introducing his Halting argument.
		Whatever the reason, he offered an alternative argument that was, essentially, based on defining an uncomputable number-theoretic function, rather than on non-constructively postulating that a period, followed by a non-terminating sequence of the digits 0 and 1, necessarily defines a Dedekind real number.
	alixcomsi.com /Three_beliefs.htm (4352 words)

	Cantor's diagonal argument
		(It is also called the diagonalization argument or the diagonal slash argument.) It does this by showing that the interval (0,1), that is, the set of real numbers larger than 0 and smaller than 1, isn't countably infinite.
		Note: Strictly speaking, this argument only shows that the number of decimal expansions of real numbers between 0 and 1 isn't countably infinite.
		The diagonal argument is an example of reductio ad absurdum because it proves a certain proposition (the interval (0,1) isn't countably infinite) by showing that the assumption of its negation leads to a contradiction.
	www.wordlookup.net /ca/cantor's-diagonal-argument.html (896 words)

	Cantors Proof
		Now, take the diagonal I highlighted, and write down the number whose nth decimal digit is 1 if the nth diagonal term is anything other than 1, and 2 if the nth diagonal term is 1.
		A slight rework of this argument shows that for any set A, A and its power set P(A) [The set of all subsets of A.] are not in 1-1 correspondence.
		Here is another CantorsProof that demonstrates that the space occupied by the rational numbers is the same as the space occupied the integers (There's a related demonstration that the space of irrational numbers is larger, but I don't remember it).
	c2.com /cgi/wiki?CantorsProof (852 words)

	NationMaster - Encyclopedia: Enumeration
		Continue this process until all elements of the set have been assigned a natural number.
		have no enumeration as proved by Cantor's diagonalization argument.
		There exists an enumeration for a set if and only if the set is countable.
	www.nationmaster.com /encyclopedia/Enumeration (1093 words)

	Orðasafn: C
		Cantor continuity axiom samfellufrumsenda Cantors, samfellufrumsetning Cantors, samfelldnifrumsenda Cantors, samfelldnifrumsetning Cantors.
		Cantor ternary set Cantor-mengi, Cantorsmengi, talnamengi Cantors, þríundamengi Cantors, = Cantor's discontinuum.
		Cantor's diagonal method hornalínuaðferð Cantors, -> diagonal method.
	www.hi.is /~mmh/ord/safn/safnC.html (3824 words)

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