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Topic: Diagonal argument

	Cantor's diagonal argument - Wikipedia, the free encyclopedia
		Assume (for the sake of argument) that the interval [0,1] is countably infinite.
		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.
	en.wikipedia.org /wiki/Cantor's_diagonal_argument (961 words)

	Cantor's diagonal argument: Definition and Links by Encyclopedian.com - All about 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, is 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.
	www.encyclopedian.com /ca/Cantor%27s-diagonal-argument.html (805 words)

	Cantors Diagonal argument/Talk - Wikipedia
		If we accept the existence of uncountable infinities (and i guess you do if you accept the diagonal argument) then it is pretty clear that the set of all sets must be of uncountable size if it exists.
		That is in fact the main difference between the actual diagonalization argument and its generalization that is used in the second proof.
		I will add a remark that the second proof is not equal to the actual diagonal argument but a generalization of it.
	nostalgia.wikipedia.org /wiki/Cantors_Diagonal_argument/Talk (546 words)

	Uncountable set - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		Formally, an uncountable set is defined as one whose cardinality is strictly greater than $\aleph_0 (aleph-null), the cardinality of the natural numbers.$
		The best known example of an uncountable set is the set R of all real numbers; Cantor's diagonal argument shows that this set is uncountable.
		The diagonalization proof technique can also be used to show that several other sets are uncountable as well, for instance the set of all infinite sequences of natural numbers (and even the set of all infinite sequences consisting only of zeros and ones) and the set of all subsets of natural numbers.
	www.eastcleveland.us /project/wikipedia/index.php/Uncountable (465 words)

	PlanetMath: Cantor's diagonal argument (Site not responding. Last check: 2007-10-21)
		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).
		The proof of the second result is based on the celebrated diagonalization argument.
		This is version 6 of Cantor's diagonal argument, born on 2002-02-18, modified 2003-01-09.
	planetmath.org /encyclopedia/CantorsDiagonalArgument.html (357 words)

	[No title]
		The lower non-zero subdiagonal is the first element of the argument, whilst the highest non-zero super-diagonal is given by the last element of the argument.
		The argument should be a vector of vectors of values which determine the upper triangle of the matrix.
		The first element of the argument specifies the value of the first subdiagonal, the subsequent elements specify the value of the diagonal and subsequent super-diagonals, all other elements are zero.
	www.win.tue.nl /~amc/oz/om/cds/linalg5.html (1052 words)

	Cantor's Diagonal Argument
		The great grand-daddy of diagonal arguments is Cantor's, which proved that some infinities are bigger than others.
		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.
		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)

	[No title]
		It is therefore ironic that “Cantorian” arguments about the nature of the Transfinite have recently been appropriated by some contemporary philosophers of religion in an attempt to discredit the notion of omniscience and so to disprove the existence of God.
		Like many contemporary atheistic arguments, the argument we are examining implicitly insists that the question of the coherence of theism be raised prior to the question of the evidence for theism.
		If Grim arguments were successful, however, their success would seem to undermine his ability to state the very conclusions his arguments were supposed to establish.
	www.sunysb.edu /philosophy/faculty/gmar/cantor.txt (6245 words)

	SingaporeMoms - Parenting Encyclopedia - Ontological argument (Site not responding. Last check: 2007-10-21)
		The argument works by examining the concept of God, and arguing that it implies the actual existence of God; that is, if we can conceive of god, then god exists — it is thus self-contradictory to state that god does not exist.
		In order to understand the place this argument has in the history of philosophy, it is important to understand the essence of the argument in the context of the Influence of Hellenic philosophy on Christianity.
		Another traditional criticism of the argument (first found in Gassendi's Objections to Descartes' Meditations, and later modified by Kant) is that existence is not a perfection, because existence is not a property as such, and that referring to it as a property confuses the distinction between a concept of something and the thing itself.
	www.singaporemoms.com /parenting/Ontological_argument (3581 words)

	Cantor's Diagonal Proof
		Fortunately, the diagonal argument applied to a countably infinite list of rational numbers does not produce another rational number.
		Of course, although the diagonal argument applied to our countably infinite list has not produced a new RATIONAL number, it HAS produced a new number.
		You would then call this overall set "the real numbers", and assert that Cantor's diagonal argument does not apply (because any acceptably simple definition is presumably one of at most a countably infinite set of definitions).
	www.mathpages.com /home/kmath371.htm (1582 words)

	BBC - h2g2 - What I don't like about Cantor’s Diagonal Argument - A386787
		I think this is a misguided use of the argument because it requires you to bring a highly developed understanding of countability issues to the table in the first place.
		The situation in which the diagonal fails to generate a new number however is where it runs off the side too quickly - if the matrix of numbers is narrow and tall, the algorithm doesn't have time to generate a new number.
		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/classic/A386787 (1533 words)

	Cantor's diagonal argument -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		Cantor's diagonal argument is a proof devised by (Click link for more info and facts about Georg Cantor) Georg Cantor to demonstrate that the (Any rational or irrational number) real numbers are not (Click link for more info and facts about countably infinite) countably infinite.
		For example, the conventional proof of the unsolvability of the (Click link for more info and facts about halting problem) halting problem is essentially a diagonal argument.
		Similarly, the question of whether there exists a set whose cardinality is between s and P(s) for some s, leads to the (Click link for more info and facts about generalized continuum hypothesis) generalized continuum hypothesis.
	www.absoluteastronomy.com /encyclopedia/c/ca/cantors_diagonal_argument.htm (1097 words)

	A Look at a Flaw in Cantor's Diagonalverfahren (Site not responding. Last check: 2007-10-21)
		The famous diagonal "proof" of Danish mathematician Georg Cantor (1845-1918) appears to be flawed.
		The diagonal "proof" opens with a window into a representation of the upper-left hand corner of an array that is regarded as infinite to the right and infinite downwards.
		Since its appearance between 1874 and 1895 in the work of Cantor the diagonal argument, (construction, device, method, motto, principle, procedure, process, proof, trick) which was his second diagonal procedure, has been used as an apparently sound mathematical proof by many workers, notably Godel in 1931 and Church, Kleene, Post, Rosser, and Turing in 1936.
	www.tphta.ws /TPH_FCDP.HTM (749 words)

	Encyclopedia: Cantor's-diagonal-argument (Site not responding. Last check: 2007-10-21)
		In mathematics the term countable set is used to describe the size of a set, e.
		Contrary to what most mathematicians believe, Georg Cantors first proof that the set of all real numbers is uncountable was not his famous diagonal argument, and did not mention decimal expansions or any other numeral system.
		In computability theory the halting problem is a decision problem which can be informally stated as follows: Given a description of a program and its initial input, determine whether the program, when executed on this input, ever halts (completes).
	www.nationmaster.com /encyclopedia/Cantor%27s_diagonal_argument (1594 words)

	Physics Help and Math Help - Physics Forums - A question on Cantor's second diagonalization argument
		If we take a list of all rational numbers and we use the diagonal argument to prove the existance of a real number not on that list, then of course that number has to be irrational.
		Cantor's first diagonal argument constructs a specific list of the rational numbers that is not the list you provided.
		The point of Cantor's argument was to say "suppose we have a list of all real numbers" and then show that that leads to a contradiction.
	www.physicsforums.com /printthread.php?t=7471 (2539 words)

	Home Page (Site not responding. Last check: 2007-10-21)
		Cantor presents an argument, called the diagonal argument, to show that the cardinality of the real numbers is greater than the cardinality of the natural numbers.
		They have the same cardinality because you can use the square function, x2, to map from each element of N to each element of S. Cantor's diagonal argument shows that there is not a one to one mapping between all of the natural numbers to all of the reals.
		Cantor's says in his argument to assume that the cardinality of the reals is the same as the cardinality of the natural numbers.
	www.oswego.edu /~msmith15/PHY309 (315 words)

	The Washington Monthly
		The Cantor diagonal argument takes any "enumeration" of the reals and explicitly constructs a number which must be omitted by that enumeration.
		The Cantor diagonal argument guarantees that the constructed number is a positive distance away, at the least, from each of the numbers in the supposed enumeration of the real numbers.
		So the fact that diagonal number differs from every representation of a rational on the list doesn't mean that the diagonalization isn't a representation of a rational different from the representation of that rational that's on the list.
	www.washingtonmonthly.com /archives/individual/2004_01/003123.php (8086 words)

	Cantor's diagonal argument (Site not responding. Last check: 2007-10-21)
		(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)

	[No title]
		The "diagonal" comes from the fact that we used the nth digit from the nth number in the list to create a new number, thus going down a diagonal line in the list.
		This kind of diagonal argument has been used in other famous proofs - Godel used it for his incompleteness theorem (he used a combination of a diagonal argument and the liar's paradox "this statement is false"), and Turing used it to prove there are noncomputable problems.
		When I presented this argument to a bunch of middle school students, the response I got was "but infinity is infinity" -- making me realize that I should never have used the word infinity to begin with.
	www.math.nyu.edu /~campbelm/stuff/nylife/041201.txt (1852 words)

	BBC - h2g2 - Cantor's Diagonal Argument
		His proof is generally referred to as Cantor's Diagonal Argument.
		We will do this by looking at the diagonal which is formed by the first decimal of the first number in the list, the second decimal of the second number, and so on.
		It is the use of this diagonal that has given the proof its name.
	www.bbc.co.uk /dna/h2g2/A479180 (1429 words)

	[Phil-logic] CANTOR'S DIAGONAL ARGUMENT: A NEW ASPECT (Site not responding. Last check: 2007-10-21)
		I follow your idea that we can take the "diagonal set" D1 that is unmapped by some function f1, and design a new function f2 that has the range of f1 plus D1.
		Then there is a new diagonal set D2 that lies beyond the range of f2.
		This ONLY X* is sufficient (from the fact that diagonal method is a version of counter-example method and from the point of view of “ELEMENTARY logic”) in order to prove that the assumption is false, and to conclude that P(X)
	philo.at /pipermail/phil-logic/2004-April/003587.html (694 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)

	LME - Array Functions
		First argument is a function reference, an inline function, or the name of a function as a string.
		With a second argument, the diagonal is moved by that amount in the upper right direction for positive values, and in the lower left direction for negative values.
		The size of the matrix is specified by one integer for a square matrix, or two integers (either as two arguments or in a vector of two elements) for a rectangular matrix.
	www.calerga.com /doc/LME_arr.htm (2286 words)

	Diagonal argument - Encyclopedia, History, Geography and Biography (Site not responding. Last check: 2007-10-21)
		Diagonal argument - Encyclopedia, History, Geography and Biography
		A variety of diagonal arguments are used in mathematics.
		This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title.
	www.arikah.net /encyclopedia/Diagonal_argument (98 words)

	Philosophy 357
		Diagonal arguments and Goedel numbering coalesce in producing the infamous sentence which "says of itself" that it is unprovable.
		One will be a non-constructive method, based on Turing's theorem and using a diagonal argument, which shows that some sentence is true and unprovable but gives no clue as to how to find it.
		Another will be a version of the original argument which painstakingly produces the sentence in question via Goedel numbering.
	www.owlnet.rice.edu /~phil357 (930 words)

	Penrose's Goedelian Argument: A review of Roger Penrose's "Shadows of the Mind".
		All that the Gödel incompleteness theorem requires of F is the former, since that is equivalent to the consistency of F. But Penrose tends to emphasize the global notion of soundness and to tie it to his Platonistic philosophy of mathematics.
		The argument goes something as follows: how could we know that F is sound if we did not understand what F is about -- its intended interpretation -- and see that the axioms of F are all true of that interpretation and that its rules of inference all preserve truth?
		It is by such means, the argument continues, that we recognize the soundness of systems from PA all the way up to ZF set theory and beyond.
	psyche.cs.monash.edu.au /v2/psyche-2-07-feferman.html (4659 words)

	Cantor's diagonal argument Info - Encyclopedia WikiWhat.com (Site not responding. Last check: 2007-10-21)
		(It is also called the diagonalization argument or the diagonal slash argument.)
		Contrary to what many mathematicians believe, the diagonal argument was not Cantor's first proof of the uncountability of the real numbers, but was published three years later.
		Cantor's original proof proceeds by showing that the interval (0,1), that is, the set of real numbers larger than 0 and smaller than 1, is not countably infinite.
	www.wikiwhat.com /encyclopedia/c/ca/cantor_s_diagonal_argument.html (786 words)

	Science Fair Projects - Georg Cantor
		Cantor recognized that infinite sets can have different sizes, distinguished between countable and uncountable sets and proved that the set of all rational numbers Q is countable while the set of all real numbers R is uncountable and hence strictly bigger.
		The original proof of this, devised in December 1873 and published in early 1874, used a moderately complicated reduction argument in which one starts with a countable list of real numbers and an interval on the real line.
		His later 1891 proof uses his celebrated diagonal argument.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Georg_Cantor (696 words)

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