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	Britain.tv Wikipedia - Georg Cantor
		In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland.
		Cantor remarked that he had effectively reproved a theorem, due to Liouville, to the effect that there are infinitely many transcendental numbers in each interval.
		Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one).
	www.britain.tv /wikipedia.php?title=Georg_Cantor (3597 words)

	Cardinal number - Biocrawler (Site not responding. Last check: 2007-10-14)
		Cantor invented the one-to-one correspondence, which easily showed that two finite sets had the same cardinality if there was a one-to-one correspondence between the members of the set.
		But, later that year, Cantor succeeded in proving that there were higher-order cardinal numbers using the ingenious but simple Cantor's diagonal argument.
		This can be 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.biocrawler.com /encyclopedia/Cardinal_number (1973 words)

	Axiomatic set theory - ExampleProblems.com
		The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B.
		Cantor gave two proofs that R is not countable, and the second of these, using what is known as the diagonal construction, has been extraordinarily influential and has had manifold applications in logic and mathematics.
		Cantor's development of set theory was still "naïve" in the sense that he did not have a precise axiomatization in mind.
	www.exampleproblems.com /wiki/index.php/Axiomatic_set_theory (2514 words)

	[No title]
		Cantor laid the foundation for the mathematical theory of the infinite by demonstrating that infinite sets, while essentially different from finite sets, nevertheless share with finite sets the property of being determinable by well-defined cardinal numbers.
		A proof of Cantor’s theorem by means of the Schröder-Bernstein theorem requires two conditions: (A) every set is cardinally similar to the set of all its one-element subsets, and (B) the set of all one-elements subsets of a set is cardinally less than the set of all its subsets.
		Cantor, in fact, attempted to demonstrate that collections such as the collection of cardinal numbers were also inconsistent multiplicities by finding a subset of the collection equinumerous to the set of all ordinals.
	www.sunysb.edu /philosophy/faculty/gmar/cantor.txt (6245 words)

	nib, bernstein, architecture, depliage, design, opera, video art, art, ceramics, sculpture (Site not responding. Last check: 2007-10-14)
		Bernstein was a highly regarded conductor among many musicians, in particular the members of the vienna philharmonic orchestra and the israel philharmonic orchestra, of which he was a regular guest conductor.
		In 1953 bernstein was the first american to conduct opera romantica at the teatro alla scala in milan cherubini's medea with maria callas.
		Bernstein was equally impressed, not only by the strength and resilience of the people, but by their unbridled cultural enthusiasm that seemed to hang on every note.
	bernst.info /nibu.html (1604 words)

	Informat.io on Georg Cantor
		Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers.
		Cantor was the son of Georg Waldemar Cantor, a Danish businessman who was a broker on the St Petersburg Stock Exchange, and Maria Anna Böhm, a Russian.
		Cantor introduces the notion of "power" (a term he took from Jakob Steiner) or "equivalence" of sets; two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them.
	www.informat.io /?title=georg-cantor (3375 words)

	Bernstein Diet -- Recommendations and Resources (Site not responding. Last check: 2007-10-14)
		Bernstein was born in Lawrence, Massachusetts to a Jewish family from Rovno, Russia and studied at Harvard (including composition with Walter Piston) and the Curtis Institute of Music in Philadelphia, where his teacher of conducting was Fritz Reiner.
		Bernstein's politics were decidedly left wing, but unlike some of his contemporaries, he was not fllisted in the 1950s.
		Daniel Julius Bernstein (sometimes known simply as djb; born October 29, 1971) is a professor at the University of Illinois at Chicago, a mathematician, a cryptologist, and a programmer.
	www.becomingapediatrician.com /health/16/bernstein-diet.html (970 words)

	Historia Matematica Mailing List Archive: [HM] Did Cantor prove the Schroeder-Bernstein theorem?
		theorem was given by Bernstein in 1897 and published in 1898 in
		Cantor in fact tried to prove the result in his final double treatise.
		Cantor's argument is as follows (according to the well-known
	sunsite.utk.edu /math_archives/.http/hypermail/historia/mar99/0147.html (605 words)

	Kids.Net.Au - Encyclopedia > Cardinal number (Site not responding. Last check: 2007-10-14)
		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.kids.net.au /encyclopedia-wiki/ca/Cardinal_number (1160 words)

	monacojerry: Wittgenstein's 'cancerous growth': An Incident in the Philosophy of Mathematics:
		Cantor's (and others such as Dedekind) ideas have since provided the basis for much of the development of mathematics thereafter.
		Cantor's theories made much of what was said previously in the philosophy of mathematics hard to justify.
		Cantor (and the way others developed Cantor) was just an example of this 'cancerous growth.' To the extent that I understand the issues here I think that Wittgenstein was being dogmatic.
	monacojerry.livejournal.com /8641.html?thread=38081 (1991 words)

	Cantor-Bernstein-Schroeder theorem: Definition and Links by Encyclopedian.com
		...Cantor-Bernstein-Schroeder theorem Cantor-Bernstein-Schroeder theorem In set...set theory, the Cantor-Bernstein-Schroeder Theorem is the theorem that for if there exist...,, mbox{if }x in C g^{-1}(x) & mbox{if }x ot in C end{matrix} ight. One can then...
		In set theory, the Cantor-Bernstein-Schroeder Theorem is the theorem that for if there exist injective functions
		An earlier proof by Cantor relied, in effect, on the Axiom of Choice by inferring the result as a corollary of the well-ordering theorem.
	www.encyclopedian.com /ca/Cantor-Bernstein-Schroeder-theorem.html (220 words)

	Axiomatic set theory (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-10-14)
		Cantor's development of set theory was still "naïve" in the sense that he didn't have a precise axiomatization in mind.
		There is no paradox in Brazilian logic, but that was almost completely unknown at the time.) In order to avoid this and similar paradoxes, Ernst Zermelo put forth a system of axioms for set theory in 1908.
		In particular Godel's 2nd incompleteness theorem which asserts that no sufficiently complex recursively axiomatizable system can prove its own Consistency can be used to prove Independence results.
	axiomatic-set-theory.iqnaut.net.cob-web.org:8888 (2624 words)

	Wikipedia: Bernstein
		Bernstein is the German word for amber, the fossil resin.
		People whose family name is or was Bernstein include
		This is a disambiguation page; that is, one that just points to other pages that might otherwise have the same name.
	www.factbook.org /wikipedia/en/b/be/bernstein.html (93 words)

	Cardinal number - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-10-14)
		The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874–1884.
		Cantor identified the fact that one-to-one correspondence is the way to tell that two sets have the same size, called "cardinality", in the case of finite sets.
		By the Schroeder-Bernstein theorem, this is equivalent to there being both a one-to-one mapping from X to Y and a one-to-one mapping from Y to X.
	en.wikipedia.org.cob-web.org:8888 /wiki/Cardinal_number (2205 words)

	PlanetMath: Schroeder-Bernstein theorem
		The Schröder-Bernstein theorem is useful for proving many results about cardinality, since it replaces one hard problem (finding a bijection between
		See Also: an injection between two finite sets of the same cardinality is bijective, proof of Schroeder-Bernstein theorem using Tarski-Knaster theorem
		This is version 4 of Schroeder-Bernstein theorem, born on 2002-02-18, modified 2006-04-15.
	planetmath.org /encyclopedia/SchroederBernsteinTheorem.html (83 words)

	PlanetMath: Schroeder-Bernstein theorem, proof of
		"proof of Schroeder-Bernstein theorem" is owned by mps.
		Cross-references: cardinality, map, implies, composition, theorem, identity map, subsets, sequence, bijection, injection, lemma
		This is version 16 of proof of Schroeder-Bernstein theorem, born on 2002-07-05, modified 2005-07-11.
	planetmath.org /encyclopedia/ProofOfCantorSchroederBernsteinTheorem.html (94 words)

	Knaster–Tarski theorem - Wikipedia, the free encyclopedia
		Since complete lattices cannot be empty, the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence of a least (or greatest) fixed point.
		Often a more specialized version of the theorem is used, where L is assumed to be the lattice of all subsets of a certain set ordered by subset inclusion.
		A kind of converse of this theorem was proved by Anne C. Davis: If every order preserving function f : L → L has a fixed point, then L is a complete lattice.
	en.wikipedia.org /wiki/Knaster-Tarski_theorem (434 words)

	Supplement3
		This is answered by a theorem of Cantor.
		(Cantor) For any set S, there is no bijective correspondence between S and the power set of S.
		The theorem implies that there are different infinite cardinals, since cl(S) and cl(P(S)) must be distinct.
	bradley.bradley.edu /~delgado/404/Schroeder-Bernstein.html (679 words)

	Cantor–Bernstein–Schroeder theorem - Wikipedia, the free encyclopedia
		In set theory, the Cantor–Bernstein–Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B.
		For the path that is infinite in both directions, and for the finite cycles, we choose to map every element to its predecessor in the path.
		The theorem is also known as the Schroeder-Bernstein theorem, but the trend has been to add Cantor's name, thus crediting him for the original version.
	en.wikipedia.org /wiki/Cantor-Bernstein-Schroeder_theorem (429 words)

	Theory of Sets of Points
		There are no definitely accepted landmarks in the didactic treatment of Georg Cantor's magnificent theory, which is the subject of the present volume.
		Theorem for the (outer) content analogous to Theorem 20 of § 52
		Cantor's (1,1)-correspondence between the points of the plane, or n-dimensional space and those of the straight line
	www.agnesscott.edu /lriddle/women/abstracts/young_SetTheory.htm (1213 words)

	M3000 Homework #22 (Site not responding. Last check: 2007-10-14)
		This process is also clearly reversible and so it is a one-to-one map.
		The composition of these two maps is therefore one-to-one, and it maps each integer to an element of S. By the Cantor-Schröder-Bernstein Theorem, S is equivalent to
		This process is clearly reversible, we get the original numbers back by reading alternate digits, so it is a one-to-one map.
	www-math.cudenver.edu /~wcherowi/courses/m3000/abhw22.html (484 words)

	Georg_Cantor - The real meaning from Timesharetalk wikipedia (Site not responding. Last check: 2007-10-14)
		^ Cantor himself is quoted as referring to "his Israelite grandparents." It is unknown what Cantor meant by this
		It was interpreted by some scholars as meaning that Cantor's paternal grandparents were "Sephardic Jews." Many sources have since taken this information as meaning that Cantor was Jewish, and we find such references in, most prominently, the Encyclopedia Judaica (art.
		He was of the Lutheran faith all his life and is sometimes referred to as a "Christian in science." (Georg Cantor 1845-1914" by Walter Purkert and Hans Joachim Ilgauds, Birkhaeuser, 1987) and (Tannery's "Memoires Scientifiques: Correspondance", edited by A. Dies, and published by J.-L. Heiberg and H.-G. Zeuthen, vol.
	www.timesharetalk.co.uk /wiki.asp?k=Georg_Cantor (3638 words)

	MAT246Y Course Material Related Links (Site not responding. Last check: 2007-10-14)
		Show that all polynomials (with integer coefficients) are countable by writing that set as a countable union of countable sets.
		Use Dedekind's Theorem to show that the set of integers Z and the interval of real numbers between 0 and 2, [0, 2], are both infinite(which is of course not surprising).
		The Cantor-Schröder-Bernstein theorem A Proof to the theorem (as rigourous as the one we did in class)
	www.math.toronto.edu /jkorman/Math246Y/links.htm (334 words)

	Math 3000 Sample Final Exam Questions
		A famous theorem (the Cantor-Schröder-Bernstein Theorem) states that if there exists an injection from A to B and another injection from B to A, then A and B are equivalent.
		Use this result and the fact that the open interval (0,1) is equivalent to
		Having found one-to-one maps in both directions, by the Cantor-Schröder-Bernstein theorem, the two sets are equivalent.
	www-math.cudenver.edu /~wcherowi/courses/m3000/abexfs.html (1112 words)

	Math211
		Other topics should probably include an introduction to symbolic logic, which stresses logical implication, quantifiers, contrapositive, logical equivalence, etc. (but does NOT stress symbolic manipulation or truth tables) and some coverage of functions – surjective, injective, bijective, and inverse functions.
		Some topics that might be included or touched upon include counting and probability, algebraic structures – binary operations and properties, the Cantor/Schroeder/Bernstein Theorem, and the binomial theorem.
		An instructor may introduce any topic that contributes to the goals of the course
	www.math.emich.edu /~ocalin/IFC1/math211.htm (435 words)

	Events - Colloquium Series - 2001 - Mathematics and Computer Science, Stetson University (Site not responding. Last check: 2007-10-14)
		During this talk, we will clarify what a cardinal is, why cardinals cannot be sets, and how we generally define operations on the cardinals.
		We will learn, via the Cantor-Schroeder-Bernstein Theorem, that sets with "really big" cardinality do not cooperate with human intuition sometimes.
		We will discuss a number of interesting results, including Russell's paradox and the Axiom of Choice.
	www.stetson.edu /departments/mathcs/events/colloquium/2001/index.shtml (1468 words)

	M505: Abstract algebra
		Successful performance in the class depends critically on completion of the homework assignments.
		O: Set constructions, First Isomorphism Theorem, proof techniques, ordered sets, monoids - codes and free monoids, dynamical systems, semilattices, relations
		III: Categories and lattices, diagonalization, Tarski Fixed Point Theorem, Cantor-Schröder-Bernstein Theorem, limits, presentations, adjoint functors, Galois theory, tensor products.
	orion.math.iastate.edu /jdhsmith/class/M505S04.htm (176 words)

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