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Topic: Georg Cantor

Related Topics

Cantor set

Transfinite number

Cardinal number

Cantor dust

Aleph number

Set of uniqueness

Countably infinite

Naive set theory

Aleph null

Infinity

Set theory

Axiomatic set theory

Absolute Infinite

Successor cardinal

Real number

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	Cantor, Georg Ferdinand Ludwig Philipp (1845-1918)
		Cantor realized that irrational numbers can be represented as infinite sequences of rational numbers, so that they can be understood as geometric points on the real-number line, just as rational numbers can.
		In 1873-74 Cantor proved that the rational numbers could be paired off, one by one, with the natural numbers and were therefore countable, but that there was no such one-to-one correspondence with the real numbers.
		Henri Poincaré said that Cantor’s theory of infinite sets would be regarded by future generations as “a disease from which one has recovered.” Kronecker went further and did all he could to ridicule Cantor’s ideas, suppress publication of his results, and block Cantor’s ambition of gaining a position at the prestigious University of Berlin.
	www.daviddarling.info /encyclopedia/C/Cantor.html (679 words)

	Georg Cantor and Cantor's Theorem
		Cantor not only found a way to make sense out an actual, as opposed to a potential, infinity but showed that their are different orders of infinity.
		Georg Cantor was born March 3, 1845 in Saint Petersburg, Russia.
		Georg Cantor's academic career was at the University of Halle, a lesser level university.
	www2.sjsu.edu /faculty/watkins/cantorth.htm (514 words)

	Georg Cantor (Site not responding. Last check: 2007-10-16)
		Cantor is also known for his work on the unique representations of functions by means of trigonometric series (a generalized version of a Fourier series).
		Cantor was born in Saint Petersburg Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a Russian musician, Maria Anna Böhm.
		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.
	georg-cantor.iqnaut.net (540 words)

	GEORG CANTOR
		Georg Ferdinand Ludwig Philip Cantor was born in St. Petersburg, Russia in 1845.
		Georg was a very religious Christian and had a deep interest in medieval theology which became very influential in his motivation to investigate the Infinite.
		Cantor determined that although not all infinite sets are denumberable, the set of all positive and negative rational numbers is denumberable and the set of all algebraic numbers (roots of polynomials with integer coefficients) is denumberable.
	www.engr.iupui.edu /~orr/webpages/cpt120/mathbios/gcant.htm (1376 words)

	Cantor biography
		Cantor was promoted to Extraordinary Professor at Halle in 1872 and in that year he began a friendship with Dedekind who he had met while on holiday in Switzerland.
		Cantor continued to correspond with Dedekind, sharing his ideas and seeking Dedekind's opinions, and he wrote to Dedekind in 1877 proving that there was a 1-1 correspondence of points on the interval [0, 1] and points in p-dimensional space.
		Cantor also discussed the concept of dimension and stressed the fact that his correspondence between the interval [0, 1] and the unit square was not a continuous map.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Cantor.html (3038 words)

	No. 1484: Georg Cantor
		When he was seventeen, his father died, and Cantor went on to finish a doctorate in mathematics, in Berlin.
		Cantor asked if counting all the infinite number of points on a line would be like counting all the points in a surface.
		When he was 33 he wrote: "The essence of mathematics is freedom." Cantor had to value freedom very highly -- freedom coupled with the iron discipline of mathematics, freedom expressed as the driving curiosity of a bright child, freedom to pursue innocent fascination until it finally touched the world we all live in.
	www.uh.edu /engines/epi1484.htm (534 words)

	Cantor's Concept of Infinity:
		Cantor advanced infinite series representations of irrationals to claim that their existence was equivalent to that of the transfinites.
		Cantor produced a classic example of contingent rationality when he drew the distinction between transfinite numbers, which exist in the human mind, and Absolute Infinity, which is beyond all human determination, and exists only in the mind of God.
		Cantor was explicitly opposed to the prevailing materialism of his scientific community, which regarded the physical universe as eternal and unbounded.
	www.asa3.org /ASA/PSCF/1993/PSCF3-93Hedman.html (5863 words)

	Georg Cantor Biography \| World of Scientific Discovery
		Cantor, however, was poorly paid by the university, and he strove to obtain a better, more prestigious, teaching appointment in Berlin but was blocked by jealous professional rivals.
		Cantor's chief mathematical pursuit was a deeper understanding of the concept of infinity.
		Cantor died in a mental hospital in 1918.
	www.bookrags.com /biography/georg-cantor-wsd (366 words)

	Georg Ferdinand Ludwig Philipp Cantor (Site not responding. Last check: 2007-10-16)
		Cantor was promoted to Extraordinary Professor in 1872, and in that year he began a friendship with Dedekind.
		Cantor was surprised at his own discovery and wrote: "I see it, but I don't believe it!" Of course this had implications for geometry and the notion of dimension of a space.
		Cantor published a rather strange paper in 1894 which listed the way that all even numbers up to 1000 could be written as the sum of two primes.
	www.stetson.edu /~efriedma/periodictable/html/Ca.html (608 words)

	Georg Cantor
		Cantor's own faith in his theory was partly theological.
		This fairly detailed Georg Cantor biography is from the School of Mathematics and Statistics at the University of St Andrews, Scotland.
		Extract from Cantor's Uber einen die trigonometrischen Reihen betraffenden Lerhrsatz which is one of his first publications on the theory of functions.
	www.erraticimpact.com /~19thcentury/html/cantor.htm (527 words)

	Cantor, Georg (1845-1918) -- from Eric Weisstein's World of Scientific Biography
		German mathematician who built a hierarchy of infinite sets according to their cardinal number.
		Cantor's highly original views were vigorously opposed by his contemporaries, especially Kronecker.
		Dauben, J. Georg Cantor: His Mathematics and Philosophy of the Infinite.
	scienceworld.wolfram.com /biography/CantorGeorg.html (109 words)

	LaRouche Afterward Cantor’s Transfinite
		Cantor’s general form of solution to conceptualization of the notion of infinite in a non-pathological way, is to express the Many-ness of very large arrays within a specific theorem-lattice by a One.
		Cantor’s work remains a great contribution to mankind, and his efforts to clarify this issue with a representative of the Vatican are an honorable part of that.
		Cantor’s repeated insistence on this during his writings of the 1880’s is indispensable for avoiding the commonplace blunders of the proverbial “usual generally recognized authorities” in their reading of both the Beiträge and these earlier writings.
	www.schillerinstitute.org /fid_91-96/943_LaR_transfinite_after.html (3650 words)

	Biography of Georg Ferdinand Ludwig Philipp Cantor
		Cantor is famous for founding the set theory and introducing the concept of transfinite numbers.
		Cantor's Theorem can be written as A Also closely related to Cantor's* advancements in transfinite set theory was his definition of the continuum.
		Georg Cantor was a very successful mathematician who advanced the study of mathematics in many ways.
	www.andrews.edu /~calkins/math/biograph/biocanto.htm (1154 words)

	Georg Ferdinand Ludwig Philipp Cantor (Site not responding. Last check: 2007-10-16)
		Georg had a very diverse background; his father was a Danish Jewish merchant that had converted to Protestantism, and his mother was a Danish Roman Catholic.
		Cantor is famous for founding the set theory and introducing the concept of transfinite (infinite) numbers with his discovery of cardinal numbers.
		Georg Ferdinand Ludwig Philipp Cantor was born in 1845 and died in 1918.
	www.andrews.edu /~calkins/math/biograph/199899/biocanto.htm (618 words)

	Glossary of People: Ca (Site not responding. Last check: 2007-10-16)
		Cantor wrote little of a philosophical nature, but his startling achievements in fundamental investigations of mathematics stimulated the deeper enquiry into the epistemological foundations of mathematics which had a profound influence on Western thought during the first three decades of the twentieth century.
		Of prosperous and cultured Danish-born parents, Cantor’s talent for mathematics was observed when he was only 14 and by the age of 18 he was studying under the great mathematicians Karl Weierstrass and Leopold Kronecker, and by 1867 had published a doctoral thesis on one of Gauss’s famous unsolved problems.
		Cantor’s theory quickly became controversial, leading to the study of a whole sequence of “transfinite numbers” each larger than the one before.
	www.marxists.org /glossary/people/c/a.htm (2944 words)

	Georg Cantor, Set Theory and Transfinite numbers
		Georg Cantor's set theory proof of the existence of numbers larger than infinity still fascinates me to this day.
		This is the question Cantor asked himself and found the answer is NO. That is to say, the number of members of the infinite natural set would not be equal to those in the real set.
		Cantor introduced a definition of the cardinality of power sets that leads to a strange conclusion for transfinite numbers.
	www.geocities.com /roble_wais/cantor_set_theory.htm (1619 words)

	test (Site not responding. Last check: 2007-10-16)
		Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg, Russia, on March 3, 1845, to Danish parents.
		However, Cantor’s natural abilities were so obvious that after several years of study his father finally agreed to let him study mathematics in Zurich, the capital of Switzerland.
		Cantor’s study of mathematics gained him many mathematical enemies, because it was new and revolutionary.
	www.mathsyear2000.org /timeline/test-mathinfo.php?m=georg-cantor (769 words)

	Cantor, Georg (Site not responding. Last check: 2007-10-16)
		He also advanced the study of trigonometric series, was the first to prove the nondenumerability of the real numbers, and made significant contributions to dimension theory.
		Cantor received his doctorate in 1867 and accepted a position at the University of Halle in 1869, where he remained.
		Closely related to Cantor's work in transfinite set theory was his definition of the continuum as a connected, perfect set.
	euler.ciens.ucv.ve /English/mathematics/cantor.html (143 words)

	Cantor
		Heine, one of his senior colleagues at Halle, who challenged Cantor to prove the open problem on the uniqueness of representation of a function as a trigonometric series.
		Mittag-Leffler and Cantor all but stopped shortly after this event and the flood of new ideas which had led to Cantor's rapid development of set theory over about 12 years seems to have almost stopped.
		Cantor was elected president of the Deutsche Mathematiker-Vereinigung at the first meeting and held this post until 1893.
	www.edu365.com /aulanet/comsoc/persones_tecniques/Cantor.htm (2756 words)

	Georg Cantor (Site not responding. Last check: 2007-10-16)
		He was born in Saint Petersburg Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a Russian musician, Maria Anna Böhm.
		The vast majority of working mathematicians accept Cantor's work on transfinite sets and recognize it as a paradigm shift of major importance.
		All the vessels route, did not reach San Diego till twenty days after the San Antonio, her crew but one sailor and the cook were left alive; the rest, along Antonio also lost eight of her crew from the same dreadful disease.
	georg-cantor.kiwiki.homeip.net (632 words)

	Quotes by Georg Cantor
		The final problem which Cantor grappled with was the realization that there could be no set containing everything, since, given any set, there is always a larger set -- its set of subsets.
		Cantor came to the conclusion that the Absolute was beyond man's reach, and identified this concept with God.
		Subsequently, he believed that he is doing theology by introducing transfinite numbers into mathematics.
	www.braungardt.com /Mathematica/quotes_by_georg_cantor.htm (443 words)

	10.5. Cantor, Georg (1845-1918)
		Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg, Russia, on March 3, 1845.
		Georg was not at all happy about this idea but he lacked the assertiveness to stand up to his father and relented.
		Among other things, he delayed or suppressed completely Cantor's and his followers' publications, raged both written and verbal personal attacks against him, belittled his ideas in front of his students and blocked Cantor's life ambition of gaining a position at the prestigious University of Berlin.
	web01.shu.edu /projects/reals/history/cantor.html (967 words)

	Georg Cantor Page
		Cantor showed that almost all numbers are transcendental by proving the real numbers were not countable while proving that the algebraic numbers were countable.
		Cantor was talking about lack of acceptance of Hyperbolic Geometry as opposed to Euclidean Geometry.
		Cantor's doctoral thesis was titled "In mathematics the art of asking questions is more valuable than solving problems.
	www.geocities.com /CollegePark/Union/3461/cantor.htm (402 words)

	Cantor, Georg - Famous mathematicians pictures, posters, gifts items, note cards, greeting cards, and prints
		Cantor's image is flanked by the "Aleph", the first letter of the Hebrew alphabet, which Cantor used (accompanied by subscripts) in his descriptions of transfinite numbers -- quite simply numbers which were not finite.
		The graphic set which backs Cantor's image began with an algorithm to generate the Cantor set, to which color was applied, and then universal operators related to color transition and magnification, ultimately resulting in a unique image whose essence was the Cantor set.
		Cantor came came to the conclusion that the Absolute was beyond man's reach, and identified this concept with God.
	www.mathematicianspictures.com /Mathematicians/Cantor.htm (534 words)

	Amazon.com: Georg Cantor: Books: Joseph Warren Dauben (Site not responding. Last check: 2007-10-16)
		Georg Cantor's creation of transfinite set theory was an achievement of major consequence in the history of mathematics.
		As Dauben is careful to point out, his research indicates that Cantor would have suffered from the bouts of depression independent of the degree of opposition to his work and most likely independent of what kind of work he did.
		Cantor was in fact a very strong personality, he stood up well against the opposition and ultimately was proven to be correct.
	www.amazon.com /Georg-Cantor-Joseph-Warren-Dauben/dp/0691024472 (1671 words)

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