Georg Cantor Concepts

The Haskell Road to Logic, Maths and Programming

by Kees Doets, Jan van Eijck and Jan Eijck · 15 Jan 2004

a set, that of a function can be defined. However, in a context that is not set-theoretic, it could well be an undefined notion. Georg Cantor (1845-1915), the founding father of set theory, gave the following description. The Comprehension Principle. A set is a collection into a whole of definite

Thinking in Numbers

by Daniel Tammet · 15 Aug 2012 · 212pp · 68,754 words

in size? How can a part of any collection not be smaller than the whole? Borges’s taxonomy is clearly inspired by the work of Georg Cantor, a nineteenth-century German mathematician whose important discoveries in the study of infinity provide us with an answer to the paradox. Cantor showed, among other

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our body could ever lose its count. Like the ox, each knows intimately the moment when to stop. Higher than Heaven On 22 January 1886 Georg Cantor, who had discovered the existence of an infinite number of infinities, wrote a letter to Cardinal Johannes Franzen of the Vatican Council, defending his ideas

The Beginning of Infinity: Explanations That Transform the World

by David Deutsch · 30 Jun 2011 · 551pp · 174,280 words

infinity. The reach of explanations cannot be limited by fiat. One expression of this within mathematics is the principle, first made explicit by the mathematician Georg Cantor in the nineteenth century, that abstract entities may be defined in any desired way out of other entities, so long as the definitions are unambiguous

Infinite Ascent: A Short History of Mathematics

by David Berlinski · 2 Jan 2005 · 158pp · 49,168 words

used one concept that he could not precisely define to explore other concepts that he could not precisely see. Two centuries were to pass before Georg Cantor was to discover the words that in 1684 Leibniz lacked; during all that time, mathematicians continued to use functions of the most remarkable variety, indifferent

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-confidence. All eyes on Berlin, of course. Let me interrupt myself to ask: Is a crack-up coming? And to answer: Of course it is. Georg Cantor was born in St. Petersburg in 1845 and spent the first eleven years of his life in the rich, warm, syrupy Russian milieu made accessible

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romantic temperaments, both concerned to please the other. These details might suggest a young man embarking on a modest mathematical career. Nothing of the sort. Georg Cantor initiated a great upheaval in nineteenth-century thought, carrying out one of those revolutions that like certain earthquakes survive in the form of aftershocks long

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to replace numbers. The success of this definition at once suggests the possibility that in some unspeakable way, nothing exists beyond the sets themselves. Although Georg Cantor spent a few dutiful months at the Zurich Polytechnique, the death of his father in 1863 and the acquisition of his inheritance allowed him to

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beyond the kiwis; nominalists have always demurred, perhaps from a sense that in matters of being as in matters of sin, nothing exceeds like excess. Georg Cantor was a mathematical Platonist, and more, a mathematical Plotinist, his unacknowledged master the Greek philosopher Plotinus, and the universe that he constructed from the philosophical

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before and after. Open any contemporary mathematical text at random, and the theorems, proofs, and definitions are all expressed in terms of the ideas that Georg Cantor created. The tools and the techniques of set theory have so completely been adopted by the mathematical community as to become almost identified with the

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unfair as the initial condemnation of Cantor’s purely mathematical ideas by mathematicians too timid to take large chances. By the turn of the century, Georg Cantor had nonetheless achieved at least a part of his heart’s desire: He was at last admired. No one doubted that he had changed the

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that in 1900 he was prepared to address to the future: He asked mathematicians to provide a proof that the axioms of arithmetic were consistent. Georg Cantor had twenty years before defended consistency as the single probative standard for all of mathematics, the free creations of the human mind, like homeopathic medicine

Wireless

by Charles Stross · 7 Jul 2009

Turing would be nearly a hundred by now. All our long-term residents are named for famous mathematicians. We’ve got Alan Turing, Kurt Godel, Georg Cantor, and Benoit Mandelbrot. Turing’s the oldest, Benny is the most recent—he actually has a payroll number, sixteen.” I’m in five digits—I

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don’t know whether to laugh or cry. “Who’s the nameless one?” I ask. “That would be Georg Cantor,” she says slowly. “He’s probably in room four.” I bend over the indicated periscope, remove the brass cap, and peer into the alien world

What We Cannot Know: Explorations at the Edge of Knowledge

by Marcus Du Sautoy · 18 May 2016

third of all the black lines that you see. Do this to infinity. The resulting picture is called the Cantor set, after the German mathematician Georg Cantor, whom we will encounter in the last Edge, when I explore what we can know about infinity. Suppose this Cantor set was actually controlling the

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the end of the nineteenth century, an intellectual shift occurred. Thanks to one man’s finite brain, the infinite seemed to be within reach. For Georg Cantor, infinity was not simply a way of speaking. It was a tangible mathematical object: The fear of infinity is a form of myopia that destroys

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our minds. At the end of the nineteenth century you might expect a separation between those practising science and those who were religious. And yet Georg Cantor was both, and he writes about religion influencing his mathematical ideas. In a similar vein to Bruno, who contemplated an infinite universe, Cantor’s belief

The Music of the Primes

by Marcus Du Sautoy · 26 Apr 2004 · 434pp · 135,226 words

this question about the existence of a machine to test for provability. Turing’s idea was based on a startling discovery made in 1873 by Georg Cantor, a mathematician from Halle in Germany. He had found that there were different sorts of infinities. It may seem a strange proposition, but it is

The Infinite Book: A Short Guide to the Boundless, Timeless and Endless

by John D. Barrow · 1 Aug 2005 · 292pp · 88,319 words

SAXONY’S PARADOX GALILEO’S PARADOX CADMUS AND HARMONIA TERMINATOR 0, ½ AND 1 COUNTABLE INFINITIES UNCOUNTABLE INFINITIES THE TOWERING INFINITO chapter five - The Madness of Georg Cantor CANTOR AND SON THE CHRONICLE OF KRONECKER CANTOR, GOD, AND INFINITY– THE TRINITY WITH AFFINITY ALL’S SAD THAT ENDS BAD chapter six - Infinity Comes

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ostracised because of their attempts to exclude infinities from mathematics. At the root of all the fuss was one man’s work. The genius of Georg Cantor showed how to make sense of the paradoxes of infinity that Galileo had first identified three hundred years before. What is the nature of an

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. It is only when you are infinitely rich that your assets depend totally on the order in which you count them up. Fig 4.5 Georg Cantor (1845–1918) with his wife, Vally.18 Not surprisingly, arguments like this made mathematicians very nervous about infinities. It is easy to see why infinity

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end. Out of all this ambiguity and confusion, clarity emerged suddenly in the nineteenth century, due to the single-handed efforts of one brilliant man. Georg Cantor (1845–1918) produced a theory that answered all the objections of his predecessors and revealed the unexpected richness hiding in the realm of the infinite

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were well received by scholars in these fields. Alas, within mathematics the story was quite different, as we shall see. chapter five The Madness of Georg Cantor ‘To be listened to is a nearly unique experience for most people. It is enormously stimulating. Man clamors for the freedom to express himself and

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water in so many hours. I found it quite enthralling.’ Agatha Christie2 Cantor & Co. was a successful international wholesale business, and as a result young Georg Cantor was one of six children who grew up in comfortable circumstances, attending good private schools in Frankfurt. Georg had many talents and might well have

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the infinite numbers; and above all, because I have followed its roots, so to speak, to the first infallible cause of all created things.’14 Georg Cantor was very interested in how mathematics might reveal the existence of God. In letters to Cardinal Franzelin, he indicated that the infinite, or the ‘Absolute

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Nobel and has similar monetary value. The first winner, in 2003, was French mathematician J.P. Serre. 18. Photograph of Georg Cantor with his wife, Vally, c. 1880, reproduced from Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard University Press, 1979. 19. F. Hutcheson (1694–1746), Inquiry into the

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information conveyed by the words ‘the statement T is true’ is different from that contained in the statement T itself. Chapter five The Madness of Georg Cantor 1. (1888–1973) US naturalist and environmental activist http://www.melodyonline.com/quotes2.htm 2. A. Christie, An Autobiography, HarperCollins, London, 1998. 3. Quoted in

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E. Schechter, Handbook of Analysis and its Foundations, Academic, New York, 1998. 4. J. Dauben, Georg Cantor, Princeton University Press, 1990, p. 1. 5. Photograph of Leopold Kronecker, c. 1885, copyright © akg-images. 6. D. Burton, History of Mathematics, 3rd edn, Wm

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. C. Brown, Dubuque, IA, 1995, p. 593. 7. Dauben, Georg Cantor, p. 134. 8. Ibid. 9. Ibid., p. 135. 10. Ibid., p. 136. 11. Ibid., p. 147. 12. Letter 15 Feb 1896 to Esser, H. Meschkowski

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, p. 9. 4. Cantor was unusual for his time in believing that the Universe was finite in age and in physical extent. 5. J. Dauben, Georg Cantor, Princeton University Press, 1990, p. 146. 6. Rucker, Infinity, p. 309. 7. A. Conan Doyle, ‘The Boscombe Valley Mystery’, The Adventures of Sherlock Holmes, Oxford

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Alan Lightman. Illustrations copyright © 1993 by Chris Costello. Reprinted by permission of Pantheon Books, a division of Random House, Inc. Princeton University Press: Excerpt from Georg Cantor by J. Dauben. Reprinted courtesy of Princeton University Press. Warner Bros. Publications U. S. Inc.: Excerpt from “As Time Goes By” by Herman Hupfeld. Copyright

Zero: The Biography of a Dangerous Idea

by Charles Seife · 31 Aug 2000 · 233pp · 62,563 words

do I know this? I have studied it…I have followed its roots, so to speak, to the first infallible cause of all created things. —GEORG CANTOR Infinity was no longer mystical; it became an ordinary number. It was a specimen impaled on a pin, ready for study, and mathematicians were quick

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as these singularities behave, they were no longer mysterious to mathematicians, who were learning to dissect the infinite. The master anatomist of the infinite was Georg Cantor. Though he was born in Russia in 1845, Cantor spent most of his life in Germany. And it was in Germany—the land of Gauss

Turing's Vision: The Birth of Computer Science

by Chris Bernhardt · 12 May 2016 · 210pp · 62,771 words

Machine Diverge on Its Encoding? Is Undecidable The Acceptance, Halting, and Blank Tape Problems An Uncomputable Function Turing’s Approach 8. Cantor’s Diagonalization Arguments Georg Cantor 1845–1918 Cardinality Subsets of the Rationals That Have the Same Cardinality Hilbert’s Hotel Subtraction Is Not Well-Defined General Diagonal Argument The Cardinality

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said “You don’t have to believe in God, but you should believe in The Book.” Turing’s proofs, along with those of Gödel and Georg Cantor on which they are based, are definitely in The Book. This book is for the reader who wants to understand these ideas. We start from

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contradiction, which is a method of proving a statement by assuming its negation and then deriving a contradiction. The other key idea really comes from Georg Cantor’s diagonal argument, but we will give it in terms of a paradox that was first stated by Bertrand Russell. Since many readers may not

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wanted an undecidable problem that was immediately comprehensible to mathematicians. He found one using an ingenious argument of Georg Cantor. 8 Cantor’s Diagonalization Arguments “I see it, but I don’t believe it!” Georg Cantor “I don’t know what predominates in Cantor’s theory — philosophy or theology, but I am sure that

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there is no mathematics there.” Leopold Kronecker “No one shall drive us from the paradise which Cantor has created for us.” David Hilbert Georg Cantor 1845–1918 Georg Cantor moved with his family from St. Petersberg to Germany when he was eleven years old. After completing his dissertation at the University of Berlin