81. Paradoxes Resolved, Origins Illuminated posted by jrich 123, As I said previously, I m undecided about Cantor s ideas about infinities beyond aleph0, which would include his diagonal proof. http://www.metaresearch.org/msgboard/post.asp?method=Reply&TOPIC_ID=483&FORUM_ID

82. Crank Dot Net | Philosophy The completed infinities, mathematician Georg Cantor s infinite sets, could be explained as cardinal identities, akin to qualia from which finite subsets http://www.crank.net/philosophy.html

Metaphysics of Astrology: Astrological Signs 2004 Apr 06 metaphysics astrology astronomy philosophy ... physics "So how does this knowledge affect Astrology? Well it tells us that the planets do actually have some affect on our matter, and perhaps even more significantly, have a pronounced affect on the sun (causing cycles in its energy output) and thus cycles in our weather patterns. Thus it is quite reasonable that the position of the planets does subtly affect our evolution, and in particular certain aspects of our personality. I must admit that I did find remarkable similarities in my personality and the description of a Scorpio, as described in the section below on Astrological Signs. Likewise my partner Karene has a personality very consistent with her Libra sign (though this is irrelevant in terms of Science). We hope that you will enjoy this page on Astrology, and also hope that you find the Wave Structure of Matter an interesting explanation for how these distant planets can subtly affect our evolution and personality." Federation Internationale des Nombres 2003 Oct 30 mathematics philosophy "Download your FREE 50 files of geometric constructions ... NEW Industry benchmark problems. Possible NEW constructions. Academic Classical Geometry eJournal - must visit. Numbered triangle centers. e.g. create and use Euklides macros, new constructions etc. Your efforts. Short but noteworthy proofs. How good is YOUR Geometry. Get your ruler and compass out, (or your EUKLIDES!) ..."

83. Crank Dot Net | Cantor Was Wrong 2001 Jul 21 Cantor was wrong All infinite sets have the same number of members All infinities are equal (ie, there is only one infinity) http://www.crank.net/cantor.html

An Editor Recalls Some Hopeless Papers 2004 Feb 26 Cantor was wrong resources "I dedicate this essay to the two-dozen-odd people whose refutations of Cantor's diagonal argument have come to me either as referee or as editor in the last twenty years or so. Sadly these submissions were all quite unpublishable; I sent them back with what I hope were helpful comments. A few years ago it occurred to me to wonder why so many people devote so much energy to refuting this harmless little argument what had it done to make them angry with it? So I started to keep notes of these papers, in the hope that some pattern would emerge. These pages report the results. They might be useful for editors faced with similar problem papers, or even for the authors of the papers themselves. But the main message to reach me is that there are several points of basic elementary logic that we usually teach and explain very badly, or not at all." In PostScript format. Dilworth v. Dudley 2001 Oct 09 Cantor was wrong legal "The decision handed down by Judge Posner in the lawsuit brought by William Dilworth against Underwood Dudley, author of Mathematical cranks . The plaintiff was upset about being referenced in Dudley's book regarding his (cranky) refutation of Cantor's diagonal construction, and sued for defamation. The suit was dismissed 'for failure to state a claim.'"

84. Symmetry Forms The Basis Of Mathematical Truth The completed infinities, mathematician Georg Cantor s infinite sets, could be explained as cardinal identities, akin to qualia from which finite subsets http://www.webspawner.com/users/monolithiclogic/

Symmetry Forms the Basis of Mathematical Truth Some thoughts... The laws of nature are explained in terms of symmetry. The completed infinities, mathematician Georg Cantor's infinite sets, could be explained as cardinal identities, akin to "qualia" from which finite subsets, and elements of subsets, can be derived. Completed infinities, called "alephs" are distributive in nature, similar to the way that a set of "red" objects has the distributive property of redness. Predicates like "red" are numbers in the sense that they interact algebraically according to the laws of Boolean algebra. Take one object away from the set of red objects and the distributive number "red" still describes the set. The distributive identity "natural number" or "real number" describes an entire collection of individual objects. These alephs can be set into a one to one correspondence with a proper subset of of themselves. The "infinite" Cantorian alephs are really distributive. Yet, if we have a finite set of 7 objects, the cardinal number 7 does not really distribute over its individual subsets. Take anything away from the set and the number 7 ceases to describe it. Symmetry is analogous to a self evident truth and is distributive via the laws of nature, being distributed over the entire set called universe. A stratification of Cantorian alephs with varying degrees of freedom. More freedom = greater symmetry = higher infinity-alephs. So the highest aleph, the "absolute-infinity" distributes over the entire set called universe and gives it "identity".

85. UCAS Forms - Example Personal Statement :: InterStudent.co.uk :: One-stop Site F interests, including the history and foundations of mathematics (for instance Gödel s Incompleteness Theorem, Cantor s different infinities Aleph_0 and c http://www.interstudent.co.uk/displayarticle566.html

@import url("themes/interstudent_std/style/style.css"); Careers Community Education Finance ... REGISTER HERE Main Menu Home Login Register Resources Accommodation Advice Campus Life City Guides ... Student Web Directory Cool Stuff Euro 2004 Online Games Hottest Student 100 Sexiest ... Ecards Messages Registered users only! Register here for free Sponsored Cheap prices and discounts on student and you... Student flights from UK Net Guide For all your travel needs, including cheap an... Find flight deals for students from Skydeals Skydeals.co.uk offers thousands of flights on... City of the week - Oxford!! Student's Guide to Oxford (Jan 14, 2004) Kebab Vans the core of student life (Jul 25, 2002) Oxford - A General Guide (Aug 08, 2001) Academic Advice UCAS Forms - Example Personal Statement Posted by: gorgeousbeth on Tuesday, April 23, 2002 - 03:27 (1883 Reads) My only real advice is to start filling it in as soon as you have got it, and have decided which universities/courses you want to apply for. It might help if you make a rough draft of it all first, before filling in the proper form. Hopefully you'll avoid messing it up! Here's a personal statment that you may want to take inspiration from. The personal statement is probably the hardest bit to fill in. Did you do one for your personal record of achievement? If you did, take a look at that for some inspiration. Above all, sell yourself - but try not to repeat things that you have written elsewhere. For instance, there's not much point writing a lot about your academic achievements. What do you do outside of school/college? Why should the university accept you instead of someone else?

86. Set Theory: Cantor Cantor solved these difficulties for himself by saying there were two kinds of infinities, the consistent ones and the inconsistent ones. http://www.thoralf.uwaterloo.ca/htdocs/scav/cantor/cantor.html

Previous: Dedekind Next: Frege Up: Supplementary Text Topics Set Theory: Cantor typing . Another, more popular solution would be introduced by Zermelo. But first let us say a few words about the achievements of Cantor. We include Cantor in our historical overview, not because of his direct contribution to logic and the formalization of mathematics, but rather because he initiated the study of infinite sets and numbers which have provided such fascinating material, and difficulties, for logicians. After all, a natural foundation for mathematics would need to talk about sets of real numbers, etc., and any reasonably expressive system should be able to cope with one-to-one correspondences and well-orderings. Cantor started his career by working in algebraic and analytic number theory. Indeed his PhD thesis, his Habilitation, and five papers between 1867 and 1880 were devoted to this area. At Halle, where he was employed after finishing his studies, Heine persuaded him to look at the subject of trigonometric series, leading to eight papers in analysis. In two papers 1870/1872 Cantor studied when the sequence converges to 0. Riemann had proved in 1867 that if this happened on an interval and the coefficients were Fourier coefficients then the coefficients converge to as well. Consequently a Fourier series converging on an interval must have coefficients converging to 0. Cantor first was able to drop the condition that the coefficients be Fourier coefficients consequently any trigonometric series convergent on an interval had coefficients converging to 0. Then in 1872 he was able to show the same if the trigonometric series converged on

87. Deron's Planet 2 The conjecture that there are no intermediate infinities is most famously stated by Cantor s Continuum Hypothesis, expressed by the equation http://w3.iac.net/~dmeranda/topics/infinity.html

skip to main content Home Topics Fixpoints ... Purple Infinity What's larger than Infinity? Aleph Zero, transfinite numbers, and the Continuum Hypothesis S oon after we learn to count in our early childhood we begin to consider the task of seeing how high we can count. Much like trying to hold our breath forever, it is not long afterwards that we learn of the futility and contemplate the idea of infinity , the seemingly largest number that can ever be imagined. That supposed number symbolized by a strange sideways-8 symbol , like a Moebius strip with a neverending surface. Most of us leave it at that never pondering if there is more to it, or even realizing that we as mere humans are allowed to contemplate infinity. It turns out that infinity has an incredibly rich and structured existance. That symbol may be fine for casual use or even modern physics, but it is woefully imprecise and ill-defined for mathematics. For that a different symbol will become important, the aleph, , the first letter of the Hebrew alphabet. The modern concepts of infinity are primarily due to the lifelong compassion of German mathmatician Georg Cantor (1845-1918). His theory of the

88. Fatal_Mistake_of_Cantor_ENG The one and only basis for such differentiation of infinities is the famous George Cantor s theorem about the uncountablity of the set of all real numbers. http://www.com2com.ru/alexzen/papers/Cantor/Fatal_Mistake_of_Cantor.html

INFINITUM ACTU NON DATUR " - ARISTOTLE FATAL MISTAKE OF GEORG CANTOR by Alexander Zenkin e-mail: alexzen@com2com.ru An extended version of the paper " Mistake of Georg Cantor - Voprosy Filosofii (Philosophy Problems), 2000, No. 2, 163-168. The theory of sets is one of the basic disciplines of modern mathematics. On the other hand, the problem of the foundations of that sets theory is one of the most important problems of the philosophy of mathematics. Some historical, logical and philosophical aspects of this important and topical scientific problem are examined in this work. The Cantor's proof still has one more unique peculiarity. The point is that today it is quite difficult to surprise anybody with a complexity and a volume of mathematical proofs. For instance, the famous solution of the four-color problem occupies about 100 pages of a mathematical text (plus about 1000 hours of computer calculations). The famous solution of the Great Fermate's Theorem takes about 1000 pages of a mathematical text. Finally, the solution of the simple finite groups' classification problem takes about 15000 journal pages. Against this background, the proof of Cantor's theorem looks simply fantastic: it takes in all  10

89. Cognitive Computer Visualization Some new notes against Cantor’s set theory at the FOMArchive. AA.Zenkin, AS TO STRICT DEFINITIONS OF POTENTIAL AND ACTUAL infinities http//www.cs.nyu.edu http://www.com2com.ru/alexzen/

COGNITIVE COMPUTER VISUALIZATION FOR SCIENTIFIC DISCOVERIES in Mathematics, Logic, Philosophy, Psychology, and Education, AND FOUNDATIONS OF MATHEMATICS by Alexander A.Zenkin, Anton A.Zenkin e-mail: alexzen@com2com.ru I. COGNITIVE COMPUTER GRAPHICS (CCG). VIRTUAL REALITY WORLD OF THE NATURAL NUMBERS p There are two (computer) graphics: the illustrative one and the cognitive one. First graphics visualizes a knowledge which is already known (at least, for the "painter" though); the second one visualizes a knowledge which can be not known even for the author. Under certain conditions, such CCG-pictures can prompt new knowledge to a human-being. That is the Cognitive Computer Graphics (CCG) develops creative capabilities of a human-being, his intuition and visual, right-hemispheric thinking, and help him to make unexpected, non-trivial scientific discoveries even in well examined areas of Classical Theory of Natural Numbers and Classical Aristotle's Logic. LAST NEWS AS TO THE MOST ANCIENT MATHEMATICAL OBJECT: SEMIOTICS OF CCG-IMAGES OR THE BOTH-HEMISPHERICAL MAN-COMPUTER CCG-SYSTEM "

90. Body Cantor s diverse infinities , furthermore in calculus domains, etc.), while, on the. other side of the edge , a mathematical quantizing is on the side of the. http://pages.prodigy.net/jamikes/Influence.htm

ABOUT WHOLENESS AND REDUCTIONISM I n t r o d u c t i o n I never denied that my 'wholeness' ideas are written in a vague fashion. I also did fight against taking the reductionist model-based views as 'final', mostly as equationally formalized and quantized. I never denied the usefulness of the reductionist sciences, as the possibility of learning the world within our mental capabilities and establishing a technology, yet I preferred to include a 'scientific agnosticism' into our theoretical conclusions, citing the unknowable inter- connections of the 'models' by influences from 'beyond their boundaries' - set f or those 'models' (as limits for their observational domain). This writing is an attempt to realize the two divergent positions: the 'useful' reductionism and the vague wholism, - the first is hardly leading to full theoretical all-explaining knowledge, the second is (at least as of today) - not (yet) practical in concreto. The goal is to explore whether there is - maybe - an alternative view, a better platform, where the two may meet(? 1 Is reductionism a partner or a foe?

91. Everything And More - BooksReview - Www.smh.com.au Cantor s work solved the continuity problems that were first suggested by Zeno are, incredibly, different orders of infinity an infinity of infinities, in fact http://www.smh.com.au/articles/2004/03/05/1078464641725.html

national world opinion business ... place an ad extra crosswords personal finance travel education subscribe home delivery eNewsletter archives today's edition: am past 10 days site guide contact us ... Review Everything and More Reviewed by James Ley March 6, 2004 Print this article Email to a friend Everything and More A Compact History of Infinity By David Foster Wallace Orion, 320pp, $39.95 About 470BC the Greek philosopher Zeno of Elea developed a series of paradoxes which asserted that Achilles was incapable of overtaking a tortoise, that a runner can never finish a race, and that an arrow fired at a target never really moves. In each case, Zeno's reasoning exploited the fact that the distance between two points is infinitely divisible. The runner must first cover half the set distance, then half the remaining distance, then half the remainder, and so on. Since there is no end to this halving, the runner never arrives. Aristotle countered Zeno by arguing that infinite quantities were potential, not actual. The concept of infinity was therefore abstract, beyond comprehension and mathematically invalid. This effectively consigned infinity to the realm of metaphysics and, while advances gradually made this Aristotelian position more and more untenable, a coherent theory of the infinite remained elusive. It was not until the 19th century, when a Russian-born German named Georg Cantor proved infinite quantities could truly be grasped by the human mind, that infinity became a legitimate part of mathematics. Cantor's work solved the continuity problems that were first suggested by Zeno. To achieve this, he developed a form of set theory that could "denumerate" the members of infinite sets by establishing a one-to-one correspondence with other sets, and in the process he discovered that there are, incredibly, different orders of infinity - an infinity of infinities, in fact.

92. CommonSense in fact refers to an earlier article by Ellen Eischen and Robert Lipshitz2 on the politics of mathematics around the work of Georg Cantor on infinities, a very http://www.cs-journal.org/lll1/lll1science1.html

93. Ask NRICH an easy one below which was originally thought up by Cantor We suppose But, we can observe the higher infinities everywhere around us for example, all areas http://nrich.maths.org/discus/messages/2069/6866.html?1066857210

94. Constructive Mathematics Sets such as the set of all real numbers are by their very nature actual infinities. Considering such objects led Cantor to the astonishing conclusion that http://digitalphysics.org/Publications/Cal79/html/cmath.htm

Constructive Mathematics This approach is based on the belief that mathematics can have real meaning only if its concepts can be constructed by the human mind, an issue that has divided mathematicians for more than a century by Allan Calder It is commonly held that if human beings ever encounter another intelligent form of life in the universe, the two civilizations will share a basic mathematics that might well serve as a means of communication. In fact, since the time of Plato it has been generally believed that mathematics exists independently of man's knowledge of it and thus possesses a kind of absolute truth. The work of the mathematician, then, is to discover that truth. Not all mathematicians, however, have shared this belief in a "God-given" mathematics. For example, the 19th-century German mathematician Leopold Kronecker maintained that only counting was predetermined. "God made the integers," he wrote (to translate from the German). "All else is the work of man." From this point of view the work of the mathematician is not to discover mathematics but to invent it. Elements

95. TheologyWeb Campus - Cantor And The Omniscience Of God They might better understand it as there being infinities larger than infinite , ie I want to attack premise 4, ie, Cantor s proof for the uncountability of http://www.theologyweb.com/forum/showthread.php?t=1872

96. Infinity Nineteenthcentury dissension brought to the stage One infinities scenario has an immobile, wheelchair-bound Georg Cantor assailed by Leopold Kronecker. http://www.siam.org/siamnews/09-03/infinity.htm

search: From SIAM News, Volume 36, Number 7, September 2003 Hilbert's Hotel, Other Paradoxes, Come to Life in New "Math Play" By Kirsten Shepherd-Barr "Mathematics provides a new language for the theatre," says Luca Ronconi, director of John D. Barrow's exciting play Infinities , which finished a second successful run in Milan in May. Barrow's play does for mathematics what Michael Frayn's Copenhagen did for physics. Infinities actually enacts some of the great paradoxes or "thought experiments" about infinity: the Hotel Infinity in all its vastness, the notion of time-travel, the idea of living forever, Borges's Library of Babel with its endless corridors of books. Watching Infinities brings such concepts to life in a stunning combination of mathematics, philosophy, science, and theatre. For those who felt that David Auburn's play Proof was only tangentially and incidentally about mathematics, here is a play that truly engages the subject, for specialists and general audiences alike. Barrow provided the text, a mixture of original passages and selections from essays on infinity of his own and other people, ranging from Nietzsche to Borges to Hawking. The acclaimed Italian director Ronconi developed the staging in conjunction with the Teatro Piccolo and Sigma Tau Foundation in Italy. The result is a play that demonstrates the very concepts it deals with, and takes the genre of "science plays"-so popular in recent years-to a new level.

97. Solution from discrete wholes (Tiles 27, pp68, 151); and she realised, with Cantor, that Cantor s Paradox demonstrates the possibility of infinities with no number http://www.arts.uwa.edu.au/PhilosWWW/Staff/solution.html

THE UNIFORM SOLUTION 1. Introduction 2. Indeterminate Sense true? It is true, of course, if the sentence named 's' is translated 'p' - but then, why isn't this presumption made explicit? The logical truth is that which immediately resolves the many paradoxes involving Truth, in the same manner as above (see, e.g. Slater [18], [21]). In fact it also resolves Curry's Paradox, which Priest classifies differently. The central question for Tarskians is thus why they take to be necessary something which is plainly contingent, and whose very contingency removes the paradoxes that have bedevilled them. 3. Indeterminate Reference holds for all sets x, and so, in particular, it holds for where DN19 is the set of finite ordinals definable in less than 19 words. But he also wants to say since 'the least ordinal not in the set of finite ordinals definable in less than 19 words' defines a finite ordinal in less than 19 words - because of its length. any least ordinal not in a set is not in that set, i.e. it is necessary that But that does not employ a referential term 'the least ordinal'. So if we want to make a referential remark of the kind Priest had in mind, we must say, instead

98. Math Forum: Cantor's Solution: Denumerability Cantor s Solution Denumerability. A Math Forum Project 1, 2, 3, 4, 5, 0, 1, 1, -2, 2, Now, Cantor made the following definition http://mathforum.org/isaac/problems/cantor2.html

Cantor's Solution: Denumerability A Math Forum Project Table of Contents: Famous Problems Home The Bridges of Konigsberg The Value of Pi Prime Numbers ... Links In the example on the previous page, student B matched each number with its double, which resulted in the following correspondence: The integers can be put into correspondence with the natural numbers like this: Now, Cantor made the following definition: Definition : Two sets are equal in magnitude (i.e. size) if their elements can be put into one-to-one correspondence with each other. This means that the natural numbers, the integers, and the even integers all have the 'same number' of elements. Cantor denoted the number of natural numbers by the transfinite number (pronounced aleph-nought or aleph-null). For ease of notation, we will call this number d, since the set of all natural numbers (and all sets of equal magnitude) are often called denumerable , a , a corresponds to the natural number 1, a to 2, and so on. Theorem: The set of rational numbers is denumerable, that is, it has cardinal number d.

99. Math Forum:Infinite Sets In 1874 Georg Cantor worked out a system of degrees of infinitythat solved the problem once and for all and greatly increased mathematicians understanding of http://mathforum.org/isaac/problems/cantor1.html

Infinite Sets A Math Forum Project Table of Contents: Famous Problems Home The Bridges of Konigsberg The Value of Pi Prime Numbers ... Links Are there more integers or more even integers? Seems like a simple question, right? After all, every even integer is an integer but what about all the even integers? So there are more integers than there are even integers, right? But wait a second. How many even integers are there? An infinite number. And how many integers are there? An infinite number. Hmmmm.... "Infinity," says math student A, "is just a term... there's no way you can actually show me that there is the same number of each." "Okay, lets play..." says math student B. "Give me an integer, and I'll give you an even integer that corresponds to it. And if two of your integers are different, I guarantee that my two even integers will be different." Math Student A: Okay... 1 Math Student B: 2 A: 2 B: 4 A: 18 B: 36 A: -100 B: -200 A: n B: 2n A: I'm beginning to see what you mean. But let's consider some of the set theory we learned in math class. The set of even integers is contained in the set of integers, but is not equal to that set. So the two sets can't be the same size. (Who's right? What kind of sets did the teacher put on the board in class? How do these sets differ from those?)

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