1. Math Forum:Infinite Sets At its core lay that troubling concept that haunts all of mathematics infinity. In 1874 Georg Cantor worked out a system of degrees of infinitythat solved 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, let's 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?)

2. Infinity - Wikipedia, The Free Encyclopedia Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, http://en.wikipedia.org/wiki/Infinity

Infinity From Wikipedia, the free encyclopedia Jump to: navigation search The lemniscate , ∞, in several typefaces. For other uses, see Infinity (disambiguation) Infinity (sometimes symbolically represented by ) is a concept in many fields, most predominantly mathematics and physics , that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. The word comes from the Latin infinitas or "unboundedness". In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers . In number systems incorporating infinitesimals , the reciprocal of an infinitesimal is an infinite number, i.e. a number greater than any real number. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities For example, the set of

3. Mathematics And The Emperor's New Clothes. 9/21/2008. Not until the late twentieth century did Cohen prove the relationship between Cantor's infinities and the Axiom of Choice. So what's the big deal? http://www.netautopsy.org/jharempr.htm

MATHEMATICS AND THE EMPEROR'S NEW CLOTHES. One of the luxuries of being an amateur mathematician, who does not earn his living at mathematics, is that one may ask questions that might get a salaried mathematician fired from his job. That is, the questions are so outrageous or apparently simple-minded, that one is fired either for blasphemy or for gross incompetence. So what does all this have to do with mathematics? Somewhere late in my graduate school training in biomathematics, it dawned on me that there are about a dozen central ideas in mathematics, all of them basically fairly simple once understood, from which one may derive all the important theories of mathematics. The amazing thing is that such simple things took such a long time to internalize in our culture. For example, why was the Greco-Latin culture so resistant to the idea of ZERO, discovered one thousand years B.C. by the Babylonians (as a place-holder on the abacus)? The idea was banished from ancient Greece, and not really embraced in Europe until the sixteenth century, by merchants not mathematicians, who could do their accounting far more easily with Arabic numerals (with zero) than with Roman numerals (without zero). A few more examples: Pythagoras's proof [Singh]; infinity; infinitesimals (Calculus; Newton/Leibniz; Weierstrass; Robinson's calculus); open/closed sets (Heine-Borel theorem); computational complexity (NP complete problem; why the Sieve of Eratosthenes takes so long to solve); symbolic logic [Boole; Lewis and Langford]; Goedel's proof [Casti and DePauli]; Hilbert's Tenth Problem [Davis]; Fermat's Last Theorem [Singh]; Riemann hypothesis [Davis]; fractals [Lauwerier], etc.

4. The Repressed Content-Requirements Of Mathematics Wittgenstein's arguments against Cantor's infinitiesin turn considered idiotic by most mathematicians. The larger issue in the foregoing is as follows. http://www.henryflynt.org/studies_sci/reqmath.html

Back to H.F. Philosophy contents The Repressed Content-Requirements of Mathematics Henry Flynt [started c. 1987; this draft 1994] (c) 1994 Henry A. Flynt, Jr. A. Mathematics, as it is conceived in the twentieth century, has presuppositions about perception, and about the comprehension of lived experience relative to the apprehension of apparitions, which are repressed in professional doctrine. It also has presuppositions of a supra-terrestrial import which are repressed. The latter concern abstractions whose reality-character is an incoherent composite of features of sensuous-concrete phenomena. In the twentieth century, mathematicians have been taught to say by rote, "We are beyond all that now. We are beyond psychology, and we are beyond independently subsisting abstractions." This recitation is a case of denial. Indeed, if this recitation were true, it would allow mathematics nowhere to live. Certainly social conventionsbeloved by the Vienna Circleare too unreliable to found the truths which mathematicians claim to possess. (Truths about the decimal value of [pi], or about different sizes of infinity, for example.) To uncover the repressed presuppositions, a combination of approaches is required. (One anthropologist has written about "the locus of mathematical reality"but, being an academic, he merely reproduces a stock answer outside his field, namely that the shape of mathematics is dictated by the physiology of the brain.)

5. Cantor's Infinities - SciForums.com 12 posts 9 authors - Last post Oct 6, 2008According to Cantor s Set Theory, there are different types of infinities, or in mathematical terms, the equation doesnt always hold. http://www.sciforums.com/showthread.php?t=86210

6. Georg Cantor - Wikipedia, The Free Encyclopedia Georg Ferdinand Ludwig Philipp Cantor was a mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. http://en.wikipedia.org/wiki/Georg_Cantor

Georg Cantor From Wikipedia, the free encyclopedia Jump to: navigation search Georg Cantor Born Georg Ferdinand Ludwig Philipp Cantor March 3, 1845 Saint Petersburg Russian Empire Died Halle Province of Saxony German Empire Residence Russian Empire German Empire Fields Mathematics Institutions University of Halle Alma mater ETH Zurich University of Berlin ... Karl Weierstrass Doctoral students Alfred Barneck Set theory Georg Ferdinand Ludwig Philipp Cantor (pronounced /ˈkæntɔr/ KAN -tor [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkʰantɔʁ] ; March 3 O.S. February 19] – January 6, 1918) was a German mathematician , best known as the inventor of set theory , which has become a fundamental theory in mathematics. 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 . In fact, Cantor's theorem implies the existence of an " infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware. Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even shocking—that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré and later from Hermann Weyl and L. E. J. Brouwer

7. Newsletter On Proof Famous Paradoxes Zeno's Paradox and Cantor's Infinities - The Problem of Points an age-old gambling problem that led to the development of probability by http://www.lettredelapreuve.it/OldPreuve/Newsletter/981112.html

Herbst P. G. What works as proof in the mathematics class . Ph.D. Dissertation, The University of Georgia, Athens GA. USA Lopes A. J. Raccah P.-Y. (1998) L'argumentation sans la preuve : prendre son biais dans la langue. Interaction et cognitions . II(1/2) 237-264. Arzarello F. Micheletti C. Olivero F. Robutti O. (1998) A model for analysing the transition to formal proofs in geometry. (Volume 2, pp.24-31) Arzarello F. Micheletti C. Olivero F. Robutti O. (1998) Dragging in Cabri and modalities transition from conjectures to proofs in geometry. (Volume 2, pp. 32-39) Baldino R. (1998) Dialectical proof: Should we teach it to physics students. (Volume 2, pp. 48-55) Furinghetti F. Paola D. (1998) Context influence on mathematical reasoning. (Volume 2, pp. 313-320) Gardiner J. Hudson B. (1998) The evolution of pupils' ideas of construction and proof using hand-held dynamic geometry technology. (Volume 2, pp. 337-344) Garuti R.

8. Infinity: You Can't Get There From Here -- Platonic Realms MiniText Cantor completely contradicted the Aristotelian doctrine proscribing actual, � completed� infinities, and for his boldness he was rewarded with a lifetime of http://www.mathacademy.com/pr/minitext/infinity/

INTRODUCTION HISTORY CANTOR CARDINALS ... www.mcescher.com INTRODUCTION then end? does never ends remained psychologically vexing for most of us. All children try at some point to see how high they can count, even having contests about it. Perhaps this activity is born, at least in part, of the felt need to challenge this notion of endlessness twice hundred million infinity ... to the last of which a good answer was hard to find. How could you get bigger than infinity? Infinity plus one? And what's that? Fish , by M.C. Escher Infinity, of course, infected our imaginations, and for some of us it cropped up in our conscious thoughts every now and then in new and interesting ways. I had nightmares for years in which I would think of something doubling in size. And then doubling again. And then doubling again. And then doubling again. And then until my ability to conceive of it was overwhelmed, and I woke up in a highly anxious state. and then I was awake, wide-eyed and perspiring. Only when I studied mathematics did I discover that my dream contained the seed of an important idea, an idea that the mathematician John Von Neumann had years before developed quite consciously and deliberately. It is called the Von Neumann heirarchy , and it is a construction in set theory.

9. Math Forum - Ask Dr. Math In Cantor's set theory, the idea of having a universal set or a set of everything cannot be true, due to the basic contradiction that arises from the nature of set theory. http://mathforum.org/library/drmath/view/72803.html

Associated Topics Dr. Math Home Search Dr. Math Cantor's Infinities and Universal Sets Date: 07/17/2008 at 03:27:17 From: Martin Subject: Cantor's Infinities In Cantor's set theory, the idea of having a universal set or a set of everything cannot be true, due to the basic contradiction that arises from the nature of set theory. Based on this, when looking at Cantor's Hierarchy of Infinities, does the hierarchy of infinities still hold? Truly, if an absolute infinity existed then it would accommodate everything...contradicting the idea of no universal set. Date: 07/17/2008 at 08:32:31 From: Doctor Tom Subject: Re: Cantor's Infinities Hi Martin. You're rightyou cannot have a set that, say, contains one infinite set of each size or you'll run into the problem you mention. One way around it, for the purposes of discussion of the idea, is to use the idea of a "class" that can be that large. The restriction, of course is that if such a class is too large to be a set (in other words, it can't be built from the allowed operations of set theory), then it is called a "proper class", and cannot be contained in any set or other class. That way I can talk to other mathematicians about "the class containing all the ordinal numbers", but I can't include that in any set or class. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/

10. The Actual Infinite As A Day Or The Games.(monotheism And Pantheism) - The Revie A century and a third the intense light of the infinity of God as a metaphor for the wonder of Cantor's infinities. The metaphor continues issues of mathematics and infinity http://www.highbeam.com/doc/1G1-160811936.html

11. Answers.com - Where Did The Cantors Infinities Happen Can you answer this question? Answer it or get updates discuss research share Facebook Twitter Search Related answers If the temperature is at infinity then what will http://wiki.answers.com/Q/Where_did_the_cantors_infinities_happen

12. INTEGRITY - Robust Non-Fractal Complexity - Text Of NECSI/ICCS1 Paper In an era of Relativity, Quantum Mechanics, Cantor's infinities, holography, Zadeh Logic, quark architecture, spread spectrum transmission, Complexity, and the like, we are http://www.ceptualinstitute.com/uiu_plus/necsi1paper.htm

THE INTEGRITY PAPERS - James N. Rose UIU Group http://www.ceptualinstitute.com Robust Non-Fractal Complexity Presented at the NECSI / ICCS International Conference on Complex Systems September 21-26, 1997; Nashua New Hampshire USA Section Links Abstract Complexity: Orchestration of Multiple Stochastic Systems Redefining the general structure of Mathematics The new Weltanschauung ... Summary Abstract: Primary versions of complexity to date have been considered relative to fractal models. They have tended to show that complexly ordered patternings arise or emerge after massive iterations of some relatively simple functions which, on their face, do not indicate that important relational and temporal patternings are nascently inherent in them. Corollary work (Prigogine, et al) has shown that in some cases contra-entropy plateaus of stability exist far from initial equilibrium conditions, giving secondary and tertiary conditions on which to build complex systems. These are important and pervasive factors of complexity. Yet, complexity can also be seen in situations which do not involve inordinate membership or interaction samples, and also, in situations that are not easily assessable by equilibrium statistics. It is the author's contention that there also exists a more general and robust form of complexity generating mechanism denotable in simple systems with non-homogeneous construction (that is, in systems which have independent yet interactive sub-components). These sub-components can be evaluated with their own behavior-space, independent from yet interactive with the behavior-space of the system at large.

13. Cantor's Concept Of Infinity: Such totality Cantor called Absolute Infinity; it is beyond all mathematical determination, and can be comprehended only in the mind of God. http://www.asa3.org/ASA/PSCF/1993/PSCF3-93Hedman.html

Cantor's Concept of Infinity: Implications of Infinity for Contingence by THE REVEREND BRUCE A. HEDMAN, Ph.D. Department of Mathematics University of Connecticut West Hartford, CT 06117 From Perspectives on Science and Christian Faith 46 (March 1993): 8-16 Georg Cantor (1845-1918) was a devout Lutheran whose explicit Christian beliefs shaped his philosophy of science. Joseph Dauben has traced the impact Cantor's Christian convictions had on the development of transfinite set theory. In this paper I propose to examine how Cantor's transfinite set theory has contributed to an increasingly contingent world view in modern science. The contingence of scientific theories is not just a cautious tentativeness, but arises out of the actual state of the universe itself. The mathematical entities Cantor studied, transfinite numbers, he admitted were fraught with paradoxes. But he believed that they were grounded in a reality beyond this universe, not finally determinable by any mathematical system. Introduction Contingence T owards the close of the twentieth century I believe that Christians are finding the climate of science to be more hospitable to our faith than did our forbearers in the nineteenth. I shall refer to Newtonian mechanics as it was developed in the eighteenth and nineteenth centuries as

14. Lectures Lectures. Calendar Conundrums Archimedes and His World Cantor's Infinities Goedel and Undecidability Geometry, Logic and Physics Escher and Symmetry http://www.math.princeton.edu/facultypapers/Conway/

Lectures Calendar Conundrums Archimedes and His World Cantor's Infinities Goedel and Undecidability ... (Free Will Lectures can be found here)

15. Letters - Salon.com The author, having not said anything interesting about quantum mechanics, then turns to Cantor's infinities. His formulas took mathematics and humanity to the next level. http://www.salon.com/technology/letters/2004/10/14/cancer_nano

Hot Topics Jon Stewart Rally to Restore Sanity Halloween ... About Salon Store Communities Letters Open Salon The Well Table Talk ... Letters Thursday, Oct 14, 2004 15:30 ET Letters The real problems with cancer research, nanotechnology and religion: Readers respond to Greg Barrett's "Ignoring the Big C" and Howard Lovy's "Nanotech Angels." [Read "Ignoring the Big C." While I agree with the Greg Barrett's points on the impediments to "cancer cure" research, his emphasis is fundamentally misplaced. I'll support more "cure" work when we put the cause in our crosshairs. Look at the chart of synthetic chemical production and the chart of cancer incidence. Not coincidentally they are the same chart and move rapidly upward in tandem after WWII. We eat, drink, breathe and slather our bodies with synthetic chemicals that are known carcinogens, cause birth defects, neurodegenerative disease, autoimmune dysfunction, asthma... Need I go on? Bill Moyers recently reported on this and found when doctors tested his blood that he was carrying around about 100 measurable synthetic chemicals very few of which his grandparents would have harbored. Barrett notes that reduced smoking has reduced cancers and that smarter lifestyles have reduced heart disease. Here's the Duh Theory of Cancer: Wean ourselves from petroleum (and all its derivatives), pesticides and plastics, and cancer will once again become the rarity it once was on the natural earth.

16. Welcome To The Hotel Infinity At the heart of Set Theory is a hall of mirrorsthe paradoxical infinity. Georg Cantor was known to have said, I see it, but I do not believe it, about http://www.ccs3.lanl.gov/mega-math/workbk/infinity/inbkgd.html

Infinity is for Children-and Mathematicians! How Big is Infinity? Most everyone is familiar with the infinity symbolthe one that looks like the number eight tipped over on its side. The infinite sometimes crops up in everyday speech as a superlative form of the word many . But how many is infinitely many? How far away is "from here to infinity"? How big is infinity? You can't count to infinity. Yet we are comfortable with the idea that there are infinitely many numbers to count with: no matter how big a number you might come up with, someone else can come up with a bigger one: that number plus oneor plus two, or times two. Or times itself. There simply is no biggest number. Is there? Is infinity a number? Is there anything bigger than infinity? How about infinity plus one? What's infinity plus infinity? What about infinity times infinity? Children to whom the concept of infinity is brand new, pose questions like this and don't usually get very satisfactory answers. For adults, these questions don't seem to have very much bearing on daily life, so their unsatisfactory answers don't seem to be a matter of concern. At the turn of the century, in Germany, the Russian-born mathematician Georg Cantor applied the tools of mathematical rigor and logical deduction to questions about infinity in search of satisfactory answers. His conclusions are paradoxical to our everyday experience, yet they are mathematically sound. The world of our everyday experience is finite. We can't exactly say where the boundary line is, but beyond the finite, in the realm of the

17. Re: Physical Models Of Set Theory Russell observed that Cantor's infinities do not correspond By the instructors and curriculum committes in the Dedekind and Frege both set aside the question of http://sci.tech-archive.net/Archive/sci.logic/2005-04/msg00001.html

Re: Physical models of set theory From : "mitch" < Date : Thu, 31 Mar 2005 19:53:27 GMT By the instructors and curriculum committes in the Mathematics Department at the University of Chicago. I have posted to this newsgroup because of a vested interest in the foundations of mathematics. I realize that most of these posts go unread and are basically incomprehensible to most people who might read them. So, I probably should not expand too much on this answer. Oh well... When one reads original sources, the disservice of territorial wars in academia become clarified. Dedekind and Frege both set aside the question of objectification. It was a central theme in Kant and Husserl. Klein sought invariance in automorphisms. Frege fixed on the intent behind the interpretation of constant symbols. Cantor devised the notion of set-as-object. Plucker and Klein viewed membership as a generalized geometric incidence. Carnap lamented Cantor's objectification. He promoted the fictionalism of Russell and Frege (Russell was explicit about "logical fictions" and Frege was explicit about not treating

18. Help Help: Ok This Is Gonna Sound Completly Obsurd And Science Fictional… - He That’s hard to understand mathematically unless you get into set theory and cantor’s infinities and things, but people seemed to sort of intuitively get it ……… and most http://help.com/post/202882-ok-this-is-gonna-sound-completly-ob

InsertFishHere An Unknown Location Basically yesterday i was trying to think of a cool scene for a film. After not long i had this vision of being in Central london during WW3 and the streets are deserted. Suddenly this modern version of an air raid siren goes off and these invisable aircraft appear in the distance (dont ask how) but on the roof tops of the sky scrapers theres these large AA guns that start fireing at these aircraft. I look up and see the AA guns fireing these large light blue laser projectiles at the aricraft and the noise the guns make is so lound that the ground rumbles and you can feel it in your chest like with big fireworks, also everything around me is rattling round like mad as the AA guns fire into the air. This open post was written 2 years, 1 month V/U/S Edit Post Leave a reply Report Post Reciprocity (0) Since writing this post InsertFishHere may have helped people, but has not within the last 4 days. InsertFishHere is a verified member , has been around for 2 years, 4 months

19. Satan, Cantor Infinity Mind-Boggling Puzzles Raymond M. Smullyan File Format PDF/Adobe Acrobat Quick View http://www.math.uic.edu/~kauffman/SatanCantor.pdf

20. Topological Models Of Set Theory Indeed, Russell observed that Cantor's infinities do not correspond with the infinity used by analystsan observation that seems to have http://sci.tech-archive.net/Archive/sci.logic/2005-04/msg00018.html

topological models of set theory From : "george" < Date : 1 Apr 2005 10:22:52 -0800 mitch wrote: correspond have The fact that that university treated you badly is not a good reason for you to treat US badly. If you have actually learned this lesson, then it would behoove you to conduct yourself HERE in such a manner as NOT to engender "territorial wars". Of course one can. All of them are treatable (even if incompletely) in first-order logic, from one viewpoint or another. The downward Lowenheim-Skolem theorem IS a resolution. Well, yes, that's what I just said. You could personally cure that. You could bring order to THAT chaos. YOU COULD DEVISE an instructional plan that was well-founded and that presented these allegedly conflicting "starting-points" within some larger framework that was robust enough to accommodate all of them. True, but at the undergraduate level, people ought to be discouraged from thinking that they need specific applications. They ought to specialize later, in grad school. I repeat, the fact that your university treated you badly is not a

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