1. Cantor's Diagonal Proof cantor's infinities. No recent mathematician has had more influence on the way mathematics is done today than Georg http://www.mathpages.com/home/kmath371.htm

Cantor's Diagonal Proof Simplicio: I'm trying to understand the significance of Cantor's diagonal proof. I find it especially confusing that the rational numbers are considered to be countable, but the real numbers are not. It seems obvious to me that in any list of rational numbers more rational numbers can be constructed, using the same diagonal approach. Interfering With PI Simplicio: You said there is no upper bound on the size of natural numbers, and thus the least upper bound on the naturals is infinite, even though every natural number is finite. To me this implies that there can be numbers which do not have such a bound. Is that not so? Salviati: It sounds like you're trying to invent a kind of "number" that has infinitely many digits in the direction of geometrically increasing significance, somewhat analagous to the reals, which have infinitely many digits in the direction of geometrically decreasing significance. Number systems like what you are talking about have actually been developed, Simplicio, (see p-adic numbers) but the crucial difference is that the infinite sequence of digits is in the direction of increasing, not decreasing, significance, so the resulting implied "sum" does not converge to a value that behaves consistently like a magnitude. (The valuations are said to be "non- Archimedian".) There's nothing wrong with conceiving new forms of numbers like this, but we need to be clear about how they differ from other forms of numbers.

2. Math Forum: Cantor's Solution: Denumerability · Zeno's Paradox. · cantor's infinities. · cantor's infinities, Page 2 http://forum.swarthmore.edu/~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.

3. Math Forum: Zeno's Paradox · Zeno's Paradox. · cantor's infinities. · cantor's infinities, Page 2 http://forum.swarthmore.edu/~isaac/problems/zeno1.html

Zeno's Paradox A Math Forum Project Table of Contents: Famous Problems Home The Bridges of Konigsberg The Value of Pi Prime Numbers ... Links The great Greek philosopher Zeno of Elea (born sometime between 495 and 480 B.C.) proposed four paradoxes in an effort to challenge the accepted notions of space and time that he encountered in various philosophical circles. His paradoxes confounded mathematicians for centuries, and it wasn't until Cantor's development (in the 1860's and 1870's) of the theory of infinite sets that the paradoxes could be fully resolved. Zeno's paradoxes focus on the relation of the discrete to the continuous, an issue that is at the very heart of mathematics. Here we will present the first of his famous four paradoxes. Zeno's first paradox attacks the notion held by many philosophers of his day that space was infinitely divisible, and that motion was therefore continuous. Paradox 1: The Motionless Runner A runner wants to run a certain distance - let us say 100 meters - in a finite time. But to reach the 100-meter mark, the runner must first reach the 50-meter mark, and to reach that, the runner must first run 25 meters. But to do that, he or she must first run 12.5 meters. Since space is infinitely divisible, we can repeat these 'requirements' forever. Thus the runner has to reach an infinite number of 'midpoints' in a finite time. This is impossible, so the runner can never reach his goal. In general, anyone who wants to move from one point to another must meet these requirements, and so motion is impossible, and what we perceive as motion is merely an illusion.

4. Mathematics Revenge: How Numbers Don't Behave As They Should! Cantor further defined even larger sets. Like with integers, normalmathematical operations could be applied to cantor's infinities. http://starship.python.net/crew/timehorse/new_math.html

Mathematics Revenge How numbers don't behave as they should! www What I don't understand about Cantor Infinities Cantor defines w w elements. The reason this is called an Ordinal Infinity is because the set of numbers has a specific ordering: 2 always follows 1, etc. This set of Ordinal Numbers does not itself include Infinity ( Inf ) since Inf is not a number you can count. For more information on this please visit: Robert Munafo's Large Number Pages Section 4 Eric Weisstein's MathWorld page on Ordinal Numbers Following the logical definition of w , Cantor further devised the concept of even larger sets. If you imagine w w because w is Inf . Adding w to that set would produce a set 1 bigger than w , which Cantor denoted w + 1. It must be noted however that Cantor did not consider 1 + w to be the same as w w w w w but w w Cantor further defined even larger sets. Like with integers, normal mathematical operations could be applied to Cantor's infinities. Thus, w w + 1 because w w + 1 + 1, where w w w w w w w w w w - 1 is a place holder for the last integer in the w w w in the same way we do normal integers: w 2. By analogy, Cantor went further to define

5. Media Center - Math Projects The Abacus Index. cantor's infinities Math Forum Infinite Sets Math ForumCantor's Solution Denumerability. Computation Systems Computer http://www.anderson2.k12.sc.us/schools/bhp/math_proj.htm

B-HP Media Center Math Projects Ancient Math The Ancients - Mathematicians of the African Diaspora COLOR Ancient Egyptian Math Texts Math Forum Mayan Arithmetic - Steven Fought Mathematics History ... The Abacus Index Cantor's Infinities Math Forum Infinite Sets Math Forum Cantor's Solution Denumerability Computation Systems Computer Programming Ainsworth Computer Seminar - Programs, flowcharts, and hypertext show how software works Programming: The Tech Teach - The PC Webopedia Binary numbers ... Computer Programming Basics (PowerPoint) Conic Sections Xah Special Plane Curves Conic Sections Conic Sections Conic Sections: Mathematics Encyclopedia Conic Section from Eric Weisstein's World of Mathematics Fermat's Last Theorem NOVA Online The Proof FIBONACCI NUMBER-THEORISTS Golden Ratio, Fibonacci Sequence - Math Forum Ask Dr. Math FAQ ... The Fibonacci Numbers and Golden section in Nature - 1 Flatland Flatland Flatland: Rachel's Artwork Edwin Abbott Abbott - Flatland: A Review by the Princeton Science Library Views from Flatland ... MathTrek A Stranger from Spaceland Science News Online Jan. 1 2000

6. INTEGRITY - Robust Non-Fractal Complexity -  NECSI Journal Paper an era of Relativity, Quantum Mechanics, cantor's infinities, holography, Zadeh Logic, quark architecture, spread http://www.ceptualinstitute.com/uiu_plus/necsij1send.htm

Presented at the NECSI / ICCS International Conference on Complex Systems September 21-26, 1997; Nashua New Hampshire USA Robust Non-Fractal Complexity James N Rose Ceptual Institute, 1271 Bronco Circle, Minden NV 89423 http://www.ceptualinstitute.com email: integrity@ceptualinstitute.com 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.

7. INTEGRITY - Robust Non-Fractal Complexity - Text Of NECSI/ICCS1 Paper an era of Relativity, Quantum Mechanics, cantor's infinities, holography, Zadeh Logic, quark architecture, spread http://www.ceptualinstitute.com/uiu_plus/necsijournal1.htm

Presented at the NECSI / ICCS International Conference on Complex Systems September 21-26, 1997; Nashua New Hampshire USA Robust Non-Fractal Complexity James N Rose CeptualInstitute, 1271 Bronco Circle, Minden NV 89423 http://www.ceptualinstitute.com email: integrity@ceptualinstitute.com 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.

8. Mathematics Resources Ancient Math. cantor's infinities. Computation Systems. Computer Programming http://www.anderson2.k12.sc.us/schools/bhp/Mathematics.htm

9. Links To Dr. Math Combinatorics Explanations · Zeno's Paradox. · cantor's infinities. · cantor's infinities, Page 2 http://mathforum.com/isaac/problems/prob4.html

'n Choose r' Table of Contents: Famous Problems Home The Bridges of Konigsberg The Value of Pi Prime Numbers ... Links Here are two links to questions (and their answers) in the Dr. Math archives. Probability in Flipping Coins (Kay) Six pennies are flipped. What is the probability of getting two heads and four tails? Etc. Combinatorics Basics (Scamacca) I need to prove that n chooses n-1 = n; e.g. C(n,n-1) = n. Pages Designed by Isaac Reed

10. Re: Limitations Of C*algebras functions. To clear up some apparent confusion, it was I who dismissedcantor's infinities and the Reals and Rationals; not Charles. The http://www.lns.cornell.edu/spr/2000-01/msg0020769.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: limitations of C*algebras Subject : Re: limitations of C*algebras From : "Terry Padden - news" <TCCG@bigpond.com> Date : 06 Jan 2000 00:00:00 GMT Approved : baez@math.ucr.edu Newsgroups : sci.physics.research Organization : Telstra BigPond Internet Services (http://www.bigpond.com) References OKw6EIAgV4c4EwYE@clef.demon.co.uk OKw6EIAgV4c4EwYE@clef.demon.co.uk ">news: OKw6EIAgV4c4EwYE@clef.demon.co.uk Follow-Ups Re: limitations of C*algebras From: References Re: limitations of C*algebras From: Prev by Date: Photon Tunnelling Next by Date: Re: Who put the 8PI in the bomp de bomp bomp Prev by thread: Re: limitations of C*algebras Next by thread: Re: limitations of C*algebras Index(es): Date Thread

11. Math Forum: Probability: Solution To Problem 1 · Zeno's Paradox. · cantor's infinities. · cantor's infinities, Page 2 http://mathforum.com/isaac/problems/probsol1.html

Solution to Problem 1 A Math Forum Project Table of Contents: Famous Problems Home The Bridges of Konigsberg The Value of Pi Prime Numbers ... Links With Pascal winning 9 to 6, the game will be over in 4 turns. If Pascal wins at least one of these flips, he wins the game. So, out of 16 possible outcomes, are favorable to Pascal. Therefore he should receive Francs. to Probability: Summary and Problems Suggestion Box Home The Math Library Help Desk ... Search http://mathforum.org/ The Math Forum webmaster@mathforum.org August, 1998

12. Media RelationsNews Releases They are free and open to the public. The topics are Jan. 26. cantor's infinities The lecture subject is Georg Cantor, a mathematician who has had a great http://www.northwestern.edu/univ-relations/media/news-releases/1999-00/*uwn/nemm

CONTACT: Charles Loebbaka at (847) 491-4887 or by e-mail at c-loebbaka@nwu.edu FOR RELEASE: Immediate Distinguished Mathematician John Conway Gives Nemmers Lectures EVANSTON, Ill. - John H. Conway will deliver his second talk in the Nemmers Prize Lecture series at Northwestern University Tuesday (Jan. 18 )on the topic "Archimedes and His World." Conway, the Frederic Esser Nemmers Professor of Mathematics at Northwestern University, is lecturing during the winter quarter on the theme "Thinking About Mathematics (and Many Other Things)." The talks are aimed at a lay audience with the goal of connecting people to the science of mathematics. One of the preeminent theorists in the study of finite groups (the mathematical abstraction of symmetry) and one of the world's foremost knot theorists, Conway is the author of more than 10 books and 130 journal articles on mathematical subjects. He has done pathbreaking work in number theory, game theory, coding theory, tiling and the creation of new number systems. Conway, who is the von Neumann Professor of Mathematics at Princeton University, was selected for the Nemmers honor in 1998. The lecture series is among his scholarly activities while in residence at Northwestern this quarter.

13. INTEGRITY - Robust Non-Fractal Complexity - Text Of NECSI/ICCS1 Paper In an era of Relativity, Quantum Mechanics, cantor's infinities, holography, ZadehLogic, quark architecture, spread spectrum transmission, Complexity, and the 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.

14. Math Lunt Hall Mathlogo Wednesday, January 26 cantor's infinities. No recent mathematician has had moreinfluence on the way mathematics is done today than Georg Cantor (18451918). http://www.math.nwu.edu/conway/

Nemmers Prize Lectures Thinking About Mathematics (and Many Other Things) A Series of Lectures for Those Interested in Mathematics, Philosophy, or Science By John H. Conway John Von Neumann Professor of Mathematics Princeton University Tea - 3:30 pm Lecture - 4:00 - 5:00 pm TECH Lecture Room 3 - Technological Institute 2145 Sheridan Road - Evanston Campus (Unless otherwise noted) Wednesday, January 12 Calendar Conundrums Which year had 445 days? Why are the weekdays named in the order they are? How did the months get their lengths and names? Why were there two Christmas days in 1066? Why is the 13 th of a month most likely to be a Friday? How many different lengths of months were there in 1993? And finally, just when should we have celebrated the new millennium? Tuesday, January 18 Archimedes and His World (NOTE: Lecture will be in TECH Lecture Room 2) On Thursday, October 29, 1998 an 800 year-old manuscript containg many of Archimedes' works was sold for over $2 million at Christie's Auction House in New York. This lecture will be about this palimpsest and the work of Archimedes, the most outstanding mathematician and scientist among the ancient Greeks. Wednesday, January 26

15. Nature Publishing Group concerned to clarify a few of the basic ideas underlying this astonishing production,such as the sieve of Eratosthenes, cantor's infinities and Euler's http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v398/n6723/full/

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

17. Net-happenings: 00-03-31: MATH> [netsites] Math Forum Included are problems such as The Bridges of Konigsberg, The Value of Pi, PrimeNumbers, Zeno's Paradox, cantor's infinities, Gambling Problems and others. http://scout.wisc.edu/addserv/NH/00-03/00-03-31/0002.html

Gleason Sackmann ( gleason@rrnet.com Fri, 31 Mar 2000 07:01:57 -0600 Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: Previous message: From: Frank Bohan [ mailto:franbo@globalnet.co.uk Sent: Thursday, March 30, 2000 7:55 AM To: netsites@onelist.com Subject: [netsites] Math Forum MATH FORUM Look at some of the problems found in mathematics and their solutions. Included are problems such as The Bridges of Konigsberg, The Value of Pi, Prime Numbers, Zeno's Paradox, Cantor's Infinities, Gambling Problems and others. http://forum.swarthmore.edu/~isaac/mathhist.html Next message: Previous message:

18. Non-Flashy Page scary monster party six cats using cat5 cable don't bite the lid eMpTyV xylene freewounded salt for a small fee dying giraffes cantor's infinities my clothes http://www.redbrick.dcu.ie/~cherub/non-flashy.html

My life has been too structured. Time for change. #Goodbye. #Hello. Back-pedal you psycho trolley-stealer. protractor wars similar sponge-clasping will get you nowhere last twig wrapped gifts shoestring fortress in a cereal museum turns left, looks right Anything in yellow is a hypertext link to another hypertext markup language page, except for the word "yellow" which appeared earlier in this very sentence. That isn't a link. self-referential ironic comedy intimidated witnesses retrograde kid in a crowd encased fission stickers darker side of yoghurt hide the key Nuisance skittles in a Mandelbrot set. parapgraph My girlfriend Samantha rocks. She's cool and gorgeous. She has crazy neurons. paragraph My audible preferences are gratuitously discussed here paragraph Whether it's Tottenham Hotspur, Valencia, Parma or even Longford Town, I'm a football-mad lunatic. Football is a sport non-paragraph [if you're not on the web at the moment, then something's gone wrong]

19. Hive 5. scampi 6. cantor's infinities 7. half a genotype 8. seventh Quidditch player9. bridal or baby 10. minister, reverend, priest, etc. 11. 1sphere 12. http://techhouse.brown.edu/puzzles/archive/puz/8/

Hive "I'll bet you didn't know Brown has a beekeeping department," quips an odd man standing by a buzzing hive on Lincoln Field. "Here, have a mask. I'm trying to figure out from what direction they're getting their food, but I'm not very good at reading their dancing." As he hands you the mesh mask you notice several bees flying between the hive and a patch of flowers several yards away, but rather than point this out you take a closer look. opportune; not late gesundheit nut Kazan and Hoblit's creation mover and ... scampi Cantor's infinities half a genotype seventh Quidditch player bridal or baby minister, reverend, priest, etc. 1-sphere woven basket furniture poniard, e.g. M31, e.g. "There is no spoon" "... You, Without You"

20. History the notion that mathematics is a selfconsistent system of knowledge.Presented here are Zeno's Paradox and cantor's infinities. http://westview.tdsb.on.ca/Mathematics/history.html

History Gives students a glimpse into the genesis of a topic and observe its seeds. A Source of Ideas for Mathematics Teachers Links http://members.aol.com/~jeff570/mathsym.html This page shows the names of individuals who first used various common mathematical symbols, and the dates the symbols first appeared. http://members.aol.com/~jeff570/mathword.html This page shows the earliest uses of various words used in mathematics, particularly those that would be encountered at the high school level. It is largely a compilation of citations from the Oxford English Dictionary (OED2) and Merriam-Webster's Collegiate Dictionary, Tenth Edition (MWCD10). (The MWCD10 only provides dates, and not the actual source of the citation.) The Random House Dictionary of the English Language, Second Edition Unabridged has also been consulted. Math History History of Mathematics - Univ. of St. Andrews From St. Andrews, Scotland General history, chronologies, and biographies of mathematicians Begin with the searchable MACTUTOR HISTORY OF MATHEMATICS ARCHIVE Brief history of almost all mathematicians, plus pictures of most.

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