61. When You Get To The End, Keep Going | Csmonitor.com The book culminates with Georg Cantor s creation of a set theory of While trying to parse Cantor s diagonal proof establishing that some infinities are http://www.csmonitor.com/2003/1104/p14s02-bogn.html

WORLD USA COMMENTARY LEARNING ... Text Edition Search: Special Offer: Subscribe to the Monitor and get 32 issues FREE! In Books: A half century later, a full look at Matisse This remake of 'Howards End' is a true 'Beauty' Wartime memoirs by women in vogue A love letter to the novel - complete with reading list ... 'Three Nights in August: Strategy, Heartbreak, and Joy' Annual guide to books Fiction Nonfiction Coverage of poetry in the news and original work by publishing poets. What is this? Most-viewed stories: (for 09/15/05) Bout 1: over. Bout 2: huge. Remember Daniel Ortega? He's back. New Al Qaeda tape hints at frustration Bush's buffeted leadership ... Books from the November 04, 2003 edition EVERYTHING AND MORE: A COMPACT HISTORY OF INFINITY By David Foster Wallace Atlas / Norton 320 pp., $23.95 When you get to the end, keep going The author of 'Infinite Jest' tackles infinity, seriously By Darren Abrecht It should go without saying that to reach the end of David Foster Wallace's "Everything and More: A Compact History of Infinity," you will first have to reach the book's halfway point. Before arriving there, however, you will first have to get a quarter of the way through - but not before reading the first one-eighth, the first one-sixteenth, and so on. Eventually, you will realize that to make any progress in the book whatsoever, you will have to read an infinite number of textual subdivisions in a finite period of time - which is, of course, impossible. Fortunately, by the time you reach that halfway point, Wallace will have given you the intellectual tools to overcome this hurdle. But it's likely that you'll be having too much fun to care - provided, of course, that you're not the sort of spoilsport who refuses to believe that a good time can be had in higher math.

62. Welcome To The Hotel Infinity Cantor s surprising results came when he considered sets that had an infinite number of Perhaps, thought Cantor, once you start dealing with infinities, http://www.c3.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

63. BC Business : Intellectual Pursuit.(Brief Article) @ HighBeam Research People who can do pure math contemplate problems like Cantor s infinities, which stumped math types for centuries, or prattle on about pi must be http://static.highbeam.com/b/bcbusiness/november012000/intellectualpursuitbriefa

Library Web Executives New! ... Customer Support Question / Keyword(s): Advanced Search Current Article: Intellectual Pursuit.(Brief Article) Start B BC Business November 01, 2000 ... Intellectual Pursuit.(Brief Article) Intellectual Pursuit.(Brief Article) BC Business; November 01, 2000; STALEY, ROBERTA STALEY, ROBERTA BC Business November 01, 2000 People who can do pure math contemplate 'problems' like Cantor's Infinities, which stumped math types for centuries, or prattle on about pi must be closer to God than you or me. Such lofty concepts are beyond the easy grasp of most mere mortals. Daniel Brox of West Vancouver is one of the chosen. At 17, this straight-A Sentinel Secondary student has made international waves with his mathematical abilities. This year the Grade 12 wunderkind won the 2000 Canadian Mathematical Olympiad and was a silver medalist at the (invitation only) International Mathematical Olympiad for high school students held this year in Korea. Brox's talent came to light in Grade 7 when he began acing Grade 9 and 10 exams. Now he's taking university math courses just for the fun of it. In between pondering imponderables, Brox contemplates his next step what intellectual hotbed to attend next year when he enters university.

64. /usr/local/doc/tex-inputs/latex/notes/paper.tex % An Example while other infinities are denoted $\aleph_\alpha$ where $\alpha$ is an ordinal \end{proof} This completes our discussion of Cantor s Theorem. http://www.math.psu.edu/katok_s/paper.tex.html

65. Different Infinities The point is that Cantor s condition is If every attempt to construct a oneto-one correspondence between A and B, leaves B with elements that are not http://mathcentral.uregina.ca/QQ/database/QQ.09.03/plober1.html

Quandaries and Queries Name: Plober Who is asking: Student Level: Secondary Question: I can't think of any other way, and I KNOW the real's cardinality is greater than the integer's one Hi Plober, Galileo knew that squaring produced a bijection between the integers and their squares, hence the saying "the whole is greater than the part" did not apply to infinite sets; his conclusion was that infinite sets were totaly incomprehensible and it was pointless to try to analyse them logically. Cantor later classified infinite sets in terms of bijections, but his arguments were not fully accepted in his age. In that respect, your bar room conversation is part of an long historical debate on the nature of infinity. Perhaps it is good that many people can express their opinion rather than having "the right answer" taught to everyone. Your real insight on the subject is that you stick to a formal definition of why a cardinality is larger than another one, rather than avoid definitions or make them up along the way as suits you best. We often get questions of the type "How many sides does a circle have?"

66. Bounded Infinities I ve got ideas of bound infinities, sets whose elements are bound in certain Semiotics perfect language - Cantor s paradox - Chaotic representation - http://homepages.which.net/~gk.sherman/baaaaaaa.htm

home human ecology home mathematics Human ecology Bounded infinities 20 Sept 1998 Notes about the an idea of a bounded infinity (BI). Something that has infinite forms within the bound part of another space. An extremely simple BI is a line. It is constrained to a particular part of a nD space (n>1) but has infinite points within that line. Are there more complicated forms that can describe some of the interactions between two rival theories. For example, the difference between Value Added and Labour theories of value in economics. These are very rough notes... 2nd-3rd March 1998 Bits about Injective and Surjective: i.e with sets, the mapping from S to T* is injective (as is the mapping from T to S) A' - A B' - B C' - C D' - D E' - E F' G' H' with the example before: the mapping from T* to S is surjective (as well as from T to S, and from S to T) Bijective: the mapping is both surjective and injective, i.e. from S to T, and from T to S) This is alright for finite sets. What happens if you have infinite sets? one - 1 two - 2 three - 3 four - 4 five - 5 ten - 10 eleven - 11 infinite - ?

67. Initial Points Of Translation From Formalism in which information finds itself organised. infinities Cantor s paradox -; Classification - Limitative theories; Theories of everything -; Colour - http://homepages.which.net/~gk.sherman/gbaa.htm

Human ecology Translations between biology and sociology Part I: Initial points of translation Initial points of translation from formalism (5,000 words) Trees and graphs Abstract - 5/7/99 - (1,200 words) Directories, hierarchies and set theory . Some differences between the ways in which computers like to represent information and the ways in which information finds itself organised. Infinities - Cantor's paradox - Classification - Limitative theories Theories of everything - Colour - Links... Translations between biology and sociology Part I : initial points of translation. Initial points of translation from biology Formalism: a linking theme between mathematics, computation, and cultural theory Initial points of translation from formalism Conclusion: Possible grounds for translation Part II: What questions does sociology ask Part III : What are the roles of calcium in the genotype - phenotype argument? Links to other sites... Created 5/7/99

68. The Naked Scientists Science Discussion Forum - Science And God - Does It Mix ? Panentheism and Pantheism are like Cantor s sets of infinities in that respect All is the set that contains all other sets and immediately someone asks http://www.thenakedscientists.com/forum/topic.asp?TOPIC_ID=618&whichpage=5

69. 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

Religion and Physics, XXI C. 2005 Jul 02 physics philosophy religion "Is there a God? Yes - whether or not? And if there is, who it (he)? Does the religion give the answer? ut there are a lot of religions and in different places - everyone beleives his or her own religion. Which of them to trust? Physics is another business. The laws of physics are fair everywhere. And they work irrespectively of any opinion of an individual. Is it possible to unite the laws of physics with religion? 20 years ago I was surprised with this question and doubtfully considered such an opportunity. However I had to change my point of view." In English and Russian. Pythagorean Physics 2005 Feb 12 philosophy physics "Pythagorean Physics postulates the existence of a basic unit of matter, the Pythagorean atom. It deals with discreteness in favor of continuity. It considers both time and space to be absolute. Motion is a function of space and time. Unlike classical mechanics, Pythagorean Physics considers mass to be a variable and has a different concept of what a particle is. Pythagorean Physics employs an axiomatic system that incorporates both philosophy and science in order to achieve meaning. ... Scientists sometimes have a difficult time understanding the limits and validity of what they think they know. Neither the theory of relativity nor quantum mechanics employs an axiomatic system that can guard against such errors. Rather, they both superimpose notions for convenience. Pythagorean Physics follows an axiomatic system that starts with definitions and proceeds step by step from there in a logical fashion that provides meaning in a way that other approaches can not."

70. Crank Dot Net | Cantor Was Wrong All infinities are equal (ie, there is only one infinity) . Cantor s Diagonal Theorem, which supposedly demonstrates there are more reals (R) than 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.'"

71. Citebase - Infinities In Physics And Transfinite Numbers In Mathematics Upon examining these examples in the context of infinities from Cantor s theory of transfinite numbers, the only known mathematical theory of infinities, http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:math-ph/9909033

72. Re: Isomorphism Between Mereology And Boolean Algebra Without Least Elem Hilbert called Cantor s hierarchy of infinities a paradise , but Wittgenstein called it a dismal swamp . That s a religious dispute, which I believe is http://suo.ieee.org/email/msg13174.html

Thread Links Date Links Thread Prev Thread Next Thread Index Date Prev ... Date Index Re: Isomorphism between Mereology and Boolean algebra without least element To : "John F. Sowa" < sowa@BESTWEB.NET Subject : Re: Isomorphism between Mereology and Boolean algebra without least element From Avril.Styrman@helsinki.fi Date : Thu, 7 Jul 2005 01:12:32 +0300 Cc standard-upper-ontology@listserv.ieee.org cg@CS.UAH.EDU alexander.heussner@gmx.net aapo.halko@helsinki.fi Sender owner-standard-upper-ontology@ieee.org User-Agent : Internet Messaging Program (IMP) 3.1 http://people.imise.uni-leipzig.de/alexander.heussner/files/Mereology.pdf Follow-Ups Re: Isomorphism between Mereology and Boolean algebra without least element From: "John F. Sowa" <sowa@BESTWEB.NET> Prev by Date: Re: Isomorphism between Mereology and Boolean algebra without least element Next by Date: Re: Isomorphism between Mereology and Boolean algebra without least element Prev by thread: Re: Isomorphism between Mereology and Boolean algebra without least element Next by thread: Re: Isomorphism between Mereology and Boolean algebra without least element Index(es): Date Thread

73. [Phil-logic] Cantor's Axiom This is the sort of thing I see as a supporting consideration for Cantor s view. What are actual infinities ? What sense of actual is meant here? http://philo.at/pipermail/phil-logic/2003-November/001982.html

[Phil-logic] Cantor's Axiom Paul F McNamara paulm at cisunix.unh.edu Wed Nov 26 16:55:17 CET 2003 Previous message: [Phil-logic] Cantor's Axiom Next message: [Phil-logic] Cantor's Axiom Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Paul McNamara wrote: Apropos of this, those remarks I recently quoted from Cantor linking the-square-root-of-two to w (omega) seem helpful. Sloppifying Cantor, one way to look at w is that it is the limit of the natural numbers, just as the-square-root-of-two is the limit of a certain sequence. What are "actual infinities"? What sense of "actual" is meant here? Good question. I had in mind the former as a term of art in Phil of Math, so that it is in contrast to "potential infinity". On the latter view, the set of natural numbers would not be deemed to exist, on the former it would exist ("as a totality", as they used to say). Likewise for (say) the reals. I think it is more challenging to say what "potential infinity" means than to say what "actual infinity" means, although defending the truth of latter sort of claim is clearly challenging. -paul > Phil-logic mailing list Phil-logic at philo.at

74. [Phil-logic] Cantor's Axiom This is the sort of thing I see as a supporting consideration for Cantor s view. CS What are actual infinities ? What sense of actual is meant http://philo.at/pipermail/phil-logic/2003-November/001983.html

[Phil-logic] Cantor's Axiom charles silver silver_1 at mindspring.com Wed Nov 26 17:29:38 CET 2003 Previous message: [Phil-logic] Cantor's Axiom Next message: [Phil-logic] Cantor's Axiom Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Right. This is the sort of thing I see as a supporting consideration Cantor's view. > What are "actual infinities"? What sense of > "actual" is meant here? Good question. I had in mind the former as a term of art in Phil of so that it is in contrast to "potential infinity". On the latter set of natural numbers would not be deemed to exist, on the former would exist ("as a totality", as they used to say). Likewise for reals. I think it is more challenging to say what "potential infinity" > means than to say what "actual infinity" means, although defending truth of latter sort of claim is clearly challenging. -paul Sure, I'm familiar with the adjectives "actual" and "potential" when applied to "infinity", but I'm wondering how this distinction really plays out. It seems okay to say that according to the view that there's only a potential infinity, "the set of natural numbers would not be deemed to exist." But, what is "would not be deemed to exist" supposed to mean in this context? (Where's the "deeming" coming from?) Previous message: [Phil-logic] Cantor's Axiom Next message: [Phil-logic] Cantor's Axiom Messages sorted by:

75. QFM: 2.iv Archival Structures & Fractal Subjectivities 4 and Georg Cantor s infinite levels of infinities nested within infinities, Like information, memory and Cantor s set theory with its multiple http://www.mcluhan.utoronto.ca/academy/carolynguertin/2iv.html

Quantum Feminist Mnemotechnics: The Archival Text, Digital Narrative and the Limits of Memory Carolyn G. Guertin Senior McLuhan Fellow, UofT Chapter 2. The Matrix: Information Overload iv. Archival Structures and Fractal Subjectivities "A network is a network is a network." ~ Friedrich Kittler Michel Foucault is known for his passion for mapping the foundations of particular types of knowledge. In The Order of Things, he seeks to chart the contours and coordinates of the episteme of knowledge, those foundations of classification and organizational structure that are integral to systems of thought. Specifically he looks to identify the importance of the visual science of 'resemblance' and the four key 'similitudes' in the Western world as they mattered and ruled from the late medieval period until the dawn of the age of Reason. The web of resemblance that he chartsthrough its components convenientia, aemulatio, analogy , and sympathy is a matrix of associational logic and connectivity similar to the World Wide Web that is linked, not by hyperlinks, but by the relational signatures of language. Convenientia is fixed but linked adjacent space; it designates a relationship between things or ideas (18).

76. PHILOSOPHY, MATHEMATICS AND STRUCTURE An Interview With James What about Cantor s grading of infinities? That s well beyond experience, surely? True. But Cantor only came to them by a process of abstracting the notion http://www.maths.unsw.edu.au/~jim/interview.html

77. Welcome To The Dana Centre Aleins, infinities and beyond Posted 19 Feb 2004 2318 PM on infinity that dealt with Cantor s theories then the email arrived with your response. http://www.danacentre.org.uk/default.aspx?DanaAction=ShowPost&PostID=326

78. Faith, Science, And Technology a series of infinities; that infinities come in an infinite number of sizes. Thus Cantor s Absolute is a perfect image for what we experience of God. http://www.elca.org/faithandscience/library/faith/quotations/

Faith and Science Library Faith and Science Articles Quotes Faith and Science Quotes The following quotes have been gleaned from the reading done by the members of the ELCA Alliance for Faith, Science and Technology . Some, due to their length, do not appear as a "Quote of the Week" but are offered here for your reading pleasure. If you know of a good quotation to add to this list and to be used on the FST home page's Quote of the Week, please let us know True religion is real living; living with all one's soul, with all one's goodness and righteousness. Albert Einstein Science without religion is lame, religion without science is blind. Albert Einstein Much of the conflict between science and religion [in the time of Galileo] turns out to have been a conflict between new science and the sanctified science of the previous generation. John Hedley Brooke We don't know how large a proportion of the significant evidence about the universe is excluded by science. Perhaps hardly any. Perhaps so great a proportion that any body of knowledge which excludes it is hardly more than a caricature. Perhaps something in between — so that science finds truth but not the whole truth.

79. Boston College ETD Repository:Beyond Infinity: Georg Cantor And Leopold Kronecke of transcendence in order to clearly described infinities of different sizes. Leading the opposition against Cantor s theory was Leopold Kronecker, http://dissertations.bc.edu/ashonors/200501/

Undergraduate Honors Program Previous Thesis Next Thesis Browse by Year: Enter search term: Beyond Infinity: Georg Cantor and Leopold Kronecker's Dispute over Transfinite Numbers Patrick Hatfield Carey, B.A. (Awarded in 2005) Undergraduate Honors Program, Philosophy Adviser: Patrick Byrne Download the Thesis (522 K, 52 p., PDF file) Tell a colleague about it. Printing Tips : Select 'print as image' in the Acrobat print dialog if you have trouble printing. ABSTRACT: In the late 19th century, Georg Cantor opened up the mathematical field of set theory with his development of transfinite numbers. In his radical departure from previous notions of infinity espoused by both mathematicians and philosophers, Cantor created new notions of transcendence in order to clearly described infinities of different sizes. Leading the opposition against Cantor's theory was Leopold Kronecker, Cantor's former mentor and the leading contemporary German mathematician. In their lifelong dispute over the transfinite numbers emerge philosophical disagreements over mathematical existence, consistency, and freedom. This thesis presents a short summary of Cantor's controversial theories, describes Cantor and Kronecker's philosophical ideas, and attempts to state clearly their differences of opinion. In the end, the author hopes to present the shock caused by Cantor's work and an appreciation of the two very different philosophies of mathematics represented by Cantor and Kronecker.

80. ET CETERAs :: Still More Stuff (p.3/5) Obviously, as with the Roman Catholic model of the spiritual Heavens (or Cantor s notion of higher infinities ), there are further levels of escalation http://www.users.cloud9.net/~bradmcc/moreStuff2.html

Still More Stuff "The longer one lives, the greater the chance that something odd will turn up." (Soseki Natsume, I am a Cat , p.189) Go/Return to Previous stuff Key: Red action/warning. Purple may offend some persons. Gray informational. I subscribe to a mailing list that sends me a different new vocabulary word each day ("A Word a Day"). I rarely if ever incorporate these words into my vocabulary, because I do not feel they "fit" i.e., I can't imagine a situation in which I would feel comfortable (or, perhaps less commendably: clever...) using them. Following are examples of "A Word a Day" words I have not added to my vocabulary: Atrichia : Absence of hair. Calvity : Baldness. Crispin : A shoemaker. Renitent : Resistant; recalcitrant. Pinguid : Fat; greasy; unctuous. Skookum : Powerful; first-rate; impressive. Peri : A beautiful, graceful girl or woman. Diriment adjective ] Nullifying. Holus-bolus : All at once. Ebrious : Inclined to excessive drinking; tipsy. Some words that do have special rhetorical import for me: Anent : Concerning.

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