21. Salon.com Technology | Letters anything interesting about quantum mechanics, then turns to Cantor s infinities. Again, the author does not say anything correct about what Cantor s http://www.salon.com/tech/letters/2004/10/14/cancer_nano/index1.html

Search Salon Directory Hot Topics Ask the Pilot The Matrix Computer Games Globalization ... Joyce McGreevy Articles by date All of Salon.com By department Opinion Books ... Table Talk [Spirited Salon forums] What do you need to look for in a laptop? Have you tried VOIP? Posts of the week The Well [Pioneering members-only discussions] Wondering about the golden age of groupies? Sound Off Letter to the Editor Salon Blogs Subscribe Gift Subscriptions ... Letters [Read "Nanotech Angels." One cannot judge Salon.com for publishing articles about Kabbalah or other Madonna fetishes. However, to misrepresent such ideological drivel as scientific discourse is misleading and dishonest. "Nanotech Angels" is filled with nonsensical statements such as "The smaller you got, the more order broke down" or "In its [the universe's] core, it [what is?] is energy, waves, strings." Energy, waves and strings are well-defined scientific constructs with properties that can be described and tested. Furthermore, they are defined in the context of "humankind's way of representing space and time," which H. Lovy thinks is flawed. Allow me to note the article's main flaw: Nanotechnology is an exciting new field where new and interesting phenomena are observed. However, this in no way implies that the phenomena are not described by our current physical theories. As a matter of fact, quantum mechanics describes objects far smaller than a nanometer or a buckyball. There is nothing magical or "miraculous" about what is observed at the nanometer scale and therefore no need to seek scientific enlightenment from religion. If one wishes to draw parallels with religion, one has to first understand properly the science.

22. Salon.com Books | What's Bigger Than A Kazillion? But don t jump to the conclusion that all infinities are the same size the Yet Cantor s diagonal argument, in its essence, is so beautifully simple http://www.salon.com/books/review/2003/11/12/infinity/

Search Salon Directory Hot Topics Book Reviews Garrison Keillor J.R.R. Tolkien Harry Potter ... Dave Eggers Articles by date All of Salon.com By department Opinion Books ... Table Talk [Spirited Salon forums] What's the most recent book you've hated? The best resources for compulsive readers. Posts of the week The Well [Pioneering members-only discussions] Wondering about the golden age of groupies? Sound Off Letter to the Editor Salon Blogs NYC: Short fiction NYC: Personal Essay SF: Personal Essay LA: Personal Essay Seattle: Personal Essay NYC: Poetry writing NYC: Joyce Maynard lecture Suggest a city or class Best writing submissions Subscribe ... Investor Relations What's bigger than a kazillion? David Foster Wallace provides an entertaining tour of the mind-blowingly big numbers and establishes that some infinities are larger than others. By Polly Shulman The greatest thrill I remember from my girlhood better than my first kiss, first airplane flight, first taste of mango, first circuit around the ice rink without clinging to a grown-up's sleeve was the heart-lifting moment when I first understood Georg Cantor's Diagonal Proof of the nondenumerability of the real numbers. This proof, the Mona Lisa of set theory (to my mind, the most satisfying branch of mathematics), changed the way mathematicians thought about infinity. Yet Cantor's diagonal argument, in its essence, is so beautifully simple that even someone who hasn't yet entirely mastered trigonometry can understand it. I know, because I did, and so can you, whoever you are. I've often wanted to share the thrill with my intelligent but mathematically innocent friends and family English teachers, textile designers, photo editors, Internet journalists, soccer moms, wedding guests and I've succeeded, too, whenever I can get them to stay put. Being stuck in an elevator together helps. But elevators don't stall that often, so I was delighted to learn that the gloriously articulate novelist and essayist

23. Cantor's Diagonal Argument arguments is Cantor s, which proved that some infinities are bigger than others. Cantor s Diagonal Argument. The classic form of the argument goes as http://www.chaos.org.uk/~eddy/math/diagonal.html

Diagonal arguments Various arguments prove extremely strong results by phrasing the result in such a way that some two-parameter thing can be used to supply a diagonal , the special case where the two-parameter thing has both parameters equal, analysis of which suffices to prove some property of the two-parameter thing which suffices to imply the desired strong result. The great grand-daddy of diagonal arguments is Cantor's, which proved that some infinities are bigger than others. Cantor's Diagonal Argument Cantor originally applied the argument to show that there are more real numbers between and 1 than there are natural yes more subsets than members. can can enumerate any set of subsets of the naturals as long as there's an upper (finite) bound on the size of the subsets. One may use each natural in turn as choice of upper bound and construct an enumeration of all the subsets of the naturals smaller than this. It might seem that in the limit at infinity this process should get us at least an enumeration of all finite sub-sets of the naturals; and one might hope that some analogous trickery might enable one to get all sub-sets of the naturals. In any case the diagonal argument shows only one The diagonal argument can be read as an algorithm for converting a function, from a set to its power set, into a function from the set's successor to

24. Northwestern University Mathematical Calendar Title Thinking About Mathematics (and Many Other Things) Cantor s infinities Abstract and details available. Wednesday, January 27, 2000 http://www.math.northwestern.edu/calendar/oldlist.cgi?seminar=Special Seminar

25. BBC - H2g2 - Bigger And Bigger Infinities There are infinities that really are bigger than other infinities, despite the fact Cantor s proof will get you a different one for each natural number, http://www.bbc.co.uk/dna/h2g2/A593552

@import url('/includes/tbenh.css') ; Home TV Radio Talk ... A-Z Index Friday 16th September 2005 Text only Guide ID: A593552 (Edited) Edited Guide Entry SEARCH h2g2 Advanced Search New visitors: Returning members: BBC Homepage The Guide to Life The Universe and Everything 3. Everything Mathematics Created: 21st September 2001 Bigger and Bigger Infinities Front Page What is h2g2? Who's Online Write an Entry ... Help Like this page? Send it to a friend! It's a commonly held belief that infinity is a pretty weird concept, mixed up with mysticism and staring up into the night sky until the early hours of the morning. This entry is not about that sort of infinity. It is about the idea of a set of things that never ends, that no matter how many of them you count, there's always going to be more you've not looked at yet. This kind of infinity (the mathematical kind) is a lot easier to get a hold on than the vague stargazing kind, and allows us to show some surprising things about infinity, in particular that there are different infinities. There are infinities that really are bigger than other infinities, despite the fact that they all 'go on forever'.

26. BBC - H2g2 - Applying To University of mathematics (for instance Gödel s Incompleteness Theorem, Cantor s different infinities Aleph_0 and c, Turing Machines and the Halting Problem). http://www.bbc.co.uk/dna/h2g2/A626762

@import url('/includes/tbenh.css') ; Home TV Radio Talk ... A-Z Index Friday 16th September 2005 Text only Guide ID: A626762 (Edited) Edited Guide Entry SEARCH h2g2 Advanced Search New visitors: Returning members: BBC Homepage The Guide to Life The Universe and Everything 3. Everything Deep Thought Education Created: 28th September 2001 Applying to University Front Page What is h2g2? Who's Online Write an Entry ... Help Like this page? Send it to a friend! So if you fancy the idea of studying at university, how do you find out what chances you have of getting in? Do you have to be some straight-A student to have a chance, or can us normal people apply? And are there any countries left where the government actually pays for it? In Britain the degree schemes range from asking for AAA to EE (at A-level or equivalent), and if you do a diploma sometimes you don't need anything. It all depends on the university; for Britain visit the UCAS website and browse through courses, they'll tell you what they're asking for. How Do you Know if you're Suited? Most universities have some sort of application/offers ratio, which is anything from 2:1 to 12:1, the higher being the most competitive so there's less chance of getting in. There's also the interview; if you're good at talking to people then having an interview will increase your chances, while if you're shy then one that accepts you by the application form alone might improve your chances. If you apply early enough most universities get back to you sooner, while if you leave it to the last minute, the university will be dealing with thousands of applications so you'll have to wait longer to find out.

27. What We Definitely Know About Endlessly Possible Catholic theologians welcomed Cantor s ideas, which provided a workable way of In the 17th century, Pascal believed that humans encountered infinities http://sfgate.com/cgi-bin/article.cgi?f=/c/a/2005/08/07/RVGPVE0D6I1.DTL

28. Net-happenings: 00-03-31: MATH> [netsites] Math Forum Included are problems such as The Bridges of Konigsberg, The Value of Pi, Prime Numbers, Zeno s Paradox, Cantor s infinities, Gambling Problems and http://scout.wisc.edu/Projects/PastProjects/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:

29. LA Weekly: Columns: Quark Soup: To Infinity And Beyond The apeiron, Aristotle called it, declaring infinities of any variety an abomination. Cantor s encounter with infinity ultimately drove him mad; http://www.laweekly.com/ink/03/16/quark-wertheim.php

Work in Progress: One architect’s never-ending search for the well-lived home and life . . . in which Josh Schweitzer explains why you should tear down your house. BY LINDA IMMEDIATO Also, Habitat for Humanity's response to Hurricane Katrina. DIY Wheels: An erstwhile punk rock movie star and a lanky Caltech defector meet the DARPA Challenge. BY JOHN ALBERT JEFFREY ANDERSON on the homegrown goat farmer and his terrorist threats. CHRISTOPHER LISOTTA on Arnold’s kiss-off to gay-marriage advocates. After the Deluge: JUSTIN CLARK tracks down New Orleans’ refugees in Echo Park JERVEY TREVALON on being black and poor in New Orleans BEN EHRENREICH sees a neighborhood that survived without government help.

30. Infinity: You Can't Get There From Here -- Platonic Realms MiniText infinities don t have to be large, of course – they can also be small. Cantor s proof of the “nondenumerability” of the real numbers (the http://www.mathacademy.com/pr/minitext/infinity/index.asp

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.

31. Re 'Uncountable': How Many Reals *are* There? (mapping All Onto One Line) Cantor s theorem doesn t actually say anything about constructs like 10^aleph_0; Unless of course you admit playing with uncountable infinities to be a http://home.iae.nl/users/benschop/reals.htm

Subject: Re: Allright, how many reals *are* there? Author: Nico Benschop Re: How diagonal is Cantor's diagonal? ...(sci.math 6jun98)

32. Filosofy: Intuition And Imagination In Math And In Learning 1.000* or Cantor s diagonal) I recently aquired a healthy distaste for the infinities beyond countable my private right. I ll advisedly keep my tendency http://home.iae.nl/users/benschop/filosofy.htm

Intuition and imagination in Math (NFB: "Weighing Cantor's Evidence" sci.math third art seems to be required, namely finding a balance between the two. The link to reality, as I see it, then is essentially in the axioms. For some this link is irrelevant, because the games to be played (intellectually) in such universe are gratifying in themselves, like for real mathematicians, say. For others, this link motivates moving in such intellectual universe at all, as it were: not for the fun of it, but for the use of it in another universe, say that of practical life applications. But here confusion galore, again: I once went to a math-conference on " Semigroups and their applications About the conference on "applications" (I had no occasion to interpret the program properly beforehand): most papers were on Category Theory, and not-to-be-followed by a simple engineer with some discrete math background. My point: it all depends on the context what is the meaning of words like application or real or ideal or infinite ( countable or not? ).

33. Peter Suber, "Infinite Reflections" Cantor s mathematics, however, boldly posits complete infinities. The natural numbers make a potential infinity when we think of counting them out, http://www.earlham.edu/~peters/writing/infinity.htm

Originally presented as an all-college address at St. John's College in October 1996. Published in the St. John's Review , XLIV, 2 (1998) 1-59. Peter Suber Infinite Reflections Peter Suber Philosophy Department Earlham College Galileo's Paradox ... Appendix: A Crash Course in the Mathematics of Infinite Sets Galileo's Paradox Here's a paradox of infinity noticed by Galileo in 1638. It seems that the even numbers are as numerous as the evens and the odds put together. Why? Because they can be put into one-to-one correspondence. The evens and odds put together are called the natural numbers. The first even number and the first natural number can be paired; the second even and the second natural can be paired, and so on. When two finite sets can be put into one-to-one correspondence in this way, they always have the same number of members. Supporting this conclusion from another direction is our intuition that "infinity is infinity", or that all infinite sets are the same size. If we can speak of infinite sets as having some number of members, then this intuition tells us that all infinite sets have the same number of members. Galileo's paradox is paradoxical because this intuitive view that the two sets are the same size violates another intuition which is just as strong. Clearly, the even numbers seem less numerous than the natural numbers, half as numerous to be precise. Why? Because we can obtain the evens by starting with the naturals and deleting every other member. Needless to say, when we delete every other member of a finite set, the result is a set which is half as numerous as the original set.

34. :: E R I V . N E T :: Week One Lifeboat Economics (Click here to download teaching MP3). Zeno s Paradox Cantor s infinities, Part One Cantor s infinities, Part Two http://eriv.net/temp/series.html

PARADOX You can download MP3s of the messages here, as well as other resources. Here are a few other resources that may help you as you study with us: Riverview's Discussion Board (Join other Riv people as they discuss what they are learning) How to Study the Bible (a great primer on Bible Study) Suggested Tools for Bible Study (a look at various Bible translations and other recommended tools) Other Bible Studies (Bible studies from previous series' at Riv) Week One - "Lifeboat Economics" ( Click here to download teaching MP3 Zeno's Paradox Cantor's Infinities, Part One Cantor's Infinities, Part Two ... Judging Others Week Two - "The Godman" Including Baptism Stories ( Click here to download teaching MP3 Week Three - "Mountain Biking With Mike" ( Click here to download teaching MP3 Week Four - "Why I Hate Budweiser in the Can" (

35. Infinity He then demonstrated, using what has become known as Cantor s theorem, that there is a hierarchy of infinities of which alephnull is the smallest. http://www.daviddarling.info/encyclopedia/I/infinity.html

Return to The Worlds of David Darling INDEX LATEST NEWS ARCHIVE ... Z entire Web this site infinity Mystery has its own mysteries, and there are gods above gods. We have ours, they have theirs. That is what's known as infinity. —Jean Cocteau (1889-1963), French author and filmmaker A concept that has always fascinated philosophers and theologians, linked as it is to the notions of unending distance or space, eternity, and God, but that was avoided or met with open hostility throughout most of the history of mathematics. Only within the past century or so have mathematicians dealt with it head on and accepted infinity as a number — albeit the strangest one we know. An early glimpse of the perils of the infinite came to Zeno of Elae through his paradoxes, the best known of which pits Achilles in a race against a tortoise. Confident of victory, Achilles gives the tortoise a head start. But then how can he ever overtake the sluggish reptile? asks Zeno. First he must catch up to the point where it began, by which time the tortoise will have moved on. When he makes up the new distance that separated them, he finds his adversary has advanced again. And so it goes on, indefinitely. No matter how many times Achilles reaches the point where his competitor was, the tortoise has progressed a bit further. So perplexed was Zeno by this problem that he decided not only was it best to avoid thinking about the infinite but also that motion was impossible! A similar shock lay in store for

36. Square Root Of 2.... - Physics Help And Math Help - Physics Forums His passion for Cantor s infinities is highly bizzare and contradictory to his philiosophy as well? When I said There could be a better system than http://physicsforums.com/showthread.php?t=79434

37. Set Theory Certainly Cantor s array of different infinities were impossible under this way of thinking. Cantor however continued with his work. http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Beginnings_of_set_theory.htm

History topic: A history of set theory The history of set theory is rather different from the history of most other areas of mathematics. For most areas a long process can usually be traced in which ideas evolve until an ultimate flash of inspiration, often by a number of mathematicians almost simultaneously, produces a discovery of major importance. Set theory however is rather different. It is the creation of one person, Georg Cantor. Before we take up the main story of Cantor's development of the theory, we first examine some early contributions. The idea of infinity had been the subject of deep thought from the time of the Greeks. Zeno of Elea, in around 450 BC, with his problems on the infinite, made an early major contribution. By the Middle Ages discussion of the infinite had led to comparison of infinite sets. For example Albert of Saxony, in Questiones subtilissime in libros de celo et mundi, proves that a beam of infinite length has the same volume as 3-space. He proves this by sawing the beam into imaginary pieces which he then assembles into successive concentric shells which fill space. Bolzano was a philosopher and mathematician of great depth of thought. In 1847 he considered sets with the following definition

38. Search Results For Cantor Certainly Cantor s array of different infinities were impossible under this way of thinking. Go directly to this paragraph http://www-groups.dcs.st-and.ac.uk/~history/Search/historysearch.cgi?SUGGESTION=

39. Triad The third proposition is that there are no actual infinities. and third, remember Cantor s definition of an infinite set (roughly, a set such that it is http://personal.bgsu.edu/~roberth/triad.html

Date: Sun, 27 Feb 2000 14:11:36 Subject: An Inconsistent Triad There are three propositions that many Objectivists believe and that Rand herself almost certainly believed that form an inconsistent triad. That is, they form a set that has the property that any two are consistent, but the addition of the third makes the set contradictory. The first rarely gets stated, because it is taken for granted by almost everybody. It may well be that it is so thoroughly taken for granted, that it is not even recognized as a belief to which there might be some alternative. Objectivists take it for granted, but are hardly ever pushed to defend it, because the people they engage in discussion also take it for granted. It is the assumption that time is non-cyclical. There is a definite future and past, so it is not the case that, say, a future event is the very same event as some past event. In fact, if time were cyclical, every event would be in both the future and past of every other (and even in its own future and past). There could be, say, three events, A, B and C, with A preceding B, B preceding C and C preceding A. (This is not to say that the cycle would

40. Lectures Archimedes and His World Cantor s infinities Goedel and Undecidability Geometry, Logic and Physics Escher and Symmetry Finally, Fun. http://www.math.princeton.edu/facultypapers/Conway/

Lectures Calendar Conundrums Archimedes and His World Cantor's Infinities Goedel and Undecidability ... Finally, Fun

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