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62. The Berry Paradox The Berry paradox. GJ Chaitin, IBM Research Division, PO Box 704, Yorktown Heights, NY 10598, chaitin@watson.ibm.com. What is the paradox of the liar? http://www.cs.auckland.ac.nz/CDMTCS/chaitin/unm2.html

The Berry Paradox G. J. Chaitin, IBM Research Division, P. O. Box 704, Yorktown Heights, NY 10598, chaitin@watson.ibm.com Complexity 1:1 (1995), pp. 26-30 Lecture given Wednesday 27 October 1993 at a Physics - Computer Science Colloquium at the University of New Mexico. The lecture was videotaped; this is an edited transcript. It also incorporates remarks made at the Limits to Scientific Knowledge meeting held at the Santa Fe Institute 24-26 May 1994. What is the paradox of the liar? Well, the paradox of the liar is ``This statement is false!'' Why is this a paradox? What does ``false'' mean? Well, ``false'' means ``does not correspond to reality.'' This statement says that it is false. If that doesn't correspond to reality, it must mean that the statement is true, right? On the other hand, if the statement is true it means that what it says corresponds to reality. But what it says is that it is false. Therefore the statement must be false. So whether you assume that it's true or false, you must conclude the opposite! So this is the paradox of the liar. Now let's look at the Berry paradox. First of all, why ``Berry''? Well it has nothing to do with fruit! This paradox was published at the beginning of this century by Bertrand Russell. Now there's a famous paradox which is called Russell's paradox and this is not it! This is another paradox that he published. I guess people felt that if you just said the Russell paradox and there were two of them it would be confusing. And Bertrand Russell when he published this paradox had a footnote saying that it was suggested to him by an Oxford University librarian, a Mr G. G. Berry. So it ended up being called the Berry paradox even though it was published by Russell.

63. Zeno's Paradox Of The Tortoise And Achilles (PRIME) Zeno’s paradox of the Tortoise and Achilles. eno race. Zeno s paradox may be rephrased as follows. Suppose I wish to cross the room. http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp

BROWSE ALPHABETICALLY LEVEL: Elementary Advanced Both INCLUDE TOPICS: Basic Math Algebra Analysis Biography Calculus Comp Sci Discrete Economics Foundations Geometry Graph Thry History Number Thry Physics Statistics Topology Trigonometry eno of Elea ( circa 450 b.c.) is credited with creating several famous paradoxes , but by far the best known is the paradox of the Tortoise and Achilles. (Achilles was the great Greek hero of Homer's The Illiad .) It has inspired many writers and thinkers through the ages, notably Lewis Carroll and Douglas Hofstadter, who also wrote dialogues involving the Tortoise and Achilles. The original goes something like this: The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow. Achilles said nothing. Zeno's Paradox may be rephrased as follows. Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room.

64. Search Directory Page Two articles on Software Architecture paradox, and how to defy that paradox to design resilient architectures. http://www.Subrahmanyam.com/articles/architecture/Paradox.html

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67. Salon :: :: Tech :: Feature :: The Gnutella Paradox, By Janelle Brown :: Page 1 The Gnutella paradox As soon as an online musictrading service gets big enough to be useful, it s doomed. That s the Gnutella paradox. http://www.salon.com/tech/feature/2000/09/29/gnutella_paradox/

Search All of Salon.com Directory Hot Topics Iraq Joe Conason's Journal The Sopranos Osama bin Laden ... World Trade Center and Pentagon Attacks Articles by date All of Salon.com By department Get a free Allstate quote Search our Personals ... Corrections The Gnutella paradox As soon as an online music-trading service gets big enough to be useful, it's doomed. By Janelle Brown "There's always Gnutella." If you care about music on the Net, you've probably been hearing this refrain a lot lately, repeated by traders, geek programmers and digital music industry types alike. On Monday in a courtroom in San Francisco, a judge will decide whether to uphold a preliminary injunction against Napster, potentially shutting down the music trading service but as a fallback, there's always Gnutella. Frightened by legal threats from the Recording Industry Association of America (RIAA), Scour may have laid off its staff and put its future in jeopardy but there's always Gnutella. And sure, the arcane file sharing software Freenet might not be ready for your average consumer yet but, of course, there's always

68. Mark's Paradox Page such as Blackwood s the more terrible the prospect of thermonuclear war becomes the less likely it is to happen, or the Moebius Strip a topological paradox http://www.geocities.com/CapitolHill/Lobby/3022/

These are old fond paradoxes to make fools laugh i' the alehouse. - Othello, Act 1, Scene 1 Paradoxes are as old as humankind. The ancient Greeks studied them intensely which eventually helped lead to the discovery of irrational numbers, and paradoxes are mentioned in the Bible: "It was one of their own prophets who said 'Cretans were never anything but liars, dangerous animals, all greed and laziness;' and that is a true statement." (Titus 1:12-13) Even today, we are surrounded by paradoxes such as Blackwood's "the more terrible the prospect of thermonuclear war becomes the less likely it is to happen," or the Moebius Strip - a topological paradox. For this article, we define a paradox as a statement or sentiment that is seemingly contradictory or opposed to common sense and yet is perhaps true in fact. Another way of thinking of a paradox is a statement that is actually self-contradictory and hence false even though its true character is not immediately apparent. One of the oldest paradoxes is the one cited by the Apostle Paul in his letter to Titus (see above.) The Liar Paradox is interesting because it cannot be true because it would make the speaker a liar and therefore what he says is false. Neither can it be true because that would imply that Cretans are truth-tellers, and consequently what the speaker says would be true. (For classic

69. Russell's Paradox [Internet Encyclopedia Of Philosophy] Examines selfreferential linguistics used to describe properties and sets. http://www.utm.edu/research/iep/p/par-russ.htm

Russell's Paradox Russell's paradox represents either of two interrelated logical antinomies. The most commonly discussed form is a contradiction arising in the logic of sets or classes. Some classes (or sets) seem to be members of themselves, while some do not. The class of all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself. However, suppose that we can form a class of all classes (or sets) that, like the null class, are not included in themselves. The paradox arises from asking the question of whether this class is in itself. It is if and only if it is not. The other form is a contradiction involving properties. Some properties seem to apply to themselves, while others do not. The property of being a property is itself a property, while the propery of being a cat is not itself a cat. Consider the property that something has just in case it is a property (like that of being a cat ) that does not apply to itself. Does this property apply to itself? Once again, from either assumption, the opposite follows. The paradox was named after Bertrand Russell, who discovered it in 1901. Table of Contents (Clicking on the links below will take you to that part of this article) History Possible Solutions to the Paradox of Properties Possible Solutions to the Paradox of Classes or Sets Suggested Reading History Russell's discovery came while he was working on his Principles of Mathematics . Although Russell discovered the paradox independently, there is some evidence that other mathematicians and set-theorists, including Ernst Zermelo and David Hilbert, had already been aware of the first version of the contradiction prior to Russell's discovery. Russell, however, was the first to discuss the contradiction at length in his published works, the first to attempt to formulate solutions and the first to appreciate fully its importance. An entire chapter of the

70. Paradox http://paradox.simtycoongamer.net/

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72. The Atlantic | January/February 2003 | The American Paradox | Halstead The Atlantic Monthly January/February 2003 Bests Worsts The American paradox The country with the most patents, Nobel laureates, and millionaires is also http://www.theatlantic.com/issues/2003/01/halstead.htm

Home Current Issue Back Issues The Archive ... Subscriber Help Browse >> Fiction Food Foreign Affairs Language ... More on politics and society from The Atlantic Monthly. From Atlantic Unbound Interviews: "A More Perfect Union" (January 14, 2003) Ted Halstead, the founder and CEO of the New America Foundation, argues that the time has come for Americans to devise a new social contract. The American Paradox The country with the most patents, Nobel laureates, and millionaires is also the country with the highest levels of poverty, homicide, and infant mortality among modern democracies. A case for revising our social contract by Ted Halstead Further reading selected by Ted Halstead The Two Faces of America Bests Worsts Gross domestic product Poverty Productivity Economic inequality Business start-ups Carbon-dioxide emissions Long-term unemployment Life expectancy Expenditure on education Infant mortality University graduates Homicide Health-care coverage High-tech exports HIV infection Movies exported Teen pregnancy Breadth of stock ownership Personal savings Volunteerism Voter participation Charitable giving Obesity How do we reconcile these two faces of America? To a remarkable degree the United States seems to have exchanged social cohesion and a broad-based middle class for economic dynamism and personal freedom. Have we abandoned what used to be referred to as the common good?

73. Divine Paradox - Ageless Wisdom In Action - Meditation, Art, Sacred Tarot Apply Tarot to everyday life meditate on multimedia Tarot a news navigator to help you learn to recognize the principles of Tarot in everyday events. http://www.divineparadox.com

Divine Paradox Ageless Wisdom in Action Try the Divine Paradox News Navigator Spirit And Sky Top Spiritual Site We live in a world of paradox, where things are seldom as they appear. We witness events or encounter evidence that doesn't make sense, or that seems contradictory, ugly, or painful. As a result, some become jaded realists, while others shut their eyes and escape into a private world of fantasy or despair. A few, however, somehow find a way to remain awake and engaged, and are able to reconcile and even redeem the inevitable contradictions and setbacks of life. Thus, they find a way to experience true peace, happiness, love, beauty and stability. Check out our definition of Paradox Read what the Ageless Wisdom has to say about the Divine Paradox A poem: Paradox The Rules of the Game of Life Explore the Principles and Laws of life, encapsulated in symbol and myth.

74. Peter Suber, Paradox Of Self-Amendment, Table Of Contents The paradox of SelfAmendment A Study of Law, Logic, Omnipotence, and Change. Peter Suber, Philosophy Department, Earlham College. The paradox of Self-Amendment. http://www.earlham.edu/~peters/writing/psa/

The Paradox of Self-Amendment: A Study of Law, Logic, Omnipotence, and Change Peter Suber Philosophy Department Earlham College This book was originally published by Peter Lang Publishing, 1990. It is now out of print. Peter Suber Table of Contents Now searchable Preface to the Print Edition Why bother? Outline ... Bibliography Index I omit the index from this HTML edition. Search the online edition of this book. Multiple words are linked by the selected Boolean operator. Word fragments return all completions, e.g. amend returns amend amends amended amending amendment and so on. Your search string Boolean AND OR Case insensitive sensitive Peter Suber Department of Philosophy Earlham College , Richmond, Indiana, 47374, U.S.A. peters@earlham.edu

75. Introduction To Gnosis #20 An article by Richard Smoley on the historical Gurdjieff and his teachings. Originally published in Gnosis magazine. http://www.lumen.org/intros/intro20.html

Meetings With a Remarkable Paradox by Richard Smoley He was by any account one of the most remarkable men the human race has produced. His name was George Ivanovich Gurdjieff (pronounced "gur-jeef" or "gur-jeff"), and he proved to be one of the most challenging, paradoxical, and enigmatic spiritual teachers of our time. His principal biographer called him "a fraud, a liar, a cheat, a scoundrel" - and then went on to note his "sympathy, compassion, charity," and his "eccentric code of honor."1 He is chiefly remembered for imparting, through the most extraordinary and difficult methods, the fundamentals of an esoteric system known as the Fourth Way - also called, austerely, "the Work." What is this Fourth Way? Gurdjieff said there were three comparatively common ways to spiritual attainment: the way of the fakir, that is, the man who masters his physical organism to the point where he can, say, stand "motionless in the same position for hours, days, months, or years."2 The second is the way of the monk, the man who masters his emotions through prayer and devotional practices. The third way is the way of the yogi, the man who gains control of his mind. Yet, Gurdjieff believed, none of these ways is complete in itself. A man, for example, may master his mind - he may genuinely know something - but may be incapable of putting it into action. Or his emotions may be developed, but his intellect may remain at a primitive state. To compound these difficulties, each of these first three ways requires withdrawal from the world, from day-to-day life.

76. Our Galaxy Should Be Teeming With Civilizations, But Where Are They? a simple observation. Surely there is a straightforward explanation for what has become known as the Fermi paradox. There must be http://www.space.com/searchforlife/shostak_paradox_011024.html

SEARCH: advertisement Our Galaxy Should Be Teeming With Civilizations, But Where Are They? By Seth Shostak Astronomer, Project Phoenix posted: 07:01 am ET 25 October 2001 Is there obvious proof that we could be alone in the Galaxy? Enrico Fermi thought so and he was a pretty smart guy. Might he have been right? It's been a hundred years since Fermi, an icon of physics, was born (and nearly a half-century since he died). He's best remembered for building a working atomic reactor in a squash court. But in 1950, Fermi made a seemingly innocuous lunchtime remark that has caught and held the attention of every SETI researcher since. (How many luncheon quips have you made with similar consequence?) The remark came while Fermi was discussing with his mealtime mates the possibility that many sophisticated societies populate the Galaxy. They thought it reasonable to assume that we have a lot of cosmic company. But somewhere between one sentence and the next, Fermi's supple brain realized that if this was true, it implied something profound. If there are really a lot of alien societies, then some of them might have spread out. Fermi realized that any civilization with a modest amount of rocket technology and an immodest amount of imperial incentive could rapidly colonize the entire Galaxy. Within ten million years, every star system could be brought under the wing of empire. Ten million years may sound long, but in fact it's quite short compared with the age of the Galaxy, which is roughly ten

77. The Twin Paradox Top Intro Next Original by Michael Weiss. The Twin paradox. This terminology. Also, the acceleration explanation of the twin paradox. http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.htm

[Physics FAQ] [Top] [Intro] [Next] Original by Michael Weiss. The Twin Paradox This hardy perennial seems to generate endless running threads on the sci.physics.* hierarchy. This contrasts sharply with the situation among mainstream physicists/hidebound reactionaries (take your choice), where the "paradox" has been no more than an entertaining (and educational) exercise since it first saw the light of day. It's too much to hope that this FAQ entry will change any minds, except maybe an "undecided" or two. But perhaps it can reduce the traffic on this topic. Read the sections as whim and mouse-clicks take you, though you might want to read the Introduction first. Introduction Where we introduce our heroes, Earth-bound Terence and star-bound Stella, and settle on a few things, like terminology. Also, the "acceleration" explanation of the twin paradox. The Doppler Shift Explanation Suppose Stella and Terence each film the other's clock through a telescope throughout the trip; when does Stella see Terence's clock run fast? The Spacetime Diagram Explanation Where we take an Olympian view, and gaze with the Fates upon the world lines of Terence and Stella.

78. Olbers' Paradox Original by Scott I. Chase. Olbers paradox. Why isn t the night sky as uniformly bright as the surface of the Sun? This is Olbers paradox. http://math.ucr.edu/home/baez/physics/Relativity/GR/olbers.html

[Physics FAQ] Updated 2004 by JCB. Original by Scott I. Chase. Olbers' Paradox Why isn't the night sky as uniformly bright as the surface of the Sun? If the Universe has infinitely many stars, then it should be. After all, if you move the Sun twice as far away from us, we will intercept one quarter as many photons, but the Sun will subtend one quarter of the angular area. So the areal intensity remains constant. With infinitely many stars, every angular element of the sky should have a star, and the entire heavens should be as bright as the sun. We should have the impression that we live in the center of a hollow black body whose temperature is about 6000 degrees Celsius. This is Olbers' paradox. It can be traced as far back as Kepler in 1610, and was rediscussed by Halley and Cheseaux in the eighteen century; but it was not popularized as a paradox until Olbers took up the issue in the nineteenth century. There are many possible explanations which have been considered. Here are a few: There's too much dust to see the distant stars.

79. Welcome To Includes five essays which summarize the Aquarian Gospel, the book that reunifies Judaism and Christianity. Also includes the entire text of the Aquarian Gospel, which purports to reveals the teachings of Jehoshua the prophet. http://home.netcom.com/~mokeeffe/

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