61. Axiom Of Choice - Definition And Meaning From Wordnik Axiom of choice One of the axioms in axiomatic set theory, equivalent to the statement that an arbitrary direct product of nonempty sets is non-empty. http://www.wordnik.com/words/axiom of choice

Sign up! Sign in Zeitgeist Word of the day ... Random word Username Password ( Forgot? or Using Facebook Connect logs you in to Wordnik whenever you are logged in to Facebook. Not a member yet? Sign up for an account axiom of choice Add to favorites Add to a list Link to this word [x] ... Tweet Definitions Thesaurus Examples Pronunciations Comments (2) axiom of choice in [x] Wiktionary (1 definition) –noun (set theory) One of the axioms in axiomatic set theory, equivalent to the statement that an arbitrary direct product of non-empty sets is non-empty. American Heritage Dictionary Century Dictionary GNU Webster's 1913 WordNet Elsewhere on the web axiom of choice elsewhere on the web: Ninjawords Online Etymology Dictionary OneLook Bartleby All Books ... Examples Direct attempts to extend the negative interpretation to analysis fail because the negative translation of the countable axiom of choice is not a theorem of intuitionistic analysis. Intuitionistic Logic This is already indicated by the fact that most regimentations of second-order logic adopt a version of the axiom of choice as one of its axioms.

62. The Axiom Of Choice Netherlands site with news, interviews, pictures, music samples, reviews, and links. http://www.cs.uu.nl/people/jur/progrock.html

63. Axiom Of Choice | Musings To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed. — Bertrand Russell http://golem.ph.utexas.edu/~distler/blog/archives/000283.html

@import url("/~distler/blog/styles-site.css"); Musings Thoughts on Science, Computing, and Life on Earth. Skip to the Main Content Enough, already! Skip to the content. Note: These pages make extensive use of the latest XHTML and CSS Standards only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser. :hover Craft Main January 8, 2004 Axiom of Choice To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed. Bertrand Russell latest post Axiom of Choice Homepage . The latter discusses the AC , and a whole range of related topics. Here, for instance, is his discussion of the Banach-Tarski Paradox: Banach and Tarski used the Axiom of Choice to prove that it is possible to take the 3-dimensional closed unit ball, B x y z x y z and partition it into finitely many pieces, and move those pieces in rigid motions (i.e., rotations and translations, with pieces permitted to move through one another) and reassemble them to form two copies of

64. Axiom Of Choice And Continuum Hypothesis Part of the Frequently Asked Questions in Mathematics. http://db.uwaterloo.ca/~alopez-o/math-faq/mathtext/node34.html

Next: The Axiom of Choice Up: Frequently Asked Questions in Mathematics Previous: Master Mind Axiom of Choice and Continuum Hypothesis The Axiom of Choice Cutting a sphere into pieces of larger volume The Continuum Hypothesis Alex Lopez-Ortiz Fri Feb 20 21:45:30 EST 1998

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68. PlanetMath: Axiom Of Choice The ZermeloFraenkel axioms for set theory are often accepted as a basis for an axiomatic set theory. On the other hand, the axiom of choice is somewhat controversial, and it http://planetmath.org/encyclopedia/AxiomOfChoice.html

(more info) Math for the people, by the people. donor list find out how Encyclopedia Requests ... Advanced search Login create new user name: pass: forget your password? Main Menu sections Encyclopædia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About axiom of choice (Axiom) The Zermelo-Fraenkel axioms for set theory are often accepted as a basis for an axiomatic set theory. On the other hand, the axiom of choice is somewhat controversial, and it is currently segregated from the ZF system of set theory axioms . When the axiom of choice is combined with the ZF axioms, the whole axiom system is called ``ZFC'' (for ``Zermelo-Fraenkel with Choice''). Axiom Let $C$ be a set of nonempty sets. Then there exists a function such that for all The function $f$ is sometimes called a choice function on $C$ For finite sets $C$ , a choice function can be constructed without appealing to the axiom of choice. In particular, if , then the choice function is

69. Finite Axiom Of Choice: How Do You Prove It From Just ZF? - MathOverflow The axiom of choice asserts the existence of a choice function for any family of sets F. Suppose, however, that F is finite, or even that F http://mathoverflow.net/questions/32538/finite-axiom-of-choice-how-do-you-prove-

70. Axiom Of Choice | Define Axiom Of Choice At Dictionary.com –noun Mathematics . the axiom of set theory that given any collection of disjoint sets, a set can be so constructed that it contains one element from each of the given sets http://dictionary.reference.com/browse/Axiom of Choice

71. Lingua Franca - November 2001| Hypothesis: Single-Egg Theory To understand what the axiom of choice is, start with this homely example, apparently thought up by Bertrand Russell. Suppose you have an infinite number of http://linguafranca.mirror.theinfo.org/print/0111/hypothesis.html

University Business UB Daily UB Exec Academic Partners ... Table of contents for this issue HYPOTHESES A Matter Of Choice Can a mathematical idea have political import? By Jim Holt THE LITERARY THEORIST JULIA KRISTEVA has written that "the functioning of poetic meaning obeys the principles designated by the axiom of choice." Alan Sokal, in his famous send-up of postmodern theory, "Transgressing the Boundaries," suggested that the axiom of choice had something to do with the abortion-rights movement and the feminist agenda for equality. Later, in the book Fashionable Nonsense , Sokal and his co-author, Jean Bricmont, correctively declared that the axiom of choice "has no political implications whatsoever." What is this much-invoked thing called the axiom of choice? Is it really devoid of political significance, as Sokal and Bricmont claim? Or could it turn out to pack an ideological punch beyond the imagination of even the most wild-eyed Left Bank postmodern theorist? socks and you want to select one sock from each pair. Since socks in a pair, unlike shoes, are identical, there is no rule for defining a set that consists of precisely one sock from each pair. The choice for each pair would have to be arbitrary; and since there are infinitely many pairs, that means an infinite number of arbitrary choices. Here is where the axiom of choice comes to the rescue. It allows one to assume the existence of such a "choice set," even though there is no rule for constructing it. The axiom of choice caused fierce controversy when it was first framed by the German logician Ernst Zermelo in 1904. Anti-choicers scoffed at the idea of a mathematical object existing by fiat. Pro-choicers countered by insisting that the axiom was self-evidently true and that it was crucial for mathematics and for science itself. Zermelo pointed out that many vocal anti-choicers had unwittingly been using the axiom of choice in their work before he explicitly formulated it.

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73. Axiom Of Choice (set Theory) -- Britannica Online Encyclopedia axiom of choice (set theory), statement in the language of set theory that makes it possible to form sets by choosing an element simultaneously from each member of an infinite http://www.britannica.com/EBchecked/topic/46212/axiom-of-choice

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75. Principio Del Buen Orden > Well Ordering Principle Apr 8, 2005 An important and fundamental axiom in set theory sometimes called Zermelo s axiom of choice. It was formulated by Zermelo in 1904 and http://www.proz.com/kudoz/spanish_to_english/mathematics_statistics/995162-princ

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76. Math Forum - Ask Dr. Math It turns out that the answer to this question requires a special axiom of set theory, namely the axiom of choice Wikipedia Axiom of Choice http://mathforum.org/library/drmath/view/74268.html

Associated Topics Dr. Math Home Search Dr. Math Countable and Uncountable Infinities http://mathworld.wolfram.com/Schroeder-BernsteinTheorem.html As I said, this is far from obvious. What the theorem states is that, if there are injective functions: f : A -> B g : B -> A then there is a bijective function: h : A -> B (write back if you would like a proof). For example, this can be used to show that the closed interval A = [0,1] is equipotent to the half-open interval [0,1), although a bijection h : A -> B is not obvious (well, it is not very hard to find...). Another question is whether this relation "<=" is a total order. Given two sets A and B, is it always true that at least one of the relations #A <= #B or #B <= #A holds, i.e., are any two sets "comparable"? It turns out that the answer to this question requires a special axiom of set theory, namely the axiom of choice: Wikipedia: Axiom of Choice http://en.wikipedia.org/wiki/Axiom_of_choice http://mathforum.org/dr.math/ Associated Topics College Logic Search the Dr. Math Library:

77. Math 3120 Homework 24 Equivalent Sets, Finite Sets (5.1) Due File Format PDF/Adobe Acrobat Quick View http://www.suu.edu/faculty/brown_s/courses/fall06-3120/documents/hwchapter5_002.

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79. Axiom Of Choice Definition Of Axiom Of Choice In The Free Online Encyclopedia. axiom of choice ak sē əm əv ′chȯis (mathematics) The axiom that for any family A of sets there is a function that assigns to each set S of the family A a member of S. http://encyclopedia2.thefreedictionary.com/Axiom of Choice

80. Axiom Of Choice@Everything2.com Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property http://www.everything2.com/title/Axiom of Choice

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