1. Axiom Of Choice - Wikipedia, The Free Encyclopedia Consequences of the Axiom of Choice, based on the book by Paul Howard and Jean Rubin. Retrieved from http//en.wikipedia.org/wiki/axiom_of_choice http://en.wikipedia.org/wiki/Axiom_of_choice

Axiom of choice From Wikipedia, the free encyclopedia Jump to: navigation search This article is about the mathematical concept. For the band named after it, see Axiom of Choice (band) In mathematics , the axiom of choice , or AC , is an axiom of set theory . Intuitively speaking, the axiom of choice says that given any collection of bins, each containing at least one object, exactly one object can be selected from each bin, even if there are infinitely many bins and there is no "rule" for which object to pick from each. The axiom of choice is not required if the number of bins is finite or if such a selection "rule" is available. It was formulated in 1904 by Ernst Zermelo While it was originally controversial, it is now used without reservation by most mathematicians. However, there are schools of mathematical thought, primarily within set theory, that either reject the axiom of choice or investigate consequences of axioms inconsistent with AC. Contents Statement Variants Usage Nonconstructive aspects ... edit Statement A choice function is a function f , defined on a collection X of nonempty sets, such that for every set

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5. Axiom Of Choice - Wikipedia Retrieved from http//nostalgia.wikipedia.org/wiki/axiom_of_choice . This page was last modified 0440, 15 January 2005. Content is available under GNU http://nostalgia.wikipedia.org/wiki/Axiom_of_choice

Axiom of choice From Wikipedia HomePage Recent changes View source Page history ... Log in Special pages Double redirects Broken redirects Disambiguation pages Log in Preferences My watchlist Recent changes Upload file File list Gallery of new files User list Statistics Random page Orphaned pages Uncategorized pages Uncategorized categories Uncategorized files Uncategorized templates Unused categories Unused files Wanted pages Wanted categories Most linked to pages Most linked to categories Most linked-to templates Pages with the most categories Most linked to files Pages with the most revisions Pages with the fewest revisions Short pages Long pages New pages Oldest pages Dead-end pages Protected pages Protected titles All pages Prefix index List of blocked IP addresses and usernames User contributions What links here Book sources Categories Export pages Version System messages Logs MIME search Search for duplicate files List redirects Unused templates Random redirect Pages without language links File path Search List of Wikimedia wikis Expand templates CategoryTree Gadgets Parser diff test Cross-namespace links Wikimedia Board of Trustees election Search web links Printable version The axiom of choice is an axiom in set theory . It was formulated about a century ago by Ernst Zermelo , and was quite controversial at the time. It states the following:

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7. Axiom Of Choice - Conservapedia Retrieved from http//www.conservapedia.com/axiom_of_choice . Categories Articles with unsourced statements Set theory Mathematics http://www.conservapedia.com/Axiom_of_Choice

Axiom of Choice From Conservapedia Jump to: navigation search The Axiom of Choice AC is an axiom of Zermelo-Fraenkel set theory holding that: given any collection of sets, however large, we can pick one element from each set in the collection. In layman's terms, the Axiom of Choice is "For every nonempty set there is a choice function." Or, "We can choose one element from every element of a nonempty set of disjoint sets and this process by which we choose these sets will set up a new set." More precisely, the Axiom of Choice states that: For every collection of nonempty sets S, there exists a function f such that f(S) is a member of S for every possible S. Mathworld explains the Axiom of Choice as follows: Given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. Yet another helpful explanation of the Axiom of Choice is this: If you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C. H is called the "choice set" because you are essentially going through each set in C and choosing one element from it. One feature of the Axiom of Choice is that H is simply assumed to exist; there is no algorithm given which might tell you how to construct an example of H. Contents Use of the Axiom of Choice Controversy Sources References ... edit Use of the Axiom of Choice

8. Axiom Of Choice An axiom in set theory that is one of the most controversial axioms in mathematics; it was formulated in 1904 by the German mathematician Ernst Zermelo http://www.daviddarling.info/encyclopedia/A/axiom_of_choice.html

MATHEMATICS A B C ... CONTACT entire Web this site axiom of choice An axiom in set theory S is a collection of non-empty sets , then there exists a set that has exactly one element in common with every set S of S . Put another way, there exists a function f with the property that, for each set S in the collection, f S ) is a member of S . Bertrand Russell summed it up neatly: "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. His point is that the two socks in a pair are identical in appearance, so, to pick one of them, we have to make an arbitrary choice. For shoes, we can use an explicit rule, such as "always choose the left shoe." Russell specifically mentions infinitely many pairs, because if the number is finite then AC is superfluous: we can pick one member of each pair using the definition of "nonempty" and then repeat the operation finitely many times using the rules of formal logic. AC lies at the heart of a number of important mathematical arguments and results. For example, it is equivalent to the well-ordering principle , to the statement that for any two cardinal numbers m and n , then m n or m n or m n , and to Tychonoff's theorem (the product of any collection of compact spaces in topology is compact). Other results hinge upon it, such as the assertion that every infinite set has a denumerable subset. Yet AC was strongly attacked when it was first suggested, and still makes some mathematicians uneasy. The central issue is what it means to choose something from the sets in question and what it means for the choosing function to exist. This problem is brought into sharp focus when

9. BibSonomy::user::a_olympia::Axiom_of_Choice Webapplikation des Fachgebiets Wissensverarbeitung, Universität Kassel. http://www.bibsonomy.org/user/a_olympia/Axiom_of_Choice

BibSonomy user tag user group author concept BibTeX key search:all search:a_olympia cssdropdown.startchrome("path"); A blue social bookmark and publication sharing system. tags relations groups popular ... about username: password: myFriends myRelations mySearch myPDF ... register cssdropdown.startchrome("upper_menu"); cssdropdown.startchrome("lower_menu"); bookmarks Prologue Chapter 1: The Prehistory of the Axiom of Choice Introduction The Origins of the Assumption The Boundary between the Finite and the Infinite to Zermelo math mathematik on 2006-10-02 15:49:42.0 publications Showing 10 items per page. Show items per page. as tag from all users as concept from all users related tags Zermelo mathematik math relations ... Connotea tags (Library +VORWAHL+RUFNUMMER .net a ... „perversen BibSonomy is offered by the Knowledge and Data Engineering Group of the University of Kassel, Germany. Contact: webmaster at bibsonomy.org init(0, 0, 0, "a_olympia", "", "", "BibSonomy");

10. Axiom Of Choice - Indopedia, The Indological Knowledgebase Retrieved from http//www.indopedia.org/axiom_of_choice.html . This page has been accessed 2998 times. This page was last modified 1743, http://www.indopedia.org/Axiom_of_choice.html

Indopedia Main Page FORUM Help ... Log in The Indology CMS In other languages: Deutsch Categories Set theory Printable version ... Wikipedia Article Axiom of choice ज्ञानकोश: - The Indological Knowledgebase In mathematics , the axiom of choice is an axiom of set theory . It was formulated about a century ago by Ernst Zermelo , and was quite controversial at the time. It states the following: Let X be a collection of non-empty sets . Then we can choose a member from each set in that collection. Stated more formally: There exists a function f defined on X such that for each set S in X f S ) is an element of S Another formulation of the axiom of choice (AC) states: Given any set of mutually exclusive non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets. For many years, the axiom of choice was used implicitly. For example, a proof might have, after establishing that the set S contains only non-empty sets, said "let F(X) be one of the members of X for all X in S ." Here, the existence of the function

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12. Axiom Of Choice - English Dictionary english to english dictionary containing references. http://www.online-dictionary.biz/english/vocabulary/reference/axiom_of_choice.as

var language=0; var from='english'; var to='english'; Online Dictionary Chinese to English English to Chinese ... French to English German to English to Japanese Italian to English Japanese to English to German Latin to English Russian to English ... Swedish to English If you can't find the translation you need, try our free translation Axiom Of Choice - English Dictionary 1. Axiom of Choice A B C D ... Z All content on this website is property of LocalTranslation unless stated otherwise. document.getElementById("generationTime").innerHTML='('+(0.52)+' seconds)'; document.getElementById("generationTime").innerHTML='('+(0.52)+' seconds)';

13. Axiom Of Choice - Wikipedia Axiom of choice. The axiom of choice is an axiom in set theory. It was formulated about a century ago by Ernst Zermelo, and was quite controversial at the http://facetroughgemstones.com/wikipedia/ax/Axiom_of_choice.html

Office Supplies Buy Posters A-Z Products Website Advertising ... Axiom of choice The axiom of choice is an axiom in set theory . It was formulated about a century ago by Ernst Zermelo , and was quite controversial at the time. It states the following: Let X be a collection of non-empty sets . Then we can choose a member from each set in that collection. Stated more formally, there exists a function f defined on X such that for each set S in X f S ) is an element of S It seems obvious: if you've got a bunch of boxes lying around with at least one item in each of them, the axiom simply states that you can choose one item out of each box. Where's the controversy? Well, the controversy was over what it meant to choose something from these sets. As an example, let us look at some sample sets. 1. Let X be any finite collection of non-empty sets. Then f can be stated explicitly (out of set A choose a , ...), since the number of sets is finite. Here the axiom of choice is not needed, you can simply use the rules of formal logic. 2. Let

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16. Discrete Mathematics/Axiom Of Choice - Wikibooks, Collection Of Open-content Tex is the identity (trivial) map. Lemma Every set can be wellordered. Retrieved from http//en.wikibooks.org/wiki/Discrete_mathematics/axiom_of_choice http://en.wikibooks.org/wiki/Discrete_mathematics/Axiom_of_choice

Discrete mathematics/Axiom of choice From Wikibooks, the open-content textbooks collection Discrete mathematics Jump to: navigation search Axiom of choice If is a surjective map, then there exists a map such that is the identity (trivial) map. Lemma: Every set can be well-ordered. Retrieved from " http://en.wikibooks.org/wiki/Discrete_mathematics/Axiom_of_choice Subject Discrete mathematics (book) Views Module Discussion Edit this page History Personal tools Log in / create account Navigation Main Page Help Cookbook Wikijunior ... Donations Community Bulletin Board Community portal Reading room Help cleanup ... Contact us Search Toolbox What links here Related changes Upload file Special pages ... Permanent link This page was last modified 21:14, 12 October 2007. All text is available under the terms of the GNU Free Documentation License (see for details). Wikibooks® is a registered trademark of the Wikimedia Foundation, Inc. About Wikibooks

17. Axiom Of Choice - Wiktionary Retrieved from http//en.wiktionary.org/wiki/axiom_of_choice . Categories English nouns Set theory Mathematics Logic http://en.wiktionary.org/wiki/axiom_of_choice

axiom of choice From Wiktionary Jump to: navigation search edit English Wikipedia has an article on: Axiom of choice Wikipedia edit Noun Singular axiom of choice Plural uncountable axiom of choice uncountable set theory The assumption that the direct product of any nonempty family of nonempty sets , is nonempty edit Translations Croatian: aksiom izbora (hr) m Czech: axiom v½běru (cs) m Retrieved from " http://en.wiktionary.org/wiki/axiom_of_choice Categories English nouns Set theory ... Logic Views Entry Discussion Edit History Personal tools Log in / create account Navigation Main Page Community portal Requested entries Recent changes ... Contact us Search Toolbox What links here Related changes Upload file Special pages ... Permanent link In other languages Svenska This page was last modified 15:16, 7 March 2008. Content is available under GNU Free Documentation License About Wiktionary

18. Axiom Of Choice - Interaction-Design.org: A Site About HCI, Usability, UI Design URL http//www.interactiondesign.org/dictionary/axiom_of_choice.html. Advertise here.. © Site Copyright/IP Policy Privacy Changelog http://www.interaction-design.org/dictionary/axiom_of_choice.html

About Contact Site map References ... Dictionary Axiom Of Choice Axiom Of Choice e.g. " user interface" Sorry, no results in the dictionary If you think you may be able to help us correct this error, please click here to contact the administrator Page information Page maintainer: The Editorial Team Database provider: The DICT Development Group (dict.org) Date created: Not available Date last modified: Not available URL: http://www.interaction-design.org/dictionary/axiom_of_choice.html Advertise here.. Privacy Changelog How to help ... RSS

19. Compile With -DP_VERBOSE=1 For Verbose Output. */ /* Compile With unrecognized %token sval unsigned_integer %token sval upper_word %token sval AC %token sval AXIOM %token sval axiom_of_choice %token sval http://www.soe.ucsc.edu/~avg/TPTPparser/FO/syntaxBNF-1.y

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20. Dick/maths/dictionary On Thu Feb 21 143040 PST 2008 axiom_of_choice logic_30_Sets; axiom_of_choice logic_5_Maps; Axioms intro_logic; axioms math.syntax; AXIOMS_OF_SETS_OF_STRINGS. http://www.csci.csusb.edu/dick/maths/dictionary.html

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