21. The Axiom Of Choice There are many equivalent statements of the axiom of choice. The axiom of choice (AC) is one of the most discussed axioms of mathematics, perhaps second http://db.uwaterloo.ca/~alopez-o/math-faq/mathtext/node35.html

Next: Cutting a sphere into Up: Axiom of Choice and Previous: Axiom of Choice and The Axiom of Choice There are several equivalent formulations: The Cartesian product of nonempty sets is nonempty, even if the product is of an infinite family of sets. Given any set S of mutually disjoint nonempty sets, there is a set C containing a single member from each element of S C can thus be thought of as the result of ``choosing" a representative from each set in S . Hence the name. Relevance of the Axiom of Choice THE AXIOM OF CHOICE There are many equivalent statements of the Axiom of Choice. The following version gave rise to its name: For any set X there is a function f , with domain , so that f(x) is a member of x for every nonempty x in X Such an f is called a ``choice function" on X . [Note that X (0) means X with the empty set removed. Also note that in Zermelo-Fraenkel set theory all mathematical objects are sets so each member of X is itself a set.] The Axiom of Choice (AC) is one of the most discussed axioms of mathematics, perhaps second only to Euclid's parallel postulate. The axioms of set theory provide a foundation for modern mathematics in the same way that Euclid's five postulates provided a foundation for Euclidean geometry, and the questions surrounding AC are the same as the questions that surrounded Euclid's Parallel Postulate: Can it be derived from the other axioms?

22. Zermelo's Axiom Of Choice Its Origins, Developments, And Apronus Home Mathematics Math Books Zermelo's axiom of choice Its Origins, Developments, and Influence by Gregory H. Moore. Prologue http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

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Home About Artists Albums ... Contact Album: Beyond Denial (Faraye-Enkaar) Artist(s): Axiom of Choice Style/Genre: World Indiginous Instrruments/Orchestral Arrangements Instrumentation: Vocals, Quarter Tone Guitar, Nylon String Guitar, Tar, Tarbass, Daf, Tombak, Nagada, Bass, Drums I have become a fugitive from the body, fearful as to the spirit; I swear I know not - I belong neither to this not to that - Rumi Persian émigrés (Mamak Khadem, Ramin Torkian, Pejman Hadadi) and American compatriots who have molded a sound that combines Middle Eastern melodies and rhythmic structures with progressive Western concepts. Their music features exotic and sensuous traditionally-styled Persian soaring feamale vocals, Middle Eastern and African percussion, Persian tar, nylon string guitar (performed by Yussi ), and a unusual quarter tone guitar that enables them to play the Persian modal scales that are unique to their music.

26. PlanetMath: Axiom Of Choice However, there is another axiom, the axiom of choice, which is more controversial Thus objects that are proved to exist using the axiom of choice cannot http://planetmath.org/encyclopedia/AxiomOfChoice.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia 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 more or less uncontroversial. However, there is another axiom , the axiom of choice , which is more controversial; it is therefore usually segregated from the others. When the Zermelo-Fraenkel axioms are accepted along with the axiom of choice, the whole axiom system is often called ``ZFC'' (for ``Zermelo-Frankel with Choice''). Axiom (Axiom of choice) Let be a set of nonempty sets. Then there exists a function such that for all The function is sometimes called a choice function on For finite sets , the axiom of choice is unnecessary to prove the existence of a choice function. It is only necessary for infinite (and usually uncountable ) sets that it becomes a problem. Here one can see why it is not considered ``obvious'' and always taken for an axiom by everyone: one really cannot imagine any process which makes uncountably many selections without also imagining some rule for making the selections. Given such a rule, the axiom of choice is not needed.

27. PlanetMath: Axiom Of Countable Choice The Axiom of Countable Choice (CC) is a weak form of the axiom of choice. Other names, countable axiom of choice, countable AC http://planetmath.org/encyclopedia/CountableAxiomOfChoice.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia 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 countable choice (Definition) The Axiom of Countable Choice (CC) is a weak form of the Axiom of Choice . It states that every countable set of nonempty sets has a choice function ZF+CC (that is, the Zermelo-Fraenkel axioms together with the Axiom of Countable Choice) suffices to prove that the union of countably many countable sets is countable. It also suffices to prove that every infinite set has a countably infinite subset "axiom of countable choice" is owned by yark view preamble View style: HTML with images page images TeX source Other names: countable axiom of choice, countable AC Also defines: countable choice Keywords: choice Cross-references: subset countably infinite infinite set union ... countable There are 25 references to this object.

28. 1999 Changes And Additions To ``Consequences Of The Axiom Of 1999 Changes and additions for the Consequences of the axiom of choice Project Keremedis, K./Tachtsis, E. 1999b The countable axiom of choice for http://www.math.purdue.edu/~jer/cgi-bin/changes-99.htm

1999 Changes and additions for the Consequences of the Axiom of Choice Project Changes June 1999 Add the following items to the bibliography: Kolany, A. Herrlich, H./Keremedis, K. [1998] Products, the Baire category theorem, and the axiom of dependent choice, preprint October 12. Add Goldblatt [1985] as a reference for form [345 A]. Forms [14 CM] and [43 W] through [43 AC] have been added. Also, the first definition in note 28 has been revised. For these additions and revisions choose one of the two below: TeX version pdf version Put [14 CM] in CHOICE FORMS, V. Conditional Choice and put [43 W] through [43 AC] in TOPOLOGICAL FORMS I. Baire Category Type Theorems and in II. Product Theorems. In the bibliography, change Banaschewski [1998] to: Choice functions and compactness conditions , Math. Logic Quart. change Felgner, U./Truss, J. K. [1999] to: The independence of the prime ideal theorem from the order extension theorem , Fund. Math.

29. Axiom Of Choice The axiom of choice is not part of ZF. It is however widely accepted and However Paul Cohen constructed models of ZF in which the axiom of choice was http://www.mtnmath.com/whatrh/node57.html

PDF version of this book Next: Trees of trees Up: Creative mathematics Previous: Power set axiom Contents Axiom of Choice The Axiom of Choice is not part of ZF. It is however widely accepted and critical to some proofs. The combination of this axiom and the others in ZF is called ZFC The axiom states that for any collection of non empty sets there exists a choice function that can select an element from every member of . In other words for every A mathematically complete statement of the above requires a definition in the language of set theory of function. A function is a set of ordered pairs where the first element is in the domain of the function and the second element is in the range of the function. Each pair maps an element of the domain uniquely into an element of the range. Thus each first element must occur only once as in the set that defines the function. . Essentially these are the sets one can build up by applying the axioms of ZF. In this model the axiom of choice is true. However Paul Cohen constructed models of ZF in which the Axiom of Choice was false making it clear that this axiom cannot be derived from the other axioms. It is a strange axiom since it would seem to be obvious. If one has a collection of sets then one should be able to choose one member from each set. But in general there is no way to do this using the axioms of ZF. It is one example of the strange nature of the infinite in formal mathematics. The real numbers derived from the power set allow one to search over all reals. This leads to many other strange questions and another postulate sometimes needed for theorems that is not derivable from the other axioms. This is the Continuum Hypothesis

30. Topological Equivalents Of The Axiom Of Choice And Of Weak Forms Of Choice, By E The axiom of choice (AC) has many important equivalent forms in many branches of The axiom of choice is the most wellknown nonconstructive assertion of http://at.yorku.ca/z/a/a/b/18.htm

Topology Atlas Document # zaab-18.htm Topology Atlas Invited Contributions, vol. 1, issue 4 (1996), 60-62. Topology Atlas Topological Equivalents of the Axiom of Choice and of Weak Forms of Choice by Eric Schechter (Department of Mathematics, Vanderbilt University, Nashville TN 37240-0001, U.S.A.) The Axiom of Choice (AC) has many important equivalent forms in many branches of mathematics Zorn's Lemma, the Well Ordering Principle, the Vector Basis Theorem. In general topology, perhaps the most important equivalent is Tychonov's Product Theorem: any product of compact topological spaces, when equipped with the product topology, is also compact. Some other statements about product topologies are also equivalent: the product of complete uniform spaces, when equipped with the product uniformity, is also complete; the product of closures of subsets of topological spaces is equal to the closure of the product of those subsets. The term ``constructive'' is used in different fashion by different mathematicians. For the most part, it means that we can ``find'' the object in question, and not just prove that it exists. The Axiom of Choice is the most well-known nonconstructive assertion of existence; it has important consequences for many branches of mathematics. The Axiom of Foundation (also known as the Axiom of Regularity) is also nonconstructive, but it has few applications in ``ordinary'' mathematics (i.e., outside of set theory). However, nonconstructiveness can occur not only in our axioms, but even in our reasoning:

31. Re: Re: Re: Tychonoff=Axiom Of Choice? Re Tychonoff=axiom of choice? by Henno Brandsma (Feb 28, 2003) In reply to Tychonoff=axiom of choice? , posted by pit on Feb 27, 2003 http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2003;task=show_msg;msg=00

32. Axiom Of Choice From FOLDOC mathematics (AC, or Choice ) An axiom of set theory We can also construct a choice function for most simple infinite sets of sets if they are http://foldoc.doc.ic.ac.uk/foldoc/foldoc.cgi?Axiom of Choice

33. Axiom Of Choice: Information From Answers.com axiom of choice In mathematics , the axiom of choice is an axiom of set theory . It was formulated in 1904 by Ernst Zermelo. http://www.answers.com/topic/axiom-of-choice

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping axiom of choice Wikipedia axiom of choice In mathematics , the axiom of choice is an axiom of set theory . It was formulated in by Ernst Zermelo . While it was originally controversial, it is now accepted and used casually by most mathematicians. However, there are still schools of mathematical thought, primarily within set theory, which either reject the axiom of choice, or even investigate consequences of its negation. The axiom of choice is typically abbreviated AC, or C as a suffix. Statement The axiom of choice states: Let X be a set of non-empty sets. Then we can choose a member from each set in X Stated more formally: Let X be a set of non-empty sets. Then there exists a choice function f defined on X . In other words, 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 states: Given any set of mutually disjoint non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets.

34. The Axiom Of Choice And Zorn's Lemma This seemingly trivial statement is known as the axiom of choice (since it says that we may To first understand the subtleties of the axiom of choice, http://www.math.uchicago.edu/~mileti/museum/choice.html

The Axiom of Choice and Zorn's Lemma In America, every state has a governor which represents it. Can you imagine a country where this is not possible? That is, can you picture a country (such that each state has at least one person living it to avoid trivialities) where we could not select a governor to represent each state? Such an idea seems preposterous. After all, we can simply go through each state one-by-one and choose a random person to designate as governor. If we translate the above statement into an abstract mathematical setting, we are saying that given any collection of nonempty sets, we may choose one object from each set. This seemingly trivial statement is known as the Axiom of Choice (since it says that we may choose a representative from each set) and has had a remarkable influence on the development of mathematics since the beginning of the twentieth century. How could such an obvious principle have such an impact? To first understand the subtleties of the Axiom of Choice, one has to realize that when phrased in such an general setting, the statement is not as obvious as one would originally think. When considering countries in the example above, you probably never thought about a country that has infinitely many states. What happens if you imagine such a country? If we tried to list the states in some order and one-by-one to pick out representatives, then we would never finish after a finite number of steps. Imagine the federal government trying to finalize the results of the election from an infinite number of states. It takes some nontrivial amount of time to fill out the necessary paperwork and swear in a governor, so if only one federal office was in charge of the election, then they would never finish the entire task. To further complicate the matter, we know that there are

35. Axiom Of Choice - Definition Of Axiom Of Choice In Encyclopedia 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 http://encyclopedia.laborlawtalk.com/Axiom_of_choice

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law 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 F depends on the axiom of choice. The principle seems obvious: if there are several boxes, each containing at least one item, the axiom simply states that one can choose exactly one item from each box. Although the statement sounds straightforward, there is controversy over what it means to

36. Infinite Ink For an overview of the axiom of choice see the Relevance of the axiom of choice (FAQ Launcher). Introduction. Notation and Format; Philosophy; Overview http://www.ii.com/math/ac/

Trapped in a frame? Break free now! T HE A XIOM OF C HOICE For an overview of the Axiom of Choice see the Relevance of the Axiom of Choice (FAQ Launcher) Introduction Notation and Format Philosophy Overview Axiom of Choice Quiz Foundations Formal systems, axiomatic theories, and metatheory Euclidean geometry and the parallel postulate ZFC Quiz Solutions and Weak Forms of AC Explanation of Quiz Answers Weak forms of AC Some theorems whose proofs within ZFC require Countable Choice or Dependent Choice Common Equivalents of AC Choice Principles, Ordering Principles, and Maximality Principles Less Well Known Equivalents of AC Cardinality Theorems Tychonoff's Theorem Every vectory space has a basis Theorems Whose Proofs Within ZFC Require AC Equivalents of the Ultrafilter Theorem A discontinuous additive function "Bad" sets of reals Conclusion Axioms that imply AC The axiom of determinacy: An axiom which contradicts AC Category Theory: A new foundation of mathematics? Accept AC? Appendix ZFC Axioms Notation Index Glossary Bibliography top of this page about ii d i ... thanks!

37. FAQ Launcher: Relevance Of The Axiom Of Choice Gateway to Relevance of the axiom of choice. Related Info, Eric Schechter axiom of choice Infinite Ink The Continuum Hypothesis sci.math FAQ http://www.ii.com/internet/faqs/launchers/sci-math-faq/AC/relevance/

FAQ Launcher R ELEVANCE OF T HE A XIOM OF C HOICE Description Mathematics, metamathematics, and philosophy of the Axiom of Choice. (This is part 26 of the sci.math FAQ.) Review University of Waterloo (non-graphical) University of Waterloo (graphical) Utrecht University Oxford University Smart Pages Related Info Eric Schechter: Axiom of Choice Infinite Ink: The Continuum Hypothesis sci.math FAQ Yahoo: Logic Discussion sci.math sci.logic sci.philosophy.tech FAQ Maintainer Infinite Ink faq-editor@ii.com top of this page about ii ... Infinite Ink and Nancy McGough Last content update on April 19, 1997 Last tweak on April 19, 1997 www.ii.com/internet/faqs/launchers/sci-math-faq/AC/relevance/ www.best.com/~ii/internet/faqs/launchers/sci-math-faq/AC/relevance/

38. Axiom Of Choice The axiom of choice (AC) says simply that you can always choose one item out of each box. More formally, if S is a collection of nonempty sets, http://www.daviddarling.info/encyclopedia/A/axiom_of_choice.html

Return to The Worlds of David Darling INDEX LATEST NEWS ARCHIVE ... Z 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

39. The Axiom Of Choice First, here s a real link to information on the axiom of choice. Zermelo identified the axiom of choice in 1904 when he used it to proved the wellordering http://www.andrew.cmu.edu/user/cebrown/notes/axiom-of-choice.html

Some Links to Notes on Axiom of Choice First, here's a real link to information on the Axiom of Choice. Zermelo identified the axiom of choice in 1904 when he used it to proved the well-ordering principle. Later, in 1908, he defended the use of choice in the original 1904 proof as well as in a newer proof of the well-ordering principle. The axiom of choice was also an involved in the proofs of the Lowenheim-Skolem theorem. Lowenheim's original proof contained gaps that could be filled using versions of the axiom of choice. Skolem filled in these gaps in two different ways (yeilding two slightly different results). One way used choice; the other did not. Fraenkel proved the independence of the axiom of choice using the idea of Russell's socks. This technique of involves constructing permutation models (requiring the existence of urelements) known as Fraenkel-Mostowski models. Godel's constructible universe showed the consistency of the axiom of choice. Cohen later proved the independence of the axiom of choice and the continuum hypothesis using forcing. Church included a version of the axiom of choice in his type theory.

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The Persian Student Association at Stanford University presents... AXIOM OF CHOICE Performing at Stanford University Friday, October 11, 2002 - 8PM Dinkelspiel Auditorium Tickets available by phone at the Stanford Ticket Office - (650) 725-ARTS or on-line at Tickets.com $20 general, $10 students, staff, and senior citizens THE CONCERT IS SOLD OUT! More about Axiom of Choice see the article in the Stanford Daily "Iranian band plays to packed Dinkelspiel" Watch this movie to learn more about Axiom of Choice (includes live concert footage): (32MB download - Best quality (MPEG)) (6MB download - fastest (ASF)) Concert Press Release (PDF) Concert Program (PDF) ... Directions to Dinkelspiel Auditorium Sponsors: For more info, please send email to: psa-admin@lists.stanford.edu

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