1. Axiom Of Choice - Official Site axiom of choice are musicians with their roots in traditional Persian music. axiom of choice. Look for the new album UNFOLDING In stores now! http://www.axiomofchoice.com/

Welcome to the official site of AXIOM OF CHOICE Look for the new album UNFOLDING In stores now! Axiom of Choice are emigré Persian musicians with their roots in the repertoire of traditional Persian music. Led by guitarist and Music Director Loga Ramin Torkian and featuring Mamak Khadem on vocals, their music is conceptual and combines Eastern and Western instrumentation in original arrangements bringing a new sound to Persian and world music. UNFOLDING A Trans-Global Exploration of Omar Khayyam's Mystic Vision Interpreting the spirit of Khayyam's evocative poetry, Axiom of Choice crafts progressive Persian music. Quarter-tone guitar, duduk, cello, ney, kamancheh, percussion, and exquisite vocals color a sophisticated palette of compositions and a lush landscape of sounds. Read Reviews about Axiom of Choice and UNFOLDING Persia Review The Seattle Times Stanford Daily A Conversation with Mamak Khadem of Axiom of Choice at Cranky Crow World Music CONCERT REVIEW Axiom Of Choice One positive side effect of U.S. involvement in Afghanistan has been increased interest by

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

3. 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

4. Consequences Of The Axiom Of Choice Project Project to keep the book (also named in the title), describing forms related to the axiom of choice and their implications, updated. http://www.math.purdue.edu/~jer/cgi-bin/conseq.html

Consequences of the Axiom of Choice Project Homepage The book Consequences of the Axiom of Choice by Paul Howard Send E-Mail to Paul Howard and Jean E. Rubin Send E- Mail to Jean Rubin is volume 59 in the series Mathematical Surveys and Monographs published by the American Mathematical Society in 1998. This book is a survey of research done during the last 100 years on the axiom of choice and its consequences. (Connect to The AMS Bookstore for ordering information.) The Consequences of the Axiom of Choice Project is a continuation of the research that produced the book. The authors would appreciate learning of any corrections or additions that should be made to the project. (phoward@emunix.emich.edu, jer@math.purdue.edu) To see the PDF version of a form enter its number below. The form number On this page you will find: Changes and additions to the data base that have occurred since publication of the book A TeX version of the implication table, Table 1 which may be downloaded and printed. (Hold down the shift key and click on the file name to download.) A TeX version of the auxillary table

5. Axiom Of Choice a home page for the axiom of choice an introduction and links collection by Eric Schechter, Vanderbilt University. axiom of choice. http://www.math.vanderbilt.edu/~schectex/ccc/choice.html

a home page for the AXIOM OF CHOICE an introduction and links collection by Eric Schechter , Vanderbilt University The Axiom of Choice ( AC ) was formulated about a century ago, and it was controversial for a few of decades after that; it may be considered the last great controversy of mathematics. It is now a basic assumption used in many parts of mathematics. In fact, assuming AC is equivalent to assuming any of these principles (and many others): Given any two sets, one set has cardinality less than or equal to that of the other set i.e., one set is in one-to-one correspondence with some subset of the other. ( Historical remark: It was questions like this that led to Zermelo 's formulation of AC.) Any vector space over a field F has a basis i.e., a maximal linearly independent subset over that field. ( Remark: If we only consider the case where F is the real line, we obtain a slightly weaker statement; it is not yet known whether this statement is also equivalent to AC.) Any product of compact topological spaces is compact. (This is now known as Tychonoff's Theorem , though Tychonoff himself only had in mind a much more specialized result that is not equivalent to the Axiom of Choice.)

6. Axiom Of Choice This page gives a brief explanation of the axiom of choice and links to other related websites. http://math.vanderbilt.edu/~schectex/ccc/choice.html

a home page for the AXIOM OF CHOICE an introduction and links collection by Eric Schechter , Vanderbilt University The Axiom of Choice ( AC ) was formulated about a century ago, and it was controversial for a few of decades after that; it may be considered the last great controversy of mathematics. It is now a basic assumption used in many parts of mathematics. In fact, assuming AC is equivalent to assuming any of these principles (and many others): Given any two sets, one set has cardinality less than or equal to that of the other set i.e., one set is in one-to-one correspondence with some subset of the other. ( Historical remark: It was questions like this that led to Zermelo 's formulation of AC.) Any vector space over a field F has a basis i.e., a maximal linearly independent subset over that field. ( Remark: If we only consider the case where F is the real line, we obtain a slightly weaker statement; it is not yet known whether this statement is also equivalent to AC.) Any product of compact topological spaces is compact. (This is now known as Tychonoff's Theorem , though Tychonoff himself only had in mind a much more specialized result that is not equivalent to the Axiom of Choice.)

7. Paul Howard's Web Page Eastern Michigan University axiom of choice. http://www.emunix.emich.edu/~phoward/

Paul Howard Department of Mathematics Eastern Michigan University Ypsilanti, MI 48197 Information Phone: (734) 487-1294 Office: 516 E Pray-Harrold email: phoward@emunix.emich.edu Send email: Mail Other Sites Information on the "Consequences of the Axiom of Choice" Project can be found at CONSEQUENCES OF AC EMU Math Department Web Page EMU MATH Current Classes and Schedule Current schedule

8. Axiom Of Choice -- From MathWorld axiom of choice. An important and fundamental axiom in set theory sometimes called Zermelo s axiom of choice. It was formulated by http://mathworld.wolfram.com/AxiomofChoice.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Axioms Foundations of Mathematics ... General Set Theory Axiom of Choice An important and fundamental axiom in set theory sometimes called Zermelo's axiom of choice . It was formulated by Zermelo in 1904 and states that, given any set of mutually exclusive nonempty sets , there exists at least one set that contains exactly one element in common with each of the nonempty sets . The axiom of choice is related to the first of Hilbert's problems In Zermelo-Fraenkel set theory (in the form omitting the axiom of choice), the Zorn's lemma trichotomy law , and the well ordering principle are equivalent to the axiom of choice (Mendelson 1997, p. 275). In contexts sensitive to the axiom of choice, the notation "ZF" is often used to denote Zermelo-Fraenkel without the axiom of choice, while "ZFC" is used if the axiom of choice is included.

9. Continuum Hypothesis -- From MathWorld hypothesis depends on the version of set theory being used, and is therefore undecidable (assuming the ZermeloFraenkel axioms together with the axiom of choice http://mathworld.wolfram.com/ContinuumHypothesis.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Foundations of Mathematics Set Theory Cardinal Numbers ... Szudzik Continuum Hypothesis Portions of this entry contributed by Matthew Szudzik The proposal originally made by Georg Cantor that there is no infinite set with a cardinal number between that of the "small" infinite set of integers and the "large" infinite set of real numbers c (the " continuum "). Symbolically, the continuum hypothesis is that showed that no contradiction would arise if the continuum hypothesis were added to conventional Zermelo-Fraenkel set theory . However, using a technique called forcing , Paul Cohen (1963, 1964) proved that no contradiction would arise if the negation of the continuum hypothesis was added to set theory set theory being used, and is therefore

10. Axiom Of Choice - Wikipedia, The Free Encyclopedia axiom of choice. From Wikipedia, the free encyclopedia. The axiom of choice is an axiom in set theory. It was formulated about http://en.wikipedia.org/wiki/Axiom_of_choice

Axiom of choice From Wikipedia, the free encyclopedia. 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 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. 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.

11. On The Computational Content Of The Axiom Of Choice - Berardi, Bezem, Coquand (R Article by S. Berardi, M. Bezem and T. Coquand presenting a possible computational content of the negative translation of classical analysis with the axiom of choice. http://citeseer.nj.nec.com/berardi95computational.html

On the computational content of the Axiom of Choice (1995) (Make Corrections) (6 citations) Stefano Berardi, Marc Bezem, Thierry Coquand The Journal of Symbolic Logic Home/Search Context Related View or download: phil.uu.nl/pub/logic preprint116.ps.Z Cached: PS.gz PS PDF DjVu ... Help From: phil.uu.nl/preprints (more) (Enter author homepages) Rate this article: (best) Comment on this article (Enter summary) Abstract: We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Gödel's Dialectica interpretation [10, 18]. Interestingly, this interpretation uses a refinement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis, and how to interpret the ... (Update) Context of citations to this paper: More It can be shown that the standard constructive explanations of classical logic fail in explaining such use of the Axiom of Choice 3 Constructive Interpretations In this section, we follow the approach taken in [Coq96] to interpret corollary 5. Rather than...

12. Talk:Axiom Of Choice - Wikipedia, The Free Encyclopedia Talkaxiom of choice. From Wikipedia, the free encyclopedia. Do Wikipedia articles assume the axiom of choice unless otherwise mentioned? http://en.wikipedia.org/wiki/Talk:Axiom_of_choice

Talk:Axiom of choice From Wikipedia, the free encyclopedia. Do Wikipedia articles assume the axiom of choice unless otherwise mentioned? Or should results which rely on choice be marked as such? Matthew Woodcraft They should be marked as such. If they're not, fix them. Taw Pretty much anything I write assumes AC. Zundark , 2001 Dec 16 I think it is polite, when writing about a central theorem like existence of prime ideals or Hahn-Banach, to mention that it depends on AC, but it is really too much to ask to do the detailed bookkeeping and mention AC for every result which depends on one of those theorems. AC is an accepted axiom in mathematics. AxelBoldt This last position seems to be modern practice, at least in the areas of mathematics I'm familiar with. Should we have a note to this effect on the main AC page? Matthew Woodcraft Yes, we should. If you're feeling in the mood, this article really needs an overhaul. I moved some paragraphs here from the set theory article months ago, but no-one has yet attempted to merge it all into a coherent whole. Zundark , 2001 Dec 16 Is there any treatment of "life without AC"? I would find this an interesting topic.

13. Axiom Of Choice axiom of choice. Beyond Denial displays a unique dynamic artistic vision the Ensemble. Ramin. Mamak. Pejman. What axiom of choice means! http://www.xdot25.com/artists/axiom.html

Home About Artists Albums ... Contact Axiom of Choice - Lloyd Barde, Backroads - Heartbeats "Beyond Denial" displays a unique and dynamic artistic vision. " ...A strong new sound...rich vocal textures, excellent musicianship...Axiom of Choice may well move into the front ranks of experimental world music" - CMJ. 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. Pejman Hadadi an original member of Axiom of Choice is the finest Iranian percussionist living in the United States. He's a much sought after Persian tombak and daf player to accompany the rare masters of Traditional Persian Music. Daf, the traditional frame drum of Kurdish music, is played in a very unique way. Pejman Hadadi has toured North America with Hossein Alizadeh, Kayhan Kalhor, and Shahram Nazeri, and is the premier percussionist member of the Dastan Ensemble.

14. Rubin, Jean E. Purdue University Set theory, axiom of choice. http://www.math.purdue.edu/~jer/

Jean E. Rubin Professor of Mathematics Purdue University West Lafayette, IN 47907-1395 Information It is with deep regret that we must announce the passing of Professor Jean E. Rubin on Friday, October 25, 2002. Please click below for the web page of the book ``Consequences of the Axiom of Choice'' by Paul Howard and Jean E. Rubin. "CONSEQUENCES of AC" AC PROJECT Please be advised that the links following "Other Web points of interest" are not being maintained. Please refer to the official Math Department course web pages for the most current information using the following links: Math 165, Fall 2002 Other Web points of interest: MA 165 Webpage Fall 2002 MA 387 Webpage Fall 2002 MA 166 Webpage Spring 2002 Course Rosters ... ssociation of Symbolic Logic The ASL Business office e-mail: asl@math.uiuc.edu. West Lafayette Public Library Smitty's LSO Home Page West Lafayette City Hall ... Logic Program E-MAIL ADDRESSES for MathWorks (Makes MATLAB) support@mathworks.com - Technical support

15. Beyond Denial By Axiom Of Choice axiom of choice Album Cover, Beyond Denial displays a unique dynamic artistic vision the freeflowing style fits with the http://www.xdot25.com/albums/beyond.htm

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.

16. Calculus: The Axiom Of Choice Before I tell you what the axiom of choice is and how it relates to our modern understanding of the world I will first discuss the idea of the axiom in http://www.wfu.edu/users/escodw0/axiomofchoice.html

"Why should I study calculus?" "I'm never going to use it." SiteMap Well I am glad you decided to come to this page despite its rather intimidating title. I hope that you will find this intriguing and perhaps interesting enough to show off your new found knowledge to your friends. Before I tell you what the Axiom of Choice is and how it relates to our modern understanding of the world I will first discuss the idea of the "axiom" in mathematics. In fact Euclid essentially realized this when he compiled all the knowledge about geometry into his book The Elements . Euclid had five postulates or statements that he could not prove. The first four were obviously unprovable statements like the ability to connect any two points with a line, to draw a circle of any given radius at any point, or the equivalance of any two right angles. Surely these statements just are and there is nothing else that one could ever do to "prove" them to anyones satisfaction short of simply demonstrating to the best of their abilities that a line can be drawn between two point or continued on indefinitely. ON the other hand his fifth postulate sounds like a theorem. It like any other theorem is an if conditional which is clearly based on something more fundamental than its own merit. I will rewrite it in more modern terminology: If two lines are cut by a third and the same side interior angles sum to 180 degrees, these lines are considered parallel and can never converge towards each other nor diverge away from each other for however far they are extended.

17. The Axiom Of Choice The axiom of choice. There are several equivalent formulations The Cartesian Relevance of the axiom of choice. THE axiom of choice. There are 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?

18. Axiom Of Choice From FOLDOC Free Online Dictionary of Computing. axiom of choice. mathematics (AC, or Choice ) An axiom of set theory If X is a set of sets http://wombat.doc.ic.ac.uk/foldoc/foldoc.cgi?Axiom of Choice

19. Topological Equivalents Of The Axiom Of Choice And Of Weak Forms Of Choice, By E Topological Equivalents of the axiom of choice and of Weak Forms of Choice. by. The Axiom of Dependent Choice (DC) has a few interesting equivalents. 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:

20. Axiom Of Choice axiom of choice. The axiom of choice is an axiom in set theory. It Another formulation of the axiom of choice (AC) states Given http://www.fact-index.com/a/ax/axiom_of_choice.html

Main Page See live article Alphabetical index 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 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. 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.

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