Geometry.Net - Theorems_And_Conjectures: Axiom Of Choice

The Axiom Of Choice
Netherlands site with news, interviews, pictures, music samples, reviews, and links.Category Arts Music Styles Rock Progressive Personal PagesProgressive Rock Music
http://www.cs.uu.nl/people/jur/progrock.html
Axiom Of Choice - Official Site
Official site of the persian music group.
http://www.axiomofchoice.com/

Extractions: 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. 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 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
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

Extractions: 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.)
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
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

Extractions: 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 Godel'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)
THE IRANIAN: Music, Axiom Of Choice
Watch this movie to learn more about axiom of choice (includes live concert footage)
http://www.iranian.com/Music/Axiomofchoice

Extractions: The Iranian The California-based Axiom of Choice developed quite a following after their first album, " Beyond denial ". With their latest " Niyayesh " CD, Axiom of Choice proves it's more than a flash in the pan. Every track is wonderfully balanced and incredibly beautiful. It's a definite sign of the increasing maturity of Iranian musicians abroad. For more information, go to their web site From the Niyayesh " CD Greener than God's dream
Axiom Of Choice
This page gives a brief explanation of the axiom of choice and links to other related websites.Category Science Math Logic and Foundations Set Theorya home page for the axiom of choice an introduction and links collectionby Eric Schechter, Vanderbilt University. axiom of choice.
http://www.math.vanderbilt.edu/~schectex/ccc/choice.html

Extractions: 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.)
Untitled Document
Eastern Michigan University axiom of choice.
http://www.emunix.emich.edu/~phoward/
Sci.math FAQ: The Axiom Of Choice
Subject sci.math FAQ The axiom of choice. Relevance of the axiom of choice THEaxiom of choice There are many equivalent statements of the axiom of choice.
http://www.cs.uu.nl/wais/html/na-dir/sci-math-faq/axiomchoice.html

Extractions: Note from archivist@cs.uu.nl : This page is part of a big collection of Usenet postings, archived here for your convenience. For matters concerning the content of this page , please contact its author(s); use the source , if all else fails. For matters concerning the archive as a whole, please refer to the archive description or contact the archivist. This article was archived around: 17 Feb 2000 22:55:52 GMT All FAQs in Directory: sci-math-faq Source: Usenet Version Archive-name: sci-math-faq/axiomchoice Last-modified: February 20, 1998 Version: 7.5 http://www.jazzie.com/ii/math/index.html http://www.jazzie.com/ii/math/index.html Alex Lopez-Ortiz alopez-o@unb.ca http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick
Axiom Of Choice
axiom of choice If X is a set, and S is the union of all the elements of X, then there exists a function fX S such that for all non-empty x in X, f(x) is an element of x.
http://burks.bton.ac.uk/burks/foldoc/51/9.htm

Extractions: The Free Online Dictionary of Computing ( http://foldoc.doc.ic.ac.uk/ dbh@doc.ic.ac.uk Previous: axiomatic set theory Next: Axiom of Comprehension mathematics axiom of set theory In other words, we can always choose an element from each set in a set of sets, simultaneously. Function f is a "choice function" for X - for each x in X, it chooses an element of x. Most people's reaction to AC is: "But of course that's true! From each set, just take the element that's biggest, stupidest, closest to the North Pole, or whatever". Indeed, for any finite set of sets, we can simply consider each set in turn and pick an arbitrary element in some such way. We can also construct a choice function for most simple infinite sets of sets if they are generated in some regular way. However, there are some infinite sets for which the construction or specification of such a choice function would never end because we would have to consider an infinite number of separate cases. For example, if we express the real number line R as the union of many "copies" of the rational numbers, Q, namely Q, Q+a, Q+b, and infinitely (in fact uncountably) many more, where a, b, etc. are irrational numbers no two of which differ by a rational, and
Axiom Of Choice Biographies
axiom of choice are musicians with their roots in traditional Persianmusic. Guitarist axiom of choice Artist Biographies. Poignant
http://www.axiomofchoice.com/Bios.htm

Extractions: Artist Biographies Poignant, innovative, epic, and soulfulthese are but a few of the adjectives used to describe the music of Axiom of Choice . Formed in 1992 by guitarist, composer and Artistic Director Loga Ramin Torkian and co-producer and percussionist Mammad Mohsenzadeh , the ensembles goal is to define a new sound within the context of Persian music. They were soon joined by vocalist Mamak Khadem whose prodigious vocal talent complemented the original compositions of Torkian. The name of the ensemble is a mathematical term, which clearly defines their intentional freedom of choice within the parameters of their compositions and music. As artists who immigrated to the United States, Axiom of Choice incorporates the sounds of other cultures both East and West - broadening the scope and appeal of their music while remaining loyal to their Persian roots and retaining the originality, distinctiveness and integrity of their sound. Their original compositions are mostly based on Persian melodies and modes and have universal appeal because of their cohesive, seamless and natural fusion with other styles of music. The musicians work within a unique compositional structure in which both improvisation and personal expression can freely take place. One important goal is to make Persian music more accessible to the broader audience. The other is to communicate through a progressive and new global sound, unique to world music.
Axiom Of Choice
axiom of choice. Beyond Denial displays a unique dynamic artisticvision the Ramin, Mamak, Pejman. What axiom of choice means! Axiom
http://www.xdot25.com/artists/axiom.html

Extractions: - 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.
Topological Curiosities
theorems are proven with what is known as axiom of choice whose usage (a clear intuitive appeal notwithstanding) was
http://www.cut-the-knot.org/do_you_know/banach.shtml

Extractions: Recommend this site Tarski-Banach Decompositions Two theorems I am going to state are mind boggling results associated with the names of F. Hausdorff, A. Tarski, S. Banach, J. von Neumann and R. M. Robinson. References are given in both books by Gelbaum and Olmsted. (The second of which actually proves the first theorem below.) Both theorems use the notion of a rigid motion . A rigid motion of a space is a transformation that does not change ( euclidean ) distance between two points. In the theorems below, B r R dist (x,0) There exists a decomposition of B into 5 pairwise disjoint sets A ,...,A of which the last is a single point such that there exist rigid motions R ,...,R with B = R (A R (A ) and B = R (A R (A R (A where all unions are disjoint. This means breaking a ball into five pieces such that it's possible to combine these pieces into two balls equal in size to the original one. No seams are visible after the operation. No cavities are created under the surface. I just wonder what prevents me from taking up this occupation professionally. Could have taken a few bowling balls' manufacturers out of business. The problem is one can't be sure that dropping a ball will break it into five pieces, let alone into the right pieces of which one is a point.
Beyond Denial By Axiom Of Choice
axiom of choice Album Cover, Beyond Denial displays a unique dynamicartistic vision the freeflowing style fits with the
http://www.xdot25.com/albums/beyond.htm

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

Extractions: 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: 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)
The Axiom Of Choice
The axiom of choice. There are several equivalent formulations The Cartesian Relevanceof the axiom of choice. THE axiom of choice. There are
http://db.uwaterloo.ca/~alopez-o/math-faq/mathtext/node35.html

Extractions: Next: Cutting a sphere into Up: Axiom of Choice and Previous: Axiom of Choice and There are several equivalent formulations: 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?
Axiom Of Choice - Wikipedia
axiom of choice. From Wikipedia, the free encyclopedia. The axiomof choice is an axiom in set theory. It was formulated about
http://www.wikipedia.org/wiki/Axiom_of_choice
Axiom Of Choice -- From MathWorld
axiom of choice, An important and fundamental axiom in set theorysometimes called Zermelo's axiom of choice. It was formulated by
http://mathworld.wolfram.com/AxiomofChoice.html

Extractions: 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. In 1940, proved that the axiom of choice is consistent with the axioms of (a conservative extension of Zermelo-Fraenkel set theory ). However, in 1963, Cohen (1963) unexpectedly demonstrated that the axiom of choice is also independent of Zermelo-Fraenkel set theory (Mendelson 1997; Boyer and Merzbacher 1991, pp. 610-611).
Zermelo's Axiom Of Choice -- From MathWorld
Eric's other sites. Z v. Zermelo's axiom of choice, axiom of choice. Author EricW. Weisstein © 1999 CRC Press LLC, © 19992003 Wolfram Research, Inc. logo, logo.
http://mathworld.wolfram.com/ZermelosAxiomofChoice.html
FAQ Launcher: Relevance Of The Axiom Of Choice
, Mathematics, metamathematics, andphilosophy of the axiom of choice. (This is part 26 of the sci.math FAQ.).RELEVANCE OF THE axiom of choice.
http://www.ii.com/internet/faqs/launchers/sci-math-faq/AC/relevance/

Extractions: 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

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