Geometry.Net - Theorems_And_Conjectures: Axiom Of Choice

Mathematics Encyclopedia -- Platonic Realms
axiom of choice n. An axiom of set theory, particularly ZFC, which asserts that fromany (infinite) family of sets a new set may be created containing exactly
http://www.mathacademy.com/cgi-bin/ref_main.cgi?axiom
Consistency Of The Axiom Of Choice And Of The Generalized Continuum-Hypothesis W
Consistency Of The axiom of choice And Of The Generalized ContinuumHypothesisWith The Axioms Of Se Author Godel, Kurt.
http://www.opengroup.com/mabooks/069/0691079277.shtml
What's Wrong With The Axiom Of Choice?
What's wrong with the axiom of choice? Part of our aversion to using theaxiom of choice stems from our view that it is probably not `true'.
http://hilbert.dartmouth.edu/~doyle/docs/three/three/section3_13.html

Extractions: Next: References Up: Division by three Previous: Division as repeated Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel axioms for set theory are necessarily even consistent . Indeed, we're somewhat doubtful whether large natural numbers (like , or even ) exist in any very real sense, and we're secretly hoping that Nelson will succeed in his program for proving that the usual axioms of arithmetic-and hence also of set theory-are inconsistent. (See Nelson .) All the more reason, then, for us to stick with methods which, because of their concrete, combinatorial nature, are likely to survive the possible collapse of set theory as we know it today. Next: References Up: Division by three Previous: Division as repeated Peter Doyle
Topological Curiosities
Both theorems are proven with what is known as axiom of choice whose usage (aclear intuitive appeal notwithstanding) was questioned by the stream of
http://www.cut-the-knot.com/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.
Ink 19 :: Axiom Of Choice
axiom of choice. Niya Yesh. Narada World. Recorded in narada.com. DaveAftandilian. axiom of choice on Amazon. Unfolding ($13.99/$9.99).
http://www.ink19.com/issues/august2000/wetInk/musicAB/axiomOfChoice.html

Extractions: Advertising Info Search Music A-B Ryan Adams Joseph Arthur Richard Ashcroft ... Streaks x-Reference Axiom Of Choice Recorded in Iran, India, and the U.S., Niya Yesh ("gnosis" in Greek) is dedicated to "those contemplative moments in life that make everything worthwhile." The album more than lives up to its dedication; it is deeply peaceful, restful, and meditative, though it has its moments of cascading danceable rhythms as well. Building on the core of Loga Ramin Torkian's classical Persian and flamenco guitar and unique quarter-tone guitar and Mamak Khadem's richly expressive Persian vocals, Axiom Of Choice extends its reach on Niya Yesh to encompass the sounds of many cultures, from Middle Eastern ney and kamancheh fiddle to the more familiar cello and accordion. Overall, the feel is mostly classical, in the Western sense, though the intricate drum rhythms and other percussion cast a definite Eastern accent over many tracks. I especially enjoyed "Prelude," with its lovely guitar and cello duet, joined for the last part of the track by hand drums and shakers. The nine-minute-plus "Calling," with its underlying drone, mournful kamancheh, and very gentle female vocals, feels a lot like a deeply moving lament until a skirling drone and swift-tattooed hand drum beats spin the track into a dance-frenzy, ending with Khadem's wild siren song. But probably my favorite track is "Raindrops," which begins almost like a lullaby with quiet solo guitar, joined soon by Khadem's exquisitely pure and vulnerable voice in an elegantly simple arrangement that evokes the calming sound of endless raindrops falling outside the window, chased down the panes by the high and sweet tones of the kamancheh.
Axiom Of Choice Beyond Denial At CD Universe
Buy Beyond Denial by axiom of choice CDs at CD Universe. Great service discountprices. Beyond Denial by axiom of choice Information.
http://www.cduniverse.com/search/xx/music/PID/1243601/a/Beyond Denial.htm

Extractions: (You save $2.19) Category World Rock / Pop Persian Label X Dot 25 Productions Orig Year CDU Part# Catalog# #Media Street Date Jun 18, 1996 Studio/Live Studio Mono/Stereo Stereo Producer Ramin Torkian Engineer Yousef Shahab; Mark Hermann Customers who bought this item also bought: Bhatt, Vishwa Mohan Track Listing Click for Sound Sample Sound Sample Help Track # Song Title Valeh Desert Storm Kaef Pulse Rhythm Riddle Whisper Panj Kashmir Moon Light Notes Personnel: Mamak Khadem (vocals), Ramin Torkian (quartertone guitar, vocals, acoustic guitar, tarbass), Jussi P. Wenger (acoustic guitar), Shahram Hashemi (tar, tarbass, vocals), Pejman Hadadi (daf, tombak, nagada), Mohammad Mohsenzadeh (tombak, synthesizer), Steven Graham (drums). Customer Reviews This is Axiom of Choice's first project. It is at once accessible, and captures unique vocal expression and exciting rhythmic integrity. In many ways it is quite different from their second more popular effort, "Niya Yesh", but in my opinion shares an overall equal rating. "Beyond Denial" unfortunately does not have Greg Ellis (of Vas) on percussion in the later work, but has some innovative drumming sequences by Steven Graham. There is admittedly quite a bit of experimentation here with an assortment of middle-eastern instruments used with abundant technique and expression. In all a splendid work that shouldn't be missed by any devotee of expressive, dynamic, eastern music that is both challenging and coherent.
Sci.math FAQ: The Axiom Of Choice
sci.math FAQ The axiom of choice. Relevance of the axiom of choice THE AXIOMOF CHOICE There are many equivalent statements of the axiom of choice.
http://www.uni-giessen.de/faq/archiv/sci-math-faq.axiomchoice/msg00000.html
Axiom Of Choice - Acapedia - Free Knowledge, For All
Friends of Acapedia axiom of choice. From Wikipedia, the free encyclopedia.The axiom of choice is an axiom in set theory. It
http://acapedia.org/aca/Axiom_of_choice
Philosophy And The Axiom
Philosophy and the axiom of choice All things are numbers pythagoricians, Thisis an attempt to apply the mathematical infinite to philosophical concepts
http://members.fortunecity.com/jebara/jebara/id2.html

Extractions: document.isTrellix = 1; web hosting domain names email addresses related sites ... Home Philosophy and the axiom Quality principals Links (Schecter web page...) Continuation of math philo Sur la liberalisation de l electricite ... TEMOIN DE MON EPOQUE Philosophy and the axiom Jebara The infinite and Good and Evil...
Niya Yesh
Niya Yesh axiom of choice 49289 Narada World. Visit the official website of axiom of choice www.axiomofchoice.com. axiom of choice,
http://www.narada.com/images/AlbumPage/NiyaYesh/NiyaYesh.htm

Extractions: Axiom of Choice brings together the sounds of many cultures, broadening the appeal to a wide-ranging audience while remaining loyal to their Persian roots and retaining the originality and distinctiveness of their sound. Feature on PRI's "The World". Check out the Global Hit Feature for 4/27/00 www.theworld.org NIYA YESH, their second album, means gnosis in Greek - a direct spiritual communication and knowledge. The music was inspired by the city of Pushkar, India. Loga Ramin Torkian, the music director for Axiom of Choice, traveled to Pushkar to collect live recordings of musicians and environmental sounds, using them to establish the theme for the new album. The remainder was recorded in Los Angeles, combining the talents of percussionist Greg Ellis (Vas), cellists Martin Tillman and Cameron Stone, vocalist David Stringer, leading kamancheh player Saied Farajipouri and the renowned Persian percussionist Pejman Hadadi. "Rich vocal textures, excellent musicianship and an open heart...Axiom of Choice well may move into the front ranks of world music."
Sentence Axiom-Of-Choice In Theory Kif-Sets
Sentence axiomOF-choice. Defined in theory Kif-sets; Source codekif-sets.lisp. Slots on this instance Documentation There is
http://www-ksl.stanford.edu/knowledge-sharing/ontologies/html/kif-sets/AXIOM-OF-
Sentence Axiom-Of-Choice In Theory Kif-Sets
Sentence axiomOF-choice. There is a set that associates every boundedset with a distinguished element of that set. In effect, it
http://saussure.irmkant.rm.cnr.it/onto/ON9.3-OL-HTML/kif-sets/onto-axiom24.html
Galahad - Following Ghosts
Subject sci.math FAQ The Continuum Hypothesis Maintainer alopezo@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Last Posted Fri, 17 Nov 1995 171559 GMT Last-modified December 8, 1994 Subject sci.math FAQ Cutting a sphere
http://www.cs.uu.nl/people/jur/reviews/followingghosts.html

Extractions: Sarah Quilter - clarinet, backing vocals, flute Myopia Imago A Short Reflection On Two Past Lives - Part One Karma For One Perfection Personified Bug Eye A Short Reflection On Two Past Lives - Part Two Ocean Blue Rejuvenation Easier Said Than Done Shine Try a sample of the album in or RealAudio Following Sleepers after three years, this is the third studio album of the band (in addition to five other releases). Sarah Quilter now seems to be part of the band. The artwork is very nice and professionally done. Galahad was once heralded as the new Marillion. With this album they have done two things: shaken themselves a bit loose from the progrock world at its smallest AND they seem to go the way of Marillion into finding a niche of their own in the music world. They will probably still not enjoy the sales Marillion has, but artistically this is a step forward and like they say themselves a broadening of horizons. In my opinion it shows a maturing band, both musically as well as in the lyrics. The music is certainly on the accessible, almost relaxed, side of progressive, but because of this not less appealing. The melodies range from fine to great and the only track I'm not really fond of is Easier Said Than Done. One negative point, but this could also be due to the fact that on my CDrom player right here, I cannot really turn the volume up, is that the sound could have been a bit fuller.

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