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Topic: Axiom of choice

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Axiom of infinity

Axiomatic set theory

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Totally ordered set

Continuum hypothesis

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	PlanetMath: axiom of choice
		objects that are proved to exist using the axiom of choice cannot generally be described by any kind of systematic rule, for if they could it would not be necessary to their construction.
		Some mathematicians have suggested an axiom that would result in all subsets of the real numbers being measurable; this would of course imply the negation of the axiom of choice.
		This is version 24 of axiom of choice, born on 2001-10-18, modified 2006-12-18.
	planetmath.org /encyclopedia/AxiomOfChoice.html (838 words)

	Axiom of choice - Wikipedia, the free encyclopedia
		A proof requiring the axiom of choice is always nonconstructive: even if the proof produces an object then it is impossible to say exactly what that object is. Consequently, while the axiom of choice asserts that there is a well-ordering of the real numbers, it does not give us an example of one.
		Stronger than AC The axiom of constructibility and the generalized continuum hypothesis both imply the axiom of choice, but are strictly stronger than it.
		The assumption of the axiom of choice is also sufficient to derive the law of the excluded middle in some constructive systems (where the law is not assumed).
	en.wikipedia.org /wiki/Axiom_of_choice (3385 words)

	Axiom of Choice Official Website
		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.
		Loga Ramin Torkian is a guitarist, Middle Eastern lutist, composer and the Artistic Director of Axiom of Choice.
		It was then that he decided to create the quarter-tone guitar to which he adapted many of the tar techniques and formed Axiom of Choice, which served as a common ground for other expatriates to work with him.
	www.axiomofchoice.com /bio.htm (1233 words)

	Axiom of dependent choice - Wikipedia, the free encyclopedia
		In mathematics, the axiom of dependent choice, denoted DC, is a weak form of the axiom of choice which is still sufficient to develop most of real analysis.
		Unlike the full axiom of choice (AC), DC is insufficient to prove (given ZF) that there is a nonmeasurable set of reals, or that there is a set of reals without the property of Baire or without the perfect set property.
		DC is the fragment of AC required to show the existence of a sequence constructed by transfinite recursion of countable length, if it is necessary to make a choice at each step.
	en.wikipedia.org /wiki/Axiom_of_dependent_choice (202 words)

	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 (1871-1953) and, at first, seems obvious and trivial.
		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.
		Thus there are no contradictions in choosing to reject it; among the alternatives are to adopt a contradictory axiom or to use a completely different framework for mathematics, such as category theory.
	www.daviddarling.info /encyclopedia/A/axiom_of_choice.html (563 words)

	Axiom of Choice - TunesWiki (Site not responding. Last check: 2007-10-26)
		The axiom of choice is equivalent to Zorn's Lemma or Zermelo's theorem.
		In a way, it can be argued that classical logic is verily built upon some kind of use of the axiom of choice at the meta-level: simultaneous coherent choice of a truth value for every proposition.
		Utter rejection of the Axiom of Choice and of the things that it entails has led to the discovery of Intuitionnistic Logic and Constructive Logic, and the development of Proof Theory.
	tunes.org /wiki/index.php?title=Axiom_of_Choice&printable=yes (217 words)

	THE IRANIAN: Axiom of Choice, Babak Khiavchi
		Axiom of Choice band members converse with their global audience in the universal dialect of music, but each with their culturally inherited accents from Iran, Armenia, India, Greece and the USA.
		Axiom utilizes lyrics sparingly and treats Mamak Khadem's evocative vocals and wordless but soothing ululations as an equal musical instrument, and not as the main attraction.
		Axiom of Choice's latest album,"Unfolding", clearly shows their growth in music production and their wary use of modern sound engineering techniques in fusing the rich cultural elements the band falls back on.
	www.iranian.com /Khiavchi/2002/October/Axiom (945 words)

	Topological Equivalents of the Axiom of Choice and of Weak Forms of Choice, by Eric Schechter
		The Axiom of Choice is the most well-known nonconstructive assertion of existence; it has important consequences for many branches of mathematics.
		A couple of very weak consequences of the Axiom of Choice are the existence of (i) subsets of R which are not Lebesgue measurable, and (ii) subsets of R which lack the Baire property.
		The existence of those two kinds of pathological objects can be proved using other assumptions instead of the Axiom of Choice; in particular, the existence of those pathological objects follows from the existence of many other pathological objects which are classical in the literature.
	at.yorku.ca /z/a/a/b/18.htm (848 words)

	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 choose a representative from each set) and has had a remarkable influence on the development of mathematics since the beginning of the twentieth century.
		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.
		Thus, by adopting the Axiom of Choice, we commit ourselves to a new kind of nonconstructive mathematics.
	www.math.uchicago.edu /~mileti/museum/choice.html (1415 words)

	Relevance of the Axiom of Choice
		The Axiom of Choice (AC) is one of the most discussed axioms of mathematics, perhaps second only to Euclid's parallel postulate.
		For many sets, including any finite set, the first six axioms of set theory (abbreviated ZF) are enough to guarantee the existence of a choice function but there do exist sets for which AC is required to show the existence of a choice function.
		Rubin, Herman and Rubin, Jean E. "Equivalents of the Axiom of Choice II." North-Holland, 1985.
	db.uwaterloo.ca /~alopez-o/math-faq/node69.html (1384 words)

	Axiom of Choice (Site not responding. Last check: 2007-10-26)
		The last great controversy in mathematics, The Axiom Of Choice and the choice function, could divide up a solid sphere into a finite number of sets that have fractal complexity.
		Assuming the usual (Zermelo-Fraenkel) axioms of set theory, the Axiom of Choice is equivalent to both Zorn's Lemma and the Well Ordering Principle.
		The Axiom Of Choice remains controversial because of its use in proofs that result in paradox, such as in the Banach-Tarski Paradox.
	userwww.sfsu.edu /~art511_h/abmasterf/artprop1f/axiomofchoice.html (142 words)

	Axiom of Choice \| Musings
		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.
		Banach and Tarski used the Axiom of Choice to prove that it is possible to take the 3-dimensional closed unit ball,
		We accept the Axiom of Choice, not because it is true, but because it is useful.
	golem.ph.utexas.edu /~distler/blog/archives/000283.html (620 words)

	Arrow's Theorem, Weglorz' Models and the Axiom of Choice (ResearchIndex)
		Applying Weglorz' models of set theory without the axiom of choice, we investigate Arrow-type social welfare functions for infinite societies with restricted coalition algebras.
		4 The Axiom of Choice (context) - Jech - 1973
		3 Consequences of the Axiom of Choice (context) - Howard, Rubin - 1998
	citeseer.ist.psu.edu /432543.html (396 words)

	Axiom of Choice
		Pejman Hadadi an original member of Axiom of Choice is the finest Iranian percussionist living in the United States.
		Pejman Hadadi has toured North America with Hossein Alizadeh, Kayhan Kalhor, and Shahram Nazeri, and is the premier percussionist member of the Dastan Ensemble.
		Axiom of Choice in Palace of Fine Arts - San Francisco June 1995
	www.xdot25.com /artists/axiom.html (286 words)

	Axiom of Choice
		Axiom of Choice are emigré Persian musicians with their roots in the repertoire of traditional Persian music.
		Poignant, innovative, epic, and soulful - these are but a few of the adjectives used to describe the music of Axiom of choice.
		Axiom of Choice trades in the kind of ambient, seductive, self-consciously exotic potions that waft through ethnic shops — not surprising, since Torkian owns one, on Melrose Street.
	www.hrmusic.com /artists/aocart.html (433 words)

	Zermelo's Axiom of Choice: Its Origins, Developments, and Influence by Gregory H. Moore
		The origins of Zermelo's Axiom of Choice, as well as the controversy that it engendered, certainly lie in that intersection.
		In the historical context of Zermelo's Axiom, I have explored both the vagaries and the fertility of this alternating concern.
		This ProvenMath page states the Axiom of Choice in the formal setting of axiomatic set theory together with a number of its equivalents like the Well-Ordering Principle, Hausdorff's Maximal Principle, and Zorn's Lemma.
	www.apronus.com /math/zermelos.htm (307 words)

	Amazon.com: Niya Yesh: Music: Axiom of Choice (Site not responding. Last check: 2007-10-26)
		Axiom Of Choice - Beyond Denial ~ Axiom of Choice
		Axiom of Choice are based in Los Angeles, but their music springs from the Persian classical music of Iran.
		The axiom of choice is a principle stating that the mathematician has the right to choose elements without explanation, and Torkian and Khadem have expanded this theory's freedom into their eclectic musical choices.
	www.amazon.com /exec/obidos/tg/detail/-/B00004T9SS?v=glance (1274 words)

	Consequences of the Axiom of Choice Project (Site not responding. Last check: 2007-10-26)
		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.
		The Consequences of the Axiom of Choice Project is a continuation of the research that produced the book.
	www.math.purdue.edu /~jer/cgi-bin/conseq.html (351 words)

	Misc: Axiom of Choice (Site not responding. Last check: 2007-10-26)
		The axiom of choice is a frequent topic at the weekly meetings.
		There are a number of different equivalent statements of the axiom, but a rough one is: "Given an infinite number of sets with an infinite number of elements in them, can you form a set that has one representative from each set?" (Note that a set may have uncountably infinitely many members.)
		While this is a vaguely religious topic, everyone generally accepts that this axiom is formally independent of the other axioms of set theory, and that its truth has to be decided on other grounds.
	www.utah.edu /utahlogic/misc/choice.html (193 words)

	Axiom of Choice (Site not responding. Last check: 2007-10-26)
		Function f is a "choice function" for X - for each x in X, it chooses an element of x.
		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.
	burks.bton.ac.uk /burks/foldoc/51/9.htm (462 words)

	Axiom Of Choice - CONNECT, Powered By Sony
		The melodies and rhythms of Persia's Radif tradition are combined with Western sensibilities by Southern California-based band Axiom of Choice.
		Led by Persia-born nylon-string classical guitar, quarter-tone guitar, and tarbass player and musical director Loga Ramin Torkian, the septet has continued to incorporate a global range of influences into its unique sound.
		Formed in the United States, Axiom of Choice remains rooted in the musical heritage of Torkian and fellow Persian émigrés Mamak Khadem and Pejman Hadadi.
	musicstore.connect.com /artist/100/419/3/1004193.html (98 words)

	eBay - axiom of ..., Women's Shoes, Music items on eBay.com (Site not responding. Last check: 2007-10-26)
		AXIOM OF CHOICE Mythos - Portraits In Sound Vol 1 VA CD
		AXIOM OF CHOICE- BEYOND DENIAL CD -NEW buySAFE
		Timikes UK The Consistency of the Axiom of Choice and of the *NEW
	search-desc.ebay.com /search/search.dll?query=axiom+of+...&newu=1&krd=1 (399 words)

	HanWorks Research: Axiom of choice
		This is a joke that although the axiom of choice, the well-ordering principle, and Zorn's lemma are mathematically equivalent, most mathematicians find the axiom of choice to be intuitive, the well-ordering principle to be counterintuitive, and Zorn's lemma to be too complex for any intuition.
		Notice the use of the words intuitive, counterintuitive and too complex for any intuition.
		You can skip to the end and leave a response.
	hanworks.blogspot.com /2006/02/axiom-of-choice.html (128 words)

	Independence, Randomness and the Axiom of Choice (ResearchIndex)
		Independence, Randomness and the Axiom of Choice (1992)
		The results show that these axioms, when added to ZF, imply the failure of AC.
		But the axiom of extensionality plays an essential role in the derivation, and a deeper analysis may ultimately show that randomness is incompatible with extensionality.
	citeseer.ist.psu.edu /567140.html (211 words)

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