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1. FAQ Launcher: Relevance Of The Axiom Of Choice
, Mathematics, metamathematics, and philosophy 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

2. Infinite Ink
Break free now! Infinite Ink logo, THE axiom of choice. For an overview of the axiom of choice see the Relevance of the axiom of choice (FAQ Launcher).
http://www.ii.com/math/ac/

Extractions: OF C HOICE For an overview of the Axiom of Choice see the Relevance of the Axiom of Choice (FAQ Launcher) Introduction Axiom of Choice Quiz Foundations Quiz Solutions and Weak Forms of AC Common Equivalents of AC Less Well Known Equivalents of AC Theorems Whose Proofs Within ZFC Require AC Conclusion Appendix

3. PlanetMath: Axiom Of Choice
axiom of choice, (Axiom). The ZermeloFrankel axioms for set theory are more or less uncontroversial. Axiom (axiom of choice) Let be a set of nonempty sets.
http://planetmath.org/encyclopedia/AxiomOfChoice.html

Extractions: axiom of choice (Axiom) The Zermelo-Frankel axioms for set theory 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 not necessary to prove the existence of a choice function. It is only necessary for infinite (and usually uncountable ) sets Strange objects that can be constructed using the axiom of choice include non-measurable sets (leading to the Banach-Tarski paradox ) and Hamel bases for any vector space . A Hamel basis may not seem strange, but try to imagine a set of continuous functions such that every continuous function can be expressed uniquely as a linear combination of finitely many elements of . Since in fact the existence of a basis for every vector space is equivalent to the axiom of choice, it is almost guaranteed that no such set

4. Axiom Of Choice Definition Of Axiom Of Choice In Computing. What Is Axiom Of Cho
Computer term of axiom of choice in the Computing Dictionary and Thesaurus. Axiom encyclopedia. Provides search by definition of axiom of choice.
http://computing-dictionary.thefreedictionary.com/Axiom of choice

Extractions: 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.

5. Axiom Of Choice Definition Of Axiom Of Choice In Computing. What Is Axiom Of Cho
Computer term of axiom of choice in the Computing Dictionary and Thesaurus. Axiom encyclopedia. Provides search by definition of axiom of choice.
http://computing-dictionary.thefreedictionary.com/Axiom of Choice

Extractions: 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.

6. Axiom Of Choice
axiom of choice. The axiom of choice is not part of ZF. It is however widely accepted and critical to some proofs. In this model the axiom of choice is true.
http://www.mtnmath.com/whatrh/node57.html

Extractions: 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 To make the definition complete we need to define in the language of set theory what a function is and that is a 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 a first element 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

7. Axiom Of Choice
axiom of choice. Fatal error Call to undefined function encode_cyr() in /afs/unibonn.de/home/manfear/public_php/mathdict-entry.php on line 51
http://www.uni-bonn.de/~manfear/mathdict-entry.php?term=axiom of choice&lang=en&

8. Axiom Of Choice
Previous axiomatic set theory Next Axiom of Comprehension. axiom of choice. mathematics (AC, or Choice ) An axiom of set theory
http://burks.brighton.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

9. Axiom Of Choice Definition Meaning Information Explanation
axiom of choice definition, meaning and explanation and more about axiom of choice. FreeDefinition - Online Glossary and Encyclopedia, axiom of choice.
http://www.free-definition.com/Axiom-of-choice.html

Extractions: Google News about your search term 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 set s. 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

10. PSA Presents Axiom Of Choice - 10/11/02
axiom of choice. Performing at Stanford University Friday, October 11, 2002 8PM Dinkelspiel Auditorium. More about axiom of choice.
http://psa.stanford.edu/axiom/

11. Zermelo's Axiom Of Choice: Its Origins, Developments, And Influence By Gregory H
Zermelo s axiom of choice Its Origins, Developments, and Influence by Gregory H. Moore. Prologue. Chapter 1 The Prehistory of the axiom of choice.
http://www.apronus.com/math/zermelos.htm

12. Axiom Of Choice And Its Equivalents - Apronus.com
axiom of choice AND ITS EQUIVALENTS. axiom of choice (AC) If I,Y are sets, AI Y and /\(x-I) A(x) != O then there exists a function
http://www.apronus.com/provenmath/choice.htm

Extractions: AXIOM OF CHOICE AND ITS EQUIVALENTS Axiom of choice (AC) u(Y) Axiom of choice (AC') Well-ordering principle (WO) If X is a set then there exists E c XxX such that (X,E) is a well ordered set. Definition S.AC.1 - Chain. Let (X,E) be a partially ordered set Definition S.AC.2 - Maximal element. Let (X,E) be a partially ordered set Definition S.AC.3 Let (X,E) be a partially ordered set. L is a maximal chain in (X,E) if and only if for every chain K c X if L c K then K = L. Hausdorff's maximal principle (HMP) Zorn's Lemma (ZL) partially ordered Theorem S.AC.4 If we assume Axioms ZF1, ZF2, ZF3, ZF4, ZF5, ZF6 then AC, AC', WO, HMP and ZL are equivalent. Proof Assume (AC). u(I) We have shown (AC') Assume(AC'). well oredered initial segment Theorem S.O.8 Theorem S.O.7 Thus (WO) is shown. Assume (WO) Recursion Principle Thus (HMP) is shown. Assume (HPM) Thus (LZ) is shown. Assume (LZ) Theorem S.C.14 (AC) is shown. Apronus Home Contact Page ProvenMath Notation document.write("");

13. Axiom Of Choice :: Online Encyclopedia :: Information Genius
axiom of choice. Online Encyclopedia The axiom of choice is an axiom in set theory. It was formulated about a century
http://www.informationgenius.com/encyclopedia/a/ax/axiom_of_choice.html

Extractions: 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

14. AXIOM OF CHOICE - Meaning And Definition Of The Word
Search Dictionary axiom of choice Dictionary Entry and Meaning. Computing Dictionary. Definition (AC, or Choice ) An axiom of set theory
http://www.hyperdictionary.com/dictionary/Axiom of Choice

Extractions: Search Dictionary: Computing Dictionary Definition: (AC, or "Choice") An 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

15. Axiom Of Choice - Artist Detail Information
axiom of choice is comprised of emigré Persian musicians with their roots in the repertoire of traditional Persian music. ABOUT axiom of choice
http://www.eyefortalent.com/index.cfm/fuseaction/artist.detail/artist_id/55

Extractions: Axiom of Choice is comprised of emigr© 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. Their new cd: "Unfolding" is a transglobal exploration of Omar Khayam'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.

16. HalDarling.com - Axiom Of Choice Review Of Hal Darling
axiom of choice. Darling D2R Artist Darling Title D2R Label self produced HD02 Length(s) 49 minutes Year(s) of release 2003
http://www.haldarling.com/reviews/axiom-of-choice.html

17. LMS JCM (6) 198-248
Published 13 Oct 2003. First received 07 Jan 2003. The relative consistency of the axiom of choice mechanized using Isabelle/ZF. Lawrence C. Paulson.
http://www.lms.ac.uk/jcm/6/lms2003-001/

Extractions: Abstract: The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle/ZF, building on a previous mechanization of the reflection theorem. The heavy reliance on metatheory in the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems without formalizing the metatheory. However, the present development follows a standard textbook, Kenneth Kunen's Set theory: an introduction to independence proofs , and could support the formalization of further material from that book. It also serves as an example of what to expect when deep mathematics is formalized.

18. The Axiom Of Choice
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
http://www.andrew.cmu.edu/~cebrown/notes/axiom-of-choice.html

Extractions: 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.

19. From The Axiom Of Choice To Choice Sequences
From the axiom of choice to Choice Sequences gif. Zermelo gave the standard argument that the axiom of choice implies the wellordering principle.
http://www.hf.uio.no/filosofi/njpl/vol1no1/choice/choice.html

Extractions: TeX DVI, compressed: choice-dvi.zip (7578 bytes) The theory of choice sequences is usually considered to be far from the mainstream of mathematics. In this note we show that it did not start that way. There is a continuous development from discussions around the use of axiom of choice to Brouwer's introduction of choice sequences. We have tried to trace this development starting in 1904 and ending in 1914. In his book on choice sequences, Troelstra (1977) gives the development after 1914, but does not indicate where Brouwer got his concept. This note is a first attempt at an answer. Our story starts in August 1904, with Zermelo writing a long letter to Hilbert, who thinks part of the letter deserves a wider audience. So he publishes it directly in Mathematische Annalen Zermelo 1904 The leisurely style is clear from the title, ``Proof that every set can be well-ordered, (from a letter sent to Mr. Hilbert)'', and the first sentence:

20. Axiom Of Choice
axiom of choice axiom of choice are emigré Persian musicians with their roots in the repertoire of traditional Persian music. axiom of choice
http://www.hrmusic.com/artists/aocart.html

Extractions: Poignant, innovative, epic, and soulful - these 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 ensemble's 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 group's original compositions. Axiom of Choice... mixes the haunting, warbling vocals of Mamak Khadem with a panoply of Western and Middle Eastern instruments: guitar, electric cello, ney (flute), clarinet, kamancheh (spike violin) duduk and zurna (oboes), saz and divan (strings), and percussion. Torkian also plays a custom-made, seven-string "quarter-tone" guitar with movable frets, which allows him to bridge Western and Eastern scales. Torkian, who studied violin as a child in Iran, then flamenco guitar after immigrating to Eugene, Ore., in 1978 at age 14, writes Axiom's compositions. They are based on traditional Iranian music, updated for a young audience. Western-world fusion guitarists Ralph Towner (and his group, Oregon), and John McLaughlin are major influences. "I played for a while with a traditional ensemble in L.A.," says Torkian, noting that there is significant Iranian population in Southern California. "But it was like taking a lion to the zoo. I felt I had to do something that belonged to the immigrant community, that was a crossover of all my influences, from India, Iran, Japan and Western jazz."

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