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_@import "http://livepress.com/ie50.css"; All Themes Business Education Entertainment ... Encyclopedia Database Encyclopedia Database Related: In mathematics , the axiom of choice , or AC , is an axiom of set theory . It was formulated in by Ernst Zermelo Mathematische Annalen Intuitively speaking, AC says that given a collection of bins, each containing at least one object, then exactly one object from each bin can be picked and gathered in another bin - even if there are infinitely many bins, and there is no "rule" for which object to pick from each. Contents Statement Usage Independence of AC Weaker forms of AC and strengthened forms of ¬AC ... References Statement The axiom of choice states: Let X be a set of non-empty sets. Then we can choose a member from each set in X Stated more formally: Let X be a set of non-empty sets. Then there exists a choice function f defined on X . In other words, there exists a function f defined on X , such that for each set s in X Another formulation of the axiom of choice states: Given any set of mutually disjoint non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets. Or alternatively: An arbitrary Cartesian product of non-empty sets is non-empty.

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Your American History Reference Guide! - Axiom of choice American History Search American History Browse Categories Set theory Axiom of choice (Redirected from Axiom of Choice In mathematics , the axiom of choice is an axiom of set theory . It was formulated in 1904 by Ernst Zermelo and has remained controversial to this day. It states the following: Stated more formally: Let X be a set of non-empty sets. There exists a choice 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 disjoint non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets. Until the late 19th century, the axiom of choice was often used implicitly. For example, a proof might have, after establishing that the set S contains only non-empty sets, said "let F(X) be one of the members of X for all X in S ." Here, the existence of the function F depends on the axiom of choice. The axiom might seem at first glance to be obviously true and unobjectionable: if there are several boxes, each containing at least one item, the axiom simply states that one can choose exactly one item from each box. The existence of a choice function is indeed straightforward and uncontroversial when only finite sets are concerned. In fact its existence can be proven from the other axioms of set theory, without the axiom of choice. More generally, the axiom of choice is not necessary for the existence of a choice function when one can come up with a rule to choose items from the sets. However, it is necessary when such a rule cannot be found, and applicable even when such a rule can be proven not to exist. Asserting the existence of a choice function in such cases is controversial. The controversy involves what it means to

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Discover Destinations Go TRAVEL GUIDE research your trip TRAVEL DIRECTORY find a travel company ... go away for less Member Login Login: Password: Join TripAtlas.com Axiom of choice About Axiom of choice In mathematics , the 'axiom of choice', or 'AC', is an axiom of set theory . Intuitively speaking, the axiom of choice says that given any collection of bins, each containing at least one object, exactly one object can be selected from each bin and all placed into one collecting bin—even if there are infinitely many bins and there is no "rule" for which object to pick from each. The axiom of choice is not required if the number of bins is finite or if such a selection "rule" is available. It was formulated in 1904 by Ernst Zermelo While it was originally controversial, it is now used without reservation by most mathematicians. However, there are schools of mathematical thought, primarily within set theory, that either reject the axiom of choice or investigate consequences of axioms inconsistent with AC. Contents Statement Variants Usage Independence ... External links Statement The axiom of choice states: :Let ''X'' be a set of non-empty set s. Then we can choose a single member from each set in ''X''.

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Book Info Similar Books Best Prices Price History New! Axiom of Choice Axiom of Choice (Lecture Notes in Mathematics) Horst Herrlich Publisher: Springer ISBN: 3540309896 DDC: 512 Edition: Paperback; 2006-07-06 Summary: AC, the axiom of choice, because of its non-constructive character, is the most controversial mathematical axiom, shunned by some, used indiscriminately by others. This treatise shows paradigmatically that: - Disasters happen without AC: Many fundamental mathematical results fail (being equivalent in ZF to AC or to some weak form of AC). - Disasters happen with AC: Many undesirable mathematical monsters are being created (e.g., non measurable sets and undeterminate games). - Some beautiful mathematical theorems hold only if AC is replaced by some alternative axiom, contradicting AC (e.g., by AD, the axiom of determinateness). Illuminating examples are drawn from diverse areas of mathematics, particularly from general topology, but also from algebra, order theory, elementary analysis, measure theory, game theory, and graph theory.

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Categories Set theory Axiom of choice In mathematics , the axiom of choice is an axiom of set theory . It was formulated in 1904 by Ernst Zermelo and has remained controversial to this day. It states the following: Stated more formally: Let X be a set of non-empty sets. There exists a choice 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 disjoint non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets. Until the late 19th century, the axiom of choice was often used implicitly. For example, a proof might have, after establishing that the set S contains only non-empty sets, said "let F(X) be one of the members of X for all X in S ." Here, the existence of the function F depends on the axiom of choice. The axiom might seem at first glance to be obviously true and unobjectionable: if there are several boxes, each containing at least one item, the axiom simply states that one can choose exactly one item from each box. The existence of a choice function is indeed straightforward and uncontroversial when only finite sets are concerned. In fact its existence can be proven from the other axioms of set theory, without the axiom of choice. More generally, the axiom of choice is not necessary for the existence of a choice function when one can come up with a rule to choose items from the sets. However, it is necessary when such a rule cannot be found, and applicable even when such a rule can be proven not to exist. Asserting the existence of a choice function in such cases is controversial. The controversy involves what it means to

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Strona startowa Kategorie Aktualnosci Projektory Multimedialne ... Rzutniki Multimedialne Szukaj: Wybierz jezyk: ar id bg ca ... zh Axiom of choice From Wikipedia, the free encyclopedia Jump to: navigation search This article is about the mathematical concept. For the band named after it, see Axiom of Choice (band) In mathematics , the axiom of choice , or AC , is an axiom of set theory . Intuitively speaking, the axiom of choice says that given any collection of bins, each containing at least one object, exactly one object can be selected from each bin, even if there are infinitely many bins and there is no "rule" for which object to pick from each. The axiom of choice is not required if the number of bins is finite or if such a selection "rule" is available. It was formulated in 1904 by Ernst Zermelo While it was originally controversial, it is now used without reservation by most mathematicians. However, there are schools of mathematical thought, primarily within set theory, that either reject the axiom of choice or investigate consequences of axioms inconsistent with AC. Contents Statement Variants Usage Nonconstructive aspects ... edit Statement A choice function is a function f , defined on a collection X of nonempty sets, such that for every set

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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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Arthritis Pain Joint Pain Chronic Pain Knee Pain ... Pain Medication Arthritis Pain Relief It's what many of us seek. What causes it and what are it's cures? You'll find your answers to Arthritis and other common forms of pain here. Categories Set theory Axiom of choice In mathematics , the axiom of choice is an axiom of set theory . It was formulated in 1904 by Ernst Zermelo and has remained controversial to this day. It states the following: Stated more formally: Let X be a set of non-empty sets. There exists a choice 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 disjoint non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets. Until the late 19th century, the axiom of choice was often used implicitly. For example, a proof might have, after establishing that the set S contains only non-empty sets, said "let

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