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Axiom of choice | A Wisdom Archive on Axiom of choice | | Axiom of choice A selection of articles related to Axiom of choice | |
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Axiom of choice
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| ARTICLES RELATED TO Axiom of choice | | | |  Axiom of choice: Encyclopedia II - Axiom of choice - Results requiring ¬ACThere are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. We will abbreviate "Zermelo-Fraenkel set theory plus the negation of the axiom of choice" by ZF¬C. For certain models of ZF¬C, it is possible to prove the negation of some standard facts. Note that any model of ZF¬C is also a model of ZF, so for each of the following statements, there exists a model of ZF in which that statement is true.
There exists a model of ZF¬C in which there is a function f from the real numbers to the real nu ...
See also:Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Results requiring ¬AC |
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| | | | | Axiom of choice: Encyclopedia II - Axiom of choice - Results requiring choice in intuitionistic logic, though not classically Interestingly, in various varieties of constructive logic (in particular, intuitionistic logic) in which the law of excluded middle is not assumed, the assumption of the axiom of choice is sufficient to obtain the law of excluded middle as a theorem. To see this, for any proposition let be the set and let be the set (see Set-builder notation). By the axiom of choice, there will exist a choice function for the set (note that, although the axiom of choice isn't classically required in order to obtain choice functions for finite sets, it ...
See also:Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic, though not classically, Axiom of choice - Quotes Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Results requiring choice in intuitionistic logic, though not classically |
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| | | | | Axiom of choice: Encyclopedia II - Axiom of choice - 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, f(s) is an element of s.
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 eleme ...
See also:Axiom of choice, Axiom of choice - Statement, Axiom of choice - Usage, Axiom of choice - Independence of AC, Axiom of choice - Weaker forms of AC, Axiom of choice - Results requiring AC or weaker forms, Axiom of choice - Results requiring ¬AC, Axiom of choice - Results requiring choice in intuitionistic logic though not classically, Axiom of choice - Quotes Read more here: » Axiom of choice: Encyclopedia II - Axiom of choice - Statement |
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