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| − | The '''Axiom of Choice''' ('''AC''') in [[set theory]] states that "if every element of a set ''S'' is a [[nonempty set]], then there is a function ''f'' with domain ''S'' such that ''f(x)'' is an element of ''x'' for every ''x'' in ''S''". | + | The '''Axiom of Choice''' ('''AC''') in [[set theory]] states that "for every [[nonempty set]] there is a function that chooses an element from each set." This axiom is powerful because by assuming the existence of such a function, one can then manipulate the function to prove otherwise unprovable theorems. This axiom is controversial because it assumes the existence of a function without demonstrating that it exists, or knowing all of its characteristics. This axiom has been used to prove an apparently absurd result, as in the [[Banach-Tarski Paradox]] discussed below. |
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| − | This axiom is powerful because by assuming the existence of such a function, one can then manipulate the function to prove otherwise unprovable theorems. This axiom is controversial because it assumes the existence of a function without giving any hint on how it could be constructed. This axiom has been used to prove many apparently absurd results, as in the [[Banach-Tarski Paradox]] discussed below. Notice, however, that the negation of the Axiom of Choice also has been used to prove equally absurd results. | |
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| | To better understand the Axiom of Choice, consider alternative descriptions of it: | | To better understand the Axiom of Choice, consider alternative descriptions of it: |
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| | In many undergraduate mathematics curricula, the Axiom of Choice is introduced early as a tool for solving problems. It often goes under one of its other guises, such as [[Zorn's Lemma]] or the [[Well-Ordering Theorem]]. Students are taught it as a proof method, but often with little mention of the questionable nature of the axiom. Often, a full discussion of the [[paradox]]es that accompany the axiom is left to a higher (graduate) level special topics course in [[logic]], despite the fact that mathematicians in numerous fields may be tempted to use it and thus should be aware of its controversy. Undergraduate courses also neglect to adequately mention when the axiom is being invoked, giving a false sense that use of the axiom is incidental, rather than documenting its widespread nature in recent mathematical proofs. | | In many undergraduate mathematics curricula, the Axiom of Choice is introduced early as a tool for solving problems. It often goes under one of its other guises, such as [[Zorn's Lemma]] or the [[Well-Ordering Theorem]]. Students are taught it as a proof method, but often with little mention of the questionable nature of the axiom. Often, a full discussion of the [[paradox]]es that accompany the axiom is left to a higher (graduate) level special topics course in [[logic]], despite the fact that mathematicians in numerous fields may be tempted to use it and thus should be aware of its controversy. Undergraduate courses also neglect to adequately mention when the axiom is being invoked, giving a false sense that use of the axiom is incidental, rather than documenting its widespread nature in recent mathematical proofs. |
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| − | == Negation of the Axiom of Choice ==
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| − | Many apparent absurd results also result from assuming a Set Theory that explicitly negates the Axiom of Choice.
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| − | For example, if ''S'' is a set for which there is no choice function, then the cartesian product of non-empty sets <math>\times_{x \in S} x\,</math> is the empty set.
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| − | It may be possible to construct sets ''A'' and ''B'' for which there is no way to compare their cardinalities: there is no injective function from ''A'' into ''B'', nor any injective function from ''B'' into ''A''.
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| − | There may exist an infinite set ''X'' that does not have an infinite sequence of distinct elements { x<sub>0</sub>, x<sub>1</sub>, ... } as a subset.
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| | == Sources == | | == Sources == |