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| − | The '''Axiom of Choice''' ('''AC''') states that "For every nonempty set there is a choice function." It is a powerful but controversial idea frequently used by mathematicians who are attempting to prove otherwise unprovable problems, but which has also been used to prove statements that appear absurd. See, e.g., the [[Banach-Tarski Paradox]] discussed below. | + | 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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| − | See also [[Zermelo-Fraenkel]] [[set theory]].
| + | To better understand the Axiom of Choice, consider alternative descriptions of it: |
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| | The Axiom of Choice holds that:<ref name="willard">Stephen Willard, "General Topology" 1.17, p. 9 (Dover 2004)</ref> | | The Axiom of Choice holds that:<ref name="willard">Stephen Willard, "General Topology" 1.17, p. 9 (Dover 2004)</ref> |
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| | ::Given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. | | ::Given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. |
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| − | Therefore, "We can choose one [[element]] from every element of a [[nonempty set]] of [[disjoint sets]] and this process by which we choose these sets will set up a new set." | + | Therefore, "We can choose one [[element]] from every element of a nonempty set of disjoint sets and this process by which we choose these sets will set up a new set." |
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| | Yet another helpful explanation of the Axiom of Choice is this:<ref name="kuro5hin">http://www.kuro5hin.org/story/2003/5/23/134430/275</ref> | | Yet another helpful explanation of the Axiom of Choice is this:<ref name="kuro5hin">http://www.kuro5hin.org/story/2003/5/23/134430/275</ref> |
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| | ::If you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C. H is called the "choice set" because you are essentially going through each set in C and choosing one element from it. One feature of the Axiom of Choice is that H is simply assumed to exist; there is no algorithm given which might tell you how to construct an example of H. | | ::If you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C. H is called the "choice set" because you are essentially going through each set in C and choosing one element from it. One feature of the Axiom of Choice is that H is simply assumed to exist; there is no algorithm given which might tell you how to construct an example of H. |
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| − | It has been shown that neither the Axiom of Choice nor its negation will contradict the other axioms of set theory. Thus assuming the Axiom of Choice and assuming its negation will each result in differingly consistent set theories. It is the choice of the mathematician whether to believe the axiom or believe its negation; many people choose to not take a position on this issue at all because the majority of mathematics can be proven without invoking the axiom altogether. | + | Here is another description of the Axiom of Choice:<ref>http://www.math.vanderbilt.edu/~schectex/ccc/choice.html</ref> |
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| | + | ::Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S. The function f is then called a choice function. |
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| | + | It has been shown that neither the Axiom of Choice nor its negation will contradict the other axioms of set theory. Thus assuming the Axiom of Choice and assuming its negation will each result in differing, consistent set theories. It is the choice of the mathematician whether to believe the axiom or believe its negation; many people choose to not take a position on this issue at all because the majority of mathematics can be proven without invoking the axiom altogether. |
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| | == Instruction == | | == Instruction == |
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| | *http://www.math.vanderbilt.edu/~schectex/ccc/choice.html | | *http://www.math.vanderbilt.edu/~schectex/ccc/choice.html |
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| | + | == See also == |
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| | + | *[[Zermelo-Fraenkel]] [[set theory]]. |
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| | == References == | | == References == |