Difference between revisions of "Axiom of choice"
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The '''Axiom of choice''' is an [[axiom]] of [[set theory]]. The axiom of choice says that if one is given any collection of boxes, each containing at least one object, it is possible to make a selection of exactly one object from each box, even if the collection is infinite. | The '''Axiom of choice''' is an [[axiom]] of [[set theory]]. The axiom of choice says that if one is given any collection of boxes, each containing at least one object, it is possible to make a selection of exactly one object from each box, even if the collection is infinite. | ||
| − | + | Formally, the Axiom of choice says that, given a non-empty set <math>S</math> of non-empty sets, there is a choice function on <math>S</math>. That is, there is a function <math>f:S\rightarrow\bigcup\limits_{A\in S}A</math> such that <math>f(A)\in A</math> for each <math>A\in S</math>. It is also equivalent to the statement that, given a set <math>S</math> of non-empty sets, <math>\prod\limits_{A\in S}A</math> is non-empty. An equivalent form of the Axiom of choice says, given a set <math>S</math> of non-empty pairwise disjoint sets, there exists a set <math>X</math> with one element from each set in <math>S</math>. | |
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| + | The Axiom of choice is equivalent to [[Zorn's Lemma]] and the [[Well-Ordering theorem]] assuming [[Zermelo-Fraenkel Axioms]]. | ||
It was discovered by German mathematician, Ernst Zermelo in 1904. | It was discovered by German mathematician, Ernst Zermelo in 1904. | ||
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| + | [[Category:Set theory]] | ||
| + | [[Category:Axioms]] | ||
Latest revision as of 00:25, 12 January 2024
The Axiom of choice is an axiom of set theory. The axiom of choice says that if one is given any collection of boxes, each containing at least one object, it is possible to make a selection of exactly one object from each box, even if the collection is infinite.
Formally, the Axiom of choice says that, given a non-empty set 
of non-empty sets, there is a choice function on . That is, there is a function 
such that 
for each 
. It is also equivalent to the statement that, given a set of non-empty sets, 
is non-empty. An equivalent form of the Axiom of choice says, given a set of non-empty pairwise disjoint sets, there exists a set 
with one element from each set in .
The Axiom of choice is equivalent to Zorn's Lemma and the Well-Ordering theorem assuming Zermelo-Fraenkel Axioms.
It was discovered by German mathematician, Ernst Zermelo in 1904.
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