Axiom of choice: Difference between revisions
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One formulation of the axiom is that the [[Cartesian product]] of any family of non-empty sets is again non-empty. | One formulation of the axiom is that the [[Cartesian product]] of any family of non-empty sets is again non-empty. | ||
AC is equivalent to [[Zorn's Lemma]] and to the [[Well-ordering Principle]]. | AC is equivalent to [[Zorn's Lemma]] and to the [[Well-ordering Principle]]. | ||
==References== | ==References== | ||
* {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[Van Nostrand Reinhold]] | year=1960 | pages=59-69 }} | * {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[Van Nostrand Reinhold]] | year=1960 | pages=59-69 }} | ||
* {{cite book | author=Michael D. Potter | title=Sets: An Introduction | publisher=[[Oxford University Press]] | year=1990 | isbn=0-19-853399-3 | pages=137-159 }} | * {{cite book | author=Michael D. Potter | title=Sets: An Introduction | publisher=[[Oxford University Press]] | year=1990 | isbn=0-19-853399-3 | pages=137-159 }} | ||
Revision as of 14:02, 12 February 2009
In mathematics, the Axiom of Choice or AC is a fundamental principle in set theory which states that it is possible to choose an element out of each of infinitely many sets simultaneously. The validity of the axiom is not universally accepted among mathematicians and Kurt Gödel showed that it was independent of the other axioms of set theory.
One formulation of the axiom is that the Cartesian product of any family of non-empty sets is again non-empty.
AC is equivalent to Zorn's Lemma and to the Well-ordering Principle.
References
- Paul Halmos (1960). Naive set theory. Van Nostrand Reinhold, 59-69.
- Michael D. Potter (1990). Sets: An Introduction. Oxford University Press, 137-159. ISBN 0-19-853399-3.