Axiom of choice: Difference between revisions
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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. | 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. | ||
Revision as of 00:49, 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.