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Axiom of regularity

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In set theory, the axiom of regularity, also known as the axiom of foundation, is that for every non-empty set S there is an element a in it which is disjoint from S. Under the axiom of choice, this axiom is equivalent to saying there is no infinite sequence {an} such that ai+1 is a member of ai for all i. It follows as a corollary that no set belongs to itself: if the set S were a member of itself, then {S} would violate the axiom of regularity.

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http://www.trinity.edu/cbrown/topics_in_logic/sets/sets contains an informative description of the axiom of regularity under the section on Zermelo-Fraenkel set theory.

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