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