One of the Zermelo-Fraenkel axioms, also known as the axiom of regularity (Rubin 1967, Suppes 1972). In the formal language
of set theory, it states that
where
means implies, means exists, means AND, denotes intersection,
and
is the empty set (Mendelson 1997, p. 288). More
descriptively, "every nonempty set is disjoint from one of its elements."
The axiom of foundation can also be stated as "A set contains no infinitely descending (membership) sequence," or "A set contains a (membership) minimal element," i.e., there is an element of the set that shares no member with the set (Ciesielski 1997, p. 37; Moore 1982, p. 269; Rubin 1967, p. 81; Suppes 1972, p. 53).
Mendelson (1958) proved that the equivalence of these two statements necessarily relies on the axiom of choice. The dual expression
is called -induction,
and is equivalent to the axiom itself (Itô 1986, p. 147).
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pp. 146-148, 1986.Mendelson, E. "The Axiom of Fundierung and
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67-70, 1958.Mendelson, E. Introduction
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D. "Les antinomies de Russell et de Burali-Forti et le problème fondamental
de la théorie des ensembles." Enseign. math.19, 37-52,
1917.Moore, G. H. Zermelo's
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1982.Neumann, J. von. "Über eine Widerspruchsfreiheitsfrage
in der axiomatischen Mengenlehre." J. reine angew. Math.160,
227-241, 1929.Neumann, J. von. "Eine Axiomatisierung der Mengenlehre."
J. reine angew. Math.154, 219-240, 1925.Rubin, J. E.
Set
Theory for the Mathematician. New York: Holden-Day, 1967.Suppes,
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means implies, 
means exists, 
means AND, 
denotes intersection,
and 
is the empty set (Mendelson 1997, p. 288). More
descriptively, "every nonempty set is disjoint from one of its elements."
-induction,
and is equivalent to the axiom itself (Itô 1986, p. 147).
