Axiom of Foundation

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).

REFERENCES:

Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997.

Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990.

Itô, K. (Ed.). "Zermelo-Fraenkel Set Theory." §33B in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press, pp. 146-148, 1986.

Mendelson, E. "The Axiom of Fundierung and the Axiom of Choice." Archiv für math. Logik und Grundlagenfors. 4, 67-70, 1958.

Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997.

Mirimanoff, 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 Axiom of Choice: Its Origin, Development, and Influence. New York: Springer-Verlag, 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, P. Axiomatic Set Theory. New York: Dover, 1972.

Zermelo, E. "Über Grenzzahlen und Mengenbereiche." Fund. Math. 16, 29-47, 1930.