In mathematics, the axiom of regularity (also known as the axiom of foundation) is one of the axioms of ZermeloFraenkel set theory and was introduced by von Neumann (1925). In firstorder logic the axiom reads:
Or in prose:
Two results which follow from the axiom are that "no set is an element of itself," and that "there is no infinite sequence (a_{n}) such that a_{i+1} is an element of a_{i} for all i."
With the axiom of dependent choice (which is a weakened from of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, the axiom of regularity is equivalent, given the axiom of dependent choice, to the alternative axiom that there are no downward infinite membership chains.
The axiom of regularity is arguably the least useful ingredient of ZermeloFraenkel set theory, since virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity (see chapter 3 of Kunen (1980)). However, it is used extensively in establishing results about wellordering and the ordinals in general. In addition to omitting the axiom of regularity, nonstandard set theories have indeed postulated the existence of sets that are elements of themselves.
Given the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.
Contents
Elementary implications of Regularity
No set is an element of itself
Let A be a set such that A is an element of itself and define B = {A}, which is a set by the axiom of pairing. Applying the axiom of regularity to B, we see that the only element of B, namely, A, must be disjoint from B. But A is both an element of itself and an element of B. Thus B does not satisfy the axiom of regularity and we have a contradiction, proving that A cannot exist.
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