Sets are well-founded if the axiom of Regularity is assumed.

Von Neumann introduced the axiom of regularity, which states that all sets are well-founded.

This is the axiom of regularity.

Metamath page on the axiom of Regularity.

For example, assuming the axiom of regularity, the following are equivalent for a set x:

Next he proved that if this weaker system is consistent, it remains consistent after adding the axiom of regularity.

The axiom of regularity.

Axiom of regularity: No set is a member of itself, and circular chains of membership are impossible.

Given the other Zermelo-Fraenkel set theory axioms, the axiom of regularity is equivalent to the epsilon-induction.

Axiom of regularity (also called the Axiom of foundation)

So if one forms a non-trivial ultrapower of V, then it will also satisfy the axiom of regularity.

The axioms do not include the Axiom of regularity and Axiom of replacement.

In ZFC, there is no infinite descending -sequence by the axiom of regularity.

With the axiom of choice, this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true.

While this approach can serve to preserve the axiom of extensionality, the axiom of regularity will need an adjustment instead.

Subsequently, the axiom of choice and the axiom of regularity were added to exclude models with some undesirable properties.

The axiom of regularity is stated in the form of an axiom schema of set induction.

From any model which does not satisfy axiom of regularity, a model which satisfies it can be constructed by taking only sets in .

This statement is even equivalent to the axiom of regularity (if we work in ZF with this axiom omitted).

It can be carried out in Zermelo-Fraenkel set theory, without using the axiom of choice, but making essential use of the axiom of regularity.

Axiom of regularity: Every non-empty set x contains some element y such that x and y are disjoint sets.

In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of sets that are elements of themselves.

The axiom of regularity enables defining the ordered pair (a,b) as a,a,b. See ordered pair for specifics.

The axiom of regularity was also shown to be independent from the other axioms of ZF(C), assuming they are consistent.

Within the framework of Zermelo-Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself.