Axiom of Foundation

The axiom of set theory asserting that every nonempty set contains an element disjoint from itself. It rules out membership cycles and infinite descending membership chains.

Regularity axiom: An alternative name for the set-theoretic axiom of foundation. This is the standard alternate name for the axiom and its formal principle.
Zermelo–Fraenkel set theory: A foundational axiomatic theory of sets, commonly abbreviated ZF. Foundation is one of the axioms in this standard theory.
Axiom of anti-foundation: An axiom replacing Foundation with principles that permit certain non-well-founded sets. It allows membership structures that the Axiom of Foundation rules out.
Von Neumann ordinal: An ordinal represented as the set of all smaller ordinals. These ordinals use membership as their ordering, with Foundation excluding descending membership loops.
Membership relation: The relation that holds between a set and each of its elements. Foundation constrains this relation by forbidding certain descending patterns.
Axiom of extensionality: The axiom stating that sets with the same elements are equal. Extensionality and Foundation jointly govern the identity and membership structure of sets.
Non-well-founded set theory: Set theory in which sets may belong to themselves or participate in membership cycles. Such membership patterns directly violate Foundation.
Set-theoretic universe: The collection of sets considered within a set theory, often modeled by the cumulative hierarchy. Foundation supports the cumulative, non-circular picture of this universe.
Well-founded relation: A relation in which every nonempty subset has a minimal element. Membership on sets satisfies this structural condition under Foundation.
Axiom of pairing: The axiom guaranteeing a set whose elements are any two given sets. Pairing supplies sets whose membership patterns Foundation also constrains.

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