Axiom of regularity

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In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads: ∀ x ( x ≠ ∅ → ( ∃ y ∈ x ) ( x ∩ y = ∅ ) ) . {\displaystyle \forall x\,(x\neq \varnothing \rightarrow (\exists y\in x)(x\cap y=\varnothing )).} The axiom of regularity together with the axiom of pairing implies that no set is an element of itself, and that there is no infinite sequence ( a n ) {\displaystyle (a_{n})} such that a i + 1...

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