Axiom of regularity

Axiom of set theory

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) ( y ∩ x = ∅)) . {\displaystyle \forall x\,(x\neq \varnothing \rightarrow (\exists y\in x)(y\cap x=\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 {\displaystyle a_{i+1}} is an element of a i {\displaystyle a_{i}} for all i {\displaystyle i} . With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true.

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