The Axiom of Foundation: $\mathrm{\forall }S(\mathrm{\exists }t(t\in S)\to \mathrm{\exists }s(s\in S\wedge \mathrm{\neg }\mathrm{\exists }u(u\in s\wedge u\in S)))$

The Axiom of Foundation, also called the Axiom of Regularity, is the sentence

$\mathrm{\forall }S(\mathrm{\exists }t(t\in S)\to \mathrm{\exists }s(s\in S\wedge \mathrm{\neg }\mathrm{\exists }u(u\in s\wedge u\in S)\left)\right)$

of the language of set theory (The first-order language of set theory: $\in$, $=$, formulas with parameters, and class abbreviations): every set with at least one member has a member $s$ that shares no member with it.

It is written here without abbreviations, for the same reason as The Axiom of Infinity: there is a set containing a set with no elements and closed under $y\mapsto y\cup \{y\}$: the usual statement uses notation introduced later on this page. In that notation it reads: every nonempty set $S$ has a member $s$ with $s\cap S=\mathrm{\varnothing }$. Such an $s$ is called an $\in$-minimal member of $S$.

• What it rules out. A set $S$ that had no $\in$-minimal member would let one descend forever through membership. The axiom forbids that, and Under Foundation, $x\mathrm{\notin }x$ for every set $x$, there are no sets with $x\in y\in x$, and there are no sets with $x\in y\in z\in x$ draws the consequences that this page needs: no set is a member of itself, and there is no membership cycle of length two or three.

• It is used nowhere else on this page. Every other construction below is carried out without it; the ledger at The axiom ledger for this page: which of the ZFC axioms each construction and each result actually consumes records exactly which results depend on it.