ZF without foundation (${\mathrm{ZF}}^{\ast }$)

ZF without foundation consists of the axioms of Zermelo–Fraenkel set theory with the Axiom of foundation omitted. Ordinal-indexed transfinite recursion still defines the cumulative hierarchy. Its union is the well-founded part and need not include all ambient sets.

The cumulative hierarchy is a transitive class even without the ambient Axiom of foundation. If $a$ is a nonempty member, choose the least ordinal $\alpha$ with $a\cap V_{\alpha }\ne \varnothing$. It is a successor $\beta +1$. An element $x$ of that intersection is a subset of $V_{\beta }$, while $a$ has no element in $V_{\beta }$, so $a\cap x=\varnothing$. This proves internal Foundation. If the hierarchy also satisfies the other ZF axioms, restricting every quantifier of a finite ZF proof to this definable class translates it into a proof from ZF without foundation. Thus adjoining Foundation preserves consistency relative to the remaining axioms.