Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2007 / B / 5 / iii / Solution

The class $V$ is transitive because all its levels are transitive. Let $a\in V$ be nonempty. There is a least ordinal $\alpha$ for which $a\cap V_{\alpha }\ne \varnothing$: existence follows by choosing an element of $a$, which lies in $V$, and then minimizing within some witnessing ordinal bound. The least ordinal is neither zero nor a limit, so write $\alpha =\beta +1$. Choose $x\in a\cap V_{\beta +1}$. Then $x\subseteq V_{\beta }$, while minimality gives $a\cap V_{\beta }=\varnothing$. Thus $a\cap x=\varnothing$. Also $x\in V$, proving Foundation inside $⟨V,\in ⟩$.

By the given hypothesis this same class satisfies every axiom of $Z{F}^{\ast }$, and hence it satisfies every axiom of $ZF$. Translate any finite formal $ZF$ proof by restricting each quantifier to the definable class $V$. Within $Z{F}^{\ast }$ the translated axioms have proofs, logical steps remain valid, and the translated contradiction is still a contradiction. Substituting those axiom proofs therefore converts the original derivation effectively into a finite derivation of contradiction from $Z{F}^{\ast }$. Equivalently,This is the syntactic relative-consistency construction; it does not assume that $V$ is a set or that it equals the whole ambient universe.