Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 16H / f

Takethe formula relativization to a class: replace each $\exists y\psi$ byand each $\forall y\psi$ byleaving atomic formulas unchanged. If $M$ is given by a defining formula, insert that formula wherever $y\in M$ occurs; this is why the map may depend on $M$.

For $z$ and any parameters in $M$, induction on formulas givesTherefore ${\phi }^{\ast }$-closure putsin $M$ for every $x\in M$. This is exactly the relativized closure criterion for separation, so $(M,\in )$ satisfies the $\phi$-instance of separation.