Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 128 / 4 / c / i

Assume $\phi (\vec{a})$ is a Delta-one formula in set theory. Thus ZF proves it equivalent to a ${\Sigma }_1$ formula $\sigma$ and to a ${\Pi }_1$ formula $\pi$. Only finitely many axioms of ZF occur in these two formal proofs; collect them, together with the finite fragment needed for bounded-formula absoluteness, into $T$.

Let $M$ be a transitive class containing $\vec{a}$ and satisfying $T$. If $M\models \phi$, then $M\models \sigma$, and upward absoluteness of ${\Sigma }_1$ formulas gives $V\models \sigma$, hence $V\models \phi$. If $V\models \phi$, then $V\models \pi$, and downward absoluteness of ${\Pi }_1$ formulas gives $M\models \pi$, hence $M\models \phi$. Therefore ZF proves that $\phi$ is absolute for every such $M$.