= Solution
Assume $\varphi(\vec a)$ 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 $M\models\varphi$, then $M\models\sigma$, and upward absoluteness of $\Sigma_1$ formulas gives $V\models\sigma$, hence $V\models\varphi$. If $V\models\varphi$, then $V\models\pi$, and downward absoluteness of $\Pi_1$ formulas gives $M\models\pi$, hence $M\models\varphi$. Therefore ZF proves that $\varphi$ is absolute for every such $M$.
Back to article page
