It is a standard result that $\Sigma_1^{\mathsf{ZF}}$-formulas are upward absolute between $\mathsf{ZF}$ $\in$-models, while $\Pi_1^{\mathsf{ZF}}$-formulas are downward absolute between $\mathsf{ZF}$ $\in$-models, so in particular $\Delta_1^{\mathsf{ZF}}$-formulas are absolute between $\mathsf{ZF}$ $\in$-models.
This is also sharp, there are some stronger results ($\Pi_2^{\mathsf{ZF}}$-formulas are $H(\kappa)$-$V$-downward absolute for an uncountable $\kappa$), but in general it's not hard to construct $\Sigma_2^{\mathsf{ZF}}$, $\Pi_2^{\mathsf{ZF}}$ and $\Delta_2^{\mathsf{ZF}}$-formulas for which the appropriate absoluteness result fails.
Is there a simple way to construct, given $n\in\Bbb N$, a formula $\varphi$ such that $\varphi$ is $\Delta_n^{\mathsf{ZF}}$ (and not equivalent to a simpler formula) and $\varphi$ is absolute between $\mathsf{ZF}$ $\in$-models?
from Hot Weekly Questions - Mathematics Stack Exchange
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