This is a repost of a MSE question.

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?

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    $\begingroup$ I think your question itself may not be absolute. If $V$ is the shortest model of $\mathsf{ZF} + V=L$ then there are no transitive set-sized models of $\mathsf{ZF}$ and no proper inner models. If $V$ is the second to shortest model of $\mathsf{ZF} + V=L$ then there is precisely one set-sized model. So depending on how you define it in both of these cases all formulas are absolute. $\endgroup$ – James Hanson Aug 8 '19 at 19:33
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    $\begingroup$ On the other hand, if you give yourself enough transitive models, there is a strong negative answer. For example, if ZF proves $\varphi(x)$ is upwards absolute between transitive models of ZF then $\varphi(x)$ is equivalent over ZF + every set belongs to a transitive model of ZF to the $\Sigma_1$-formula "there is a transitive model $M\vDash \text{ZF}$ such that $x\in M$ and $M\vDash \varphi(x)$." In other words, absoluteness between transitive models of ZF is the same as being simply expressible relative to a theory that proves the existence of enough transitive models. $\endgroup$ – Gabe Goldberg Aug 8 '19 at 19:36
  • $\begingroup$ @Gabe thanks, this is a very interesting result I wasn't aware of. Is there something similar for downward absoluteness? $\endgroup$ – Alessandro Codenotti Aug 9 '19 at 7:06
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    $\begingroup$ Yeah: if $\varphi(x)$ is downwards absolute, then $\varphi(x)$ is equivalent to the $\Pi_1$-formula "for all transitive $M$ containing $x$, $M\vDash \varphi(x)$" (again, assuming there are enough transitive models). $\endgroup$ – Gabe Goldberg Aug 9 '19 at 19:33

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