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Ikegami and Schlicht proposed a principle, namely generic absoluteness, which is stated below using Hamkins' and Lowe's terminology:

Working in $ZF$:

(Generic Absoluteness): For all formulas in language of set theory

$$\diamondsuit\Phi\longrightarrow\Phi$$

Which means if something is forceable then is already true.

This principle is false as Woodin gave the following counterexample:

$$\psi\equiv~~~\forall A\subseteq \omega_1~~\text{there exists a random real over}~ L[A]$$

In the sense that both $\Psi$ and $\neg \Psi$ are forceable over $ZF$.

I am not sure about the complexity of $\Psi$, but it seems it is $\Pi_3$.

Now my questions are:

1) Is it consistent to have generic absoluteness restricted to formulas with complexity less than $\Pi_3$, or more precisely less than complexity of $\Psi$?

2) Is there any nontrivial reasonable class of forcing notions avoiding a counter example?

For example, as suggested by Schlicht:

Is there a model of $ZF$ such that generic absoluteness holds for adding arbitrary many Cohen reals?

Edit: The first problem is solved by Joel's answer.

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    $\begingroup$ Your sentence $\psi$ is a local property (see jdh.hamkins.org/local-properties-in-set-theory), since we can check whether or not it is true by looking inside $V_{\omega+5}$ or so. So it has complexity $\Delta_2$. $\endgroup$
    – Joel David Hamkins
    Commented Nov 9, 2016 at 15:34
  • $\begingroup$ @JoelDavidHamkins Thank you. so now the question is meaningless. $\endgroup$
    – Rahman. M
    Commented Nov 9, 2016 at 15:39
  • $\begingroup$ Perhaps I should post my comment as an answer? $\endgroup$
    – Joel David Hamkins
    Commented Nov 9, 2016 at 15:42
  • $\begingroup$ @JoelDavidHamkins It's good idea to post your comment as an answer. I modified my question because actually I had it in my mind but I posted only one of them. $\endgroup$
    – Rahman. M
    Commented Nov 9, 2016 at 15:46
  • $\begingroup$ OK, I posted an answer, and explained the positive result for $\Sigma_1$. $\endgroup$
    – Joel David Hamkins
    Commented Nov 9, 2016 at 15:50

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Your sentence $\psi$ is a local property (see my blog post Local properties in set theory), since we can check whether or not it is true by looking inside $V_{\omega+5}$ or so. So it has complexity $\Delta_2$. So one cannot have the principle for assertions at that level of complexity.

Meanwhile, if one goes down to $\Sigma_1$, then the axiom $\Diamond\varphi\to\varphi$ for sentences $\varphi$ is consistent with ZFC, since it follows from the maximality principle, which asserts $\Diamond\square\varphi\to\varphi$, since every forceable $\Sigma_1$ statement is automatically forceably necessary, since once it becomerue it remains true in all further extensions. My article

explains why MP is relatively consistent with ZFC, and with a little strength, you can also allow real parameters.

In the case of $\Pi_1$ assertions $\varphi$, the implication $\Diamond\varphi\to\varphi$ is simply provable in ZF, since if a universal statement holds in some extension, then it is already true.

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  • $\begingroup$ Thanks for your answer. but what about the second question? $\endgroup$
    – Rahman. M
    Commented Nov 9, 2016 at 15:53
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    $\begingroup$ The last sentence implies that whenever $\Diamond\varphi\to\varphi$ holds for all $\Sigma_1$ sentences $\varphi$, it also holds for all Boolean combinations of $\Sigma_1$ sentences (by considering the DNF). Thus, the latter principle is also consistent with ZFC. $\endgroup$
    – Emil Jeřábek
    Commented Nov 9, 2016 at 17:05

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