# Generic Absoluteness restricted to formulas with low complexity or to the class of forcings

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.

• 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 $\Delta_2$. – Joel David Hamkins Nov 9 '16 at 15:34
• @JoelDavidHamkins Thank you. so now the question is meaningless. – Rahman. M Nov 9 '16 at 15:39
• Perhaps I should post my comment as an answer? – Joel David Hamkins Nov 9 '16 at 15:42
• @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. – Rahman. M Nov 9 '16 at 15:46
• OK, I posted an answer, and explained the positive result for $\Sigma_1$. – Joel David Hamkins Nov 9 '16 at 15:50

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
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.
• 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. – Emil Jeřábek Nov 9 '16 at 17:05