Preserving $\omega_1$ is Inaccessible to the reals

Question

$\omega_1$ is inaccessible to the reals if and only if for all $x \in {}^\omega\omega$, $\omega_1^{L[x]} < \omega_1$.

The question is if $\omega_1$ is inaccessible to the reals in $V$ and $\mathbb{P}$ is an $\aleph_1$-preserving forcing, in the generic $\mathbb{P}$-extension $V[G]$, is $\omega_1$ still inaccessible to the reals?

Being inaccessible to the reals is a $\mathbf{\Pi}_4^1$ statement. Under certain assumptions, any forcing will preserve $\omega_1$ is inaccessible to the reals.

Therefore, a more specific question would be if $\kappa$ is an inaccessible cardinal in $L$ and $G$ is generic for $\text{Coll}(\omega, <\kappa)$, is $\omega_1$ being inaccessible to the reals preserved in all $\aleph_1$-preserving forcings extension of $L[G]$?

In $L[G]$, the $\text{Coll}(\omega, \omega_1^{L[G]})$ extension of $L[G]$ is equal to a $\text{Coll}(\omega, \kappa)$ extension of $L$. So $L[G]$ is an example of a model where arbitrary forcings do not preserve $\omega_1$ being inaccessible to the reals.

Thanks for any information.

Answer 1

It seems that this question is addressed in Schindler's book "Set Theory Exploring Independence and Truth" Exercise 8.7.

It seems that Mohammad Golshani deleted answer using almost disjoint forcing was the basic idea although there seems to be some subtle point about starting with a ground model that has very weak large cardinals. I will sketch the idea below:

Start with a ground model such that $\omega_1$ is inaccessible to the reals and $\omega_1$ is not Mahlo in $L$. Such a model can be obtained by assuming $L$ has a single inaccessible cardinal $\kappa$ and let $V$ be the $\text{Coll}(\omega, < \kappa)$ extension.

By Exercise 8.7 in Schindler's textbook, there is a $A \subseteq \omega_1$ which is reshaped, i.e. for all $\xi < \omega_1$, $L[A \cap \xi] \models \xi \text{ is countable}$. Then using almost disjoint forcing, one obtained a c.c.c. forcing $\mathbb{P}$ such that if $G \subseteq \mathbb{P}$ is generic over $V$, then in $V[G]$ there is a real $x$ such that $A \in L[x]$. Then

$$\omega_1^{L[x]} \leq \omega_1^{V[G]} = \omega_1^V = \omega_1^{L[A]} \leq \omega_1^{L[x]}$$

using c.c.c. and that $A$ is reshaped. Hence $\omega_1^{L[x]} = \omega_1^{V[G]}$. In $V[G]$, $\omega_1$ is not inaccessible to the reals.

---

It appears that some assumption on the ground model is necessary. If $\Pi_4^1$-generic absoluteness holds, every set forcing extension will preserve $\omega_1$ is inaccessible to real. Exercise 8.6 in Schindler's book seems to imply that if $\kappa$ is weakly compact in $V$ and $G$ is generic for $\text{Coll}(\omega, <\kappa)$, then all c.c.c. extension of $V[G]$ preserve $\omega_1$ is inaccessible to the reals.

Answer 2

Assume in $V,$ there exists $A \subset \omega_1$ such that $H_{\omega_2}\subseteq L[A].$ By extra coding we can also arrange that $\omega_1^V=\omega_1^{L[A]}.$ Then by a theorem of Jensen, we can find a $\sigma$-distributive and stationary set preserving forcing $\mathbb{P}$, which adds a set $b ⊆ ω_1$ that is reshaped and codes $A$, in the sense that $A ∈ L[b]$ (see Fuchs The stationarity of the collection of the locally regulars, Theorem 2.3. Note that the proof of the fact that the forcing is stationary preserving is due to Schindler). Now as it is explained in the answer by William, we can find a $c.c.c.$ forcing notion, coding $b$ into a real $r$. Call the final extension $W.$ Now it is clear that $\omega_1^W=\omega_1^V=\omega_1^{L[A]}=\omega_1^{L[r]},$ and so in $W$, $\omega_1$ is not inaccessible to the reals.

Answer 3

If κ is the least inaccessible of L and g is Col(ω<κ) generic over L, then a.d. coding over L[g] is ccc and the extension is of the form L[x], x a real. A variant of the question is: which ω₁ preserving forcings preserve e.g. analytic determinacy provably in ZFC? Consistently, again, there is a ccc counterexample (R. David), but Sacks forcing and others provably preserve analyt. determinacy (Castiblanco and Schlicht).