Absoluteness between $L_\kappa$ and $L$

Question

Working in $L$, suppose $L \models \kappa$ is a cardinal and $(\mathbb{P}, <) \in L_\kappa$. Let $\varphi(x)$ be a $\Sigma_1^1$ formula. Let $\tau \in L_\kappa$ be a $\mathbb{P}$-name for an element of ${}^\omega\omega$. Is "$1_\mathbb{P} \Vdash \varphi(\tau)$" absolute between $L_\kappa$ and $L$?

It seems that if $L_\kappa$ satisfies enough of set theory to prove the Mostowski Absoluteness and some basic facts about the forcing relation, then this would be true. For instance if $\kappa$ was inaccessible or even regular uncountable. For any arbitrary cardinal $\kappa$, does $L_\kappa$ satisfy enough set theory for the Mostowski absoluteness.

A related question is what are the known fragments of ZF that all $L_\kappa$ (or the fine structures $J_\kappa$) where $\kappa$ is a cardinal satisfy?

Thanks for any information that can be provided.

Answer 1

The answer is yes, and much more. If $\kappa$ is any uncountable cardinal and $\mathbb{P}$ is a notion of forcing in $L_\kappa$, then for any projective statement $\varphi$, the assertion $1_{\mathbb{P}}\Vdash\varphi$ is absolute between $L_\kappa$ and $L$. To see this, note first that $G\subset\mathbb{P}$ is $L_\kappa$-generic if and only if it is $L$-generic, since all subsets of $\mathbb{P}$ in $L$ are already in $L_\kappa$. Next, observe that all the nice $\mathbb{P}$-names for reals that are in $L$ are already in $L_\kappa$, since a nice name for a real is essentially a function from $\omega$ to the antichains of $\mathbb{P}$, and all such functions that are in $L$ are already in $L_\kappa$, since subsets of set appear in $L$ before the next cardinal stage. It follows that if $\tau$ is any $\mathbb{P}$-name for a real with $\tau\in L$, then we may find a nice name $\sigma\in L_\kappa$ such that $1\Vdash \tau=\sigma$. Consequently, $$\mathbb{R}^{L[G]}=\mathbb{R}^{L_\kappa[G]},$$ and so any projective statement forced over $L$ is equally forced over $L_\kappa$ by the same condition.