Here is a naive idea for a forcing $\mathbb A(\kappa)$, for an inaccessible cardinal $\kappa$. Conditions are pairs $(P,p)$, where $P \in V_\kappa$ is a partial order and $p \in P$. We define the ordering $(Q,q) \leq (P,p)$ to hold when $P$ is a regular suborder of $Q$, and $q \leq_Q p$.

It is easy to see that whenever $G \subseteq \mathbb A(\kappa)$ is generic and $(P,p) \in G$, then $G$ induces a generic filter $G_P \subseteq P$. Also, every cardinal below $\kappa$ is collapsed to $\omega$.

Question: Is $\kappa$ preserved? Does $\mathbb A(\kappa)$ add any bounded subsets of $\kappa$ that aren't added by some $G_P$ as above?

Here is a naive idea for a forcing $\mathbb A(\kappa)$, for an inaccessible cardinal $\kappa$. Conditions are pairs $(P,p)$, where $P \in V_\kappa$ is a partial order and $p \in P$. We define the ordering $(Q,q) \leq (P,p)$ to hold when $P$ is a regular suborder of $Q$, and $q \leq_Q p$.

The Amoeba's body grows larger and its nucleus gets wiser.

It is easy to see that whenever $G \subseteq \mathbb A(\kappa)$ is generic and $(P,p) \in G$, then $G$ induces a generic filter $G_P \subseteq P$. Also, every cardinal below $\kappa$ is collapsed to $\omega$.

Question: Is $\kappa$ preserved? Does $\mathbb A(\kappa)$ add any bounded subsets of $\kappa$ that aren't added by some $G_P$ as above?