Suppose $\kappa$ is a weakly compact cardinal. Is there a $\kappa$-c.c. forcing $\mathbb{P}$ such that $\mathbb{P} \subseteq V\_\kappa$ and $\Vdash\_{\mathbb{P}} \kappa = \aleph\_1$, where $\mathbb{P}$ is provably NOT equivalent to the Levy collapse $Col(\omega,<\kappa)$?


I ask this because if $\kappa$ is weakly compact and $\mathbb{P}$ has the above properties, then one can prove:


a) There are stationarily many $\alpha < \kappa$ such that $\mathbb{P} \cap V\_\alpha$ is a regular suborder of $\mathbb{P}$.


b) There are unboundedly many $\alpha < \kappa$ such that $\mathbb{P} \cap V\_\alpha$ is equivalent to $Col(\omega,\alpha)$.


c) If $G$ is $\mathbb{P}$-generic over $V$, then there is a further forcing $\mathbb{Q}$ such that if $H \subseteq \mathbb{Q}$ is generic, then in $V[G][H]$, there is $G' \subseteq Col(\omega,<\kappa)$ which is generic over $V$, and $\mathbb{R}^{V[G]} = \mathbb{R}^{V[G']}$.


So in some sense $\mathbb{P}$ resembles $Col(\omega,<\kappa)$.



Resembling the Levy Collapse
The Silver collapse forcing $S(\omega, <\kappa)$ has the required properties. The conditions are functions $f$ such that $dom(f)=n\times X,$ for some $n<\omega$ and $X\in [\kappa]^{\omega}$ and $f(i, \alpha) <\alpha$ for all $i<n, \alpha\in X.$


For more details see Cummings paper "Iterated forcing and elementary embeddings", section 20.


**Added remarks:**


**Theroem.** (1). There are no Levy generic filters over $V$ in any generic extension via the Silver collapse over $V$.


(2). There are no Silver generic filters over $V$ in any generic extension via the Levy collapse over $V$.


So in particular the Levy collapse and the Silver collapse are not forcing equivalent. The theorem follows from the following lemma, that I state without proof.


**Lemma.** Let $G$ be a generic filter for $Col(\omega, <\kappa)$ and $H$ be a generic filter for $S(\omega, <\kappa).$ Then:


**A.** There exists $A \in [\kappa]^{\kappa} \cap V[G]$ of size $\kappa$ (which is of course $\omega\_1$ of $V[G]$) such that $A$ does not contain any countable ground model set.


**B.** For any set $A \in [\kappa]^{\kappa} \cap V[H]$ of size $\kappa$ (which is of course $\omega\_1$ of $V[H]$) there exists a countable set $B\in V$ such that $B \subseteq A$.


**C.** For any $A \in [\kappa]^{\kappa} \cap V[G]$ there exists $X \in [\kappa]^{\kappa} \cap V$ such that for all $Y \in [X]^{\omega} \cap V, Y\cap A \neq \emptyset.$


**D.** There exists $A \in [\kappa]^{\kappa} \cap V[H]$ such that for all $X \in [\kappa]^{\kappa} \cap V$ there exists $Y \in [X]^{\omega} \cap V, Y\cap A = \emptyset.$


