A Silver-like forcing on $\omega_1$ Consider the following forcing notion: $\mathbb{P}= \{p:\subset \omega_1\rightarrow 2 \mid \omega_1\setminus\text{dom}(p) \text{ is uncountable}\}$, with $p\le q$ if $p$ extends $q$.
My questions are:

*

*Does $\mathbb{P}$ preserve cardinals? If not, why?

*Does this forcing appear somewhere in the literature?

*What about $\prod_\omega \mathbb{P}$ with finite support? Does it collapse cardinals?

Ideas?
Thanks!
 A: I don't think there's much literature on this sort of forcing. I asked Halbeisen and Brendle about them a few years ago and they weren't aware of much. I worked out the basic collapsing properties of these forcings though I haven't written it up. The relevant theorem for this question is that for any uncountable $\kappa,$ $\mathbb{P}_{\kappa}:=\{p: \subset \kappa \rightarrow 2 : |\kappa \setminus \text{dom}(p)| = \kappa\}$ collapses $2^{\kappa}$ to $\omega.$ So naively generalized Silver forcings are as destructive as possible.
The proof for $\kappa=\omega_1$ is fairly simple. Fix an injection $f: \omega_1 \rightarrow \mathcal{P}(\omega)$ and let $X_n = \{\alpha<\omega_1: n \in f(\alpha)\}.$ Notice that $\langle X_n \rangle$ is a splitting family for $\omega_1,$ i.e. for any uncountable $X \subset \omega_1,$ there is $n$ such that $X \cap X_n$ and $X \setminus X_n$ are both uncountable.
Let $F$ be as in the lemma here for $\lambda=\omega_1.$ In $V[G],$ let $g = \bigcup G.$ We define a map $h: \omega \rightarrow (^{\omega_1} 2)^V$ by $h(n) = F(g \cdot 1_{X_n}).$ It's easy to check $h$ is surjective.
