Adding $\kappa$ Cohen reals, sets of hereditary cardinality $<\kappa$ are in some intermediate model.

Question

Suppose $\kappa$ is a regular cardinal and consider the forcing notion $Fn(\kappa,2)$ consisting of finite partials functions from $\kappa$ to $\{0,1\}$. Let $G$ be $Fn(\kappa,2)$-generic over $V$.

Suppose $x\in V[G]$ is of hereditary cardinality $<\kappa.$ I need the following but I haven't been able to argue its truth:

There exists a set $J\subseteq\kappa$ with $|J|<\kappa$ and (identifying $Fn(\kappa,2)\cong Fn(J,2)\times Fn(\kappa\setminus J,2)$) filters $G_1$ and $G_2$ such that $G=G_1\times G_2$ and \begin{align} &G_1\text{ is }Fn(J,2)\text{-generic over }V\text{ and }G_2\text{ is }Fn(\kappa\setminus J,2)\text{-generic over }V[G_1];\\ &x\in V[G_1];\text{ and}\\ &V[G]=V[G_1][G_2]. \end{align}

I think I need to find a name for $x$ in $V$ of both hereditary cardinality and rank $<\kappa$, but I've been tying myself into knots over this.

Answer 1

Here you can see this result (pag 11-12).

For $x\in V$ you need find a nice name and use the ccc property to obtain a upper bound for the size of $J$.

Answer 2

We first verify the following:

If $ G$ is generic for $Fn(\kappa,2)$ over $V$, then $H_{\kappa}[G] = H_{\kappa}^{V[G]}$

It is easy to see that $H_{\kappa}[G] \subseteq H_{\kappa}^{V[G]}$. For $\supseteq$ suppose for a contradiction that there is a $\in$-minimal $x \in H_{\kappa}^{V[G]} \setminus H_{\kappa}[G]$. Then $ trcl(x) \subseteq H_{\kappa}[G]$, so for each $ y \in trcl(x) $ we can define $h(y) \in H_{\kappa} \cap V^{\mathbb{P}}$ such that $ h(y)_{G} = y$. Let $ z = \{ h(y) \ | \ y \in trcl(x) \}$, since $ |trcl(\{x\})| =:\theta < \kappa $, $|z| = |x| \leq \theta < \kappa$ it follows that in $V[G]$ there is a surjection $f:\theta \longrightarrow trcl(\{z\})$. Let $ \rho $ be a name for $ x $, $\tau$ a name for $z$, $\dot{f}$ a name for $f$ and $p \in G $ such that \begin{gather*} p \Vdash \big( \rho \in H_{\kappa}^{V[\dot{G}]} \setminus H_{\kappa}[\dot{G}] \ \& \ \rho \subseteq H_{\xi}[G] \ \& \ \tau \subseteq H_{\xi} \big) \end{gather*} and \begin{gather*} p \Vdash \dot{f}:\theta \longrightarrow trcl(\{\tau\}) \ \text{is a surjection} \end{gather*} For each $\alpha < \theta$ let $B_{\alpha} = \{ q \in Fn(\kappa,2) \ | \ \exists \pi \in H_{\xi}\cap V^{\mathbb{P}} \ \& \ q \leq p \ \& \ q \Vdash \pi \in \rho \ \& \check{\pi} = \dot{f}(\alpha) \} $ and $A_{\alpha} \subseteq B_{\alpha} \cup \{ q \in Fn(\kappa,2) \ | \ q \perp p \} $ a maximal antichain. For each $ \alpha < \theta $ and $ q \in A_{\alpha} \cap B_{\alpha}$ pick $ t(\alpha,q) \in H_{\xi}$ such that $q \Vdash t(\alpha,q) \in \rho \ \& \ \check{t(\alpha,q)} = \dot{f}(\alpha)$. \begin{gather*} \sigma:= \{ (\pi,q) \ | \exists \alpha,q ( \alpha < \theta \ \& \ q \in A_{\alpha} \cap B_{\alpha} \ \& \ \pi = t(\alpha,q)) \} \end{gather*} then \begin{gather*} p \Vdash \sigma = \rho \end{gather*} a contradiction since $ \sigma \in H_{\kappa}$ (This uses $Fn(\kappa,2) \subseteq H_{\kappa}$). This gives $ H_{\kappa}[G] = H_{\kappa}^{V[G]}$

Let $ x \in H_{\kappa}[G]$, then in $V$ there is $ J \subseteq \kappa $ such that $ |J| < \kappa $ and $ x \in V[G_{0}]$ for $ G_{0} = G \cap Fn(J,2)$.

Let $ y \in H_{\kappa} \cap V^{\mathbb{P}}$ such that $ y_{G} = trcl(\{ x\}) $. Then $|trcl(y) \cap Fn(\kappa,2)| < \kappa$, let $ J = \bigcup\{ dom(r) \ | \ r \in ( trcl(y)\cap Fn(\kappa,2)) \}$

We can then verify by $\in$-induction that $ (\tau,q) \in y$ implies $ \tau_{G} = \tau_{G_{0}}$. Suppose $(\tau,q) \in y $ and for every $ (\pi,t) \in \tau $ we have $\sigma_{G} = \sigma_{G_{0}}$. Then we have \begin{gather*} \tau_{G} =\{ \pi_{G} \ | \ (\pi,t) \in \tau \ \& \ t \in G\} \stackrel{(1)}{=} \{ \pi_{G} \ | \ (\pi,t) \in \tau \ \& \ t \in G_{0} \} \stackrel{(2)}{=} \{\pi_{G_{0}} \ | \ (\pi,t) \in \tau \ \& \ t \in G_{0} \} = \tau_{G_{0}}\end{gather*} where (1) follows from the definition of $ J$ and (2) follows by induction. We can then apply 1.29 from the link cited by @Gödel.