1-Let $P=Add(\omega\_1, \kappa)$, and let $D$ be a dense open subset of $P$. Then there is a dense subset $S$ of $D$ such that for every $f \in S$ and any $g \in P$, if $domf=domg$ and the set $\{ \beta: f(\beta) \neq g(\beta) \}$ is finite then $g \in D$.


Why is this true?


2-Assume $0^{\sharp}$ exists, and let $\lambda \geq \aleph\_2^{L[0^{\sharp}]}.$ In $L$ let $P$ be the Easton support product of the forcing notions $Add(\delta^{++}, 1)$ where $\delta < \lambda$ is a limit cardinal in $L$, and for $p \in Add(\delta^{++}, 1)$ we require $domp$ be a subset of $(\delta, \delta^{++})$. Let $F\_{0}$ from the union of the intervals $(\delta, (\delta^{++})^{L})$ where $\delta$ is a limit cardinal in $L$ into $2$ be the resulting generic function. Let $I$ be the class of silver indiscernibles and for $\delta \in I$ let $\langle \alpha\_{n}^{\delta}: n < \omega \rangle \in L[0^{\sharp}]$ be a cofinal sequence through $(\delta^{++})^{L}.$ Also let $\langle r\_{\alpha}:\alpha < \lambda \rangle$ be a sequence of Cohen reals generic over $L[0^{\sharp}]$. Define $F\_{0}^{\*}$ by the same domain as $F\_{0}$ by:


$F\_{0}^{\*}(\beta)=r\_{\delta}(n)$ , if $\beta=\alpha\_{n}^{\delta}, \delta \in I$


$F\_{0}^{\*}(\beta)=F\_{0}(\beta)$ otherwise


Show that $F\_{0}^{\*}$ is $P-$generic over $L$.



Some questions from the paper "Forcing the failure of CH by adding a real" by Shelah and Woodin
Let me answer question 1. Conditions in $P$ are partial
functions $p$ from $\omega\_1\times\kappa\to 2$, with
countable domain, ordered by inclusion. For any condition
$p$, since $\text{dom}(p)$ is countable, there are only countably
many finite modifications of $p$ on this domain. For each
such finite modification $p^\ast$, there is a stronger
condition $q^\ast\leq p^\ast$ with $q^\ast\in D$, and hence
a corresponding $q\leq p$ such that the same modification
to $q\mapsto q^\ast$ places it into $D$. So we may build a
descending sequence of conditions $p\_0\geq p\_1\geq
p\_2\geq\cdots$ such that at each step, moving from $p\_n$ to
$p\_{n+1}$, we handle one finite modification of $p\_n$ to
$p\_n^\ast$, and extend $p\_n$ to $p\_{n+1}$ such that the
corresponding finite modification $p\_{n+1}^\ast$ is in $D$.
Since the domains are countable, we can arrange by suitable
bookkeeping to handle all the finite modifications that
arise (details: for example, we could handle the $k$-th
modification to $p\_r$ at stage $\langle k,r\rangle$, using
a pairing function). By the countable closure of the
forcing, the union $p\_\omega=\bigcup\_n p\_n$ is a condition,
and this condition has the property that for every finite
modification $p\_\omega^\ast$ of it, there is some $p\_n$
supporting this modification, and we already arranged that
$p\_{n+1}^\ast\in D$, and since $D$ is open this means
$p\_\omega^\ast\in D$ as well. Thus, all finite
modifications of $p\_\omega$ land into $D$. So the set $S$
of all such $p\_\omega$ is a dense subset of $P$ with the
desired property.


For question 2, perhaps you can provide more details. I don't have the paper here.


