a question on forcing

Question

Suppose $p_{n}$, $n\in \omega$ is a sequence in a partial order $P$ and $Q_{n}$ is a dense open subset of $P$ for each $n\in\omega$ such that $p_{n}\in Q_{n}$ and $\bigcap_{n}Q_{n}=\emptyset$. Is it true that no generic $G$ can contain all $p_{n}$?

Answer 1

No, it is not true. In fact, if the $Q_n$ are all of the dense subsets of $P$, the filter generated by $\{p_n:n\in\omega\}$ is $P$-generic.

Added: Let $P$ be the set of finite partial functions from $\omega$ to $\omega_1$, partially ordered by $\supseteq$. For $\alpha<\omega_1$ let $D_\alpha=\left\{p\in P:\exists n\in\operatorname{dom}p\big(p(n)=\alpha\big)\right\}$; it’s easy to see that $D_\alpha$ is dense in $P$. Suppose that $G\subseteq P$ and $G\cap D_\alpha\ne\varnothing$ for each $\alpha<\omega_1$. For $\alpha<\omega_1$ fix $p_\alpha\in G\cap D_\alpha$; there is an $n_\alpha\in\operatorname{dom}p_\alpha$ such that $p_\alpha(n_\alpha)=\alpha$. For each $n\in\omega$ let $A_n=\{\alpha<\omega_1:n_\alpha=n\}$; clearly there must be some $k\in\omega$ such that $A_k$ is uncountable. But then $p_\alpha$ and $p_\beta$ are incompatible (i.e., have no common extension) whenever $\alpha$ and $\beta$ are distinct elements of $A_k$. In particular, $G$ cannot be generic.