TITLE: Generic filters in the ground model (forcing)
QUESTION [7 upvotes]: In his book Set Theory (second edition), page 203, Thomas Jech writes

Lemma 14.4 Let $\mathfrak{M}$ be a countable model of ZFC and $P$ be a partially ordered set. If $p\in P$, there exists a generic filter on $P$.

Then, the following exercise is given:

Exercise Let $\mathfrak{M}$ as above. Let $P\in M$ be a partially ordered set with no atom (for every $p\in P$, then there exist $q,r\in P$ such that $q,r\le p$ and
  $q,r$ are incompatible). If $G\subset P$ is generic over $\mathfrak{M}$,
  then $G\not\in \mathfrak{M}$.

The short proof of Lemma 14.4 is given here:

According to the exercise, the filter $G$ constructed in this proof must fail to belong to $\mathfrak{M}$ in some cases. But why ? Since $P\in \mathfrak{M}$ it seems that $G\in \mathfrak{M}$ using comprehension, doesn't it ?

REPLY [2 votes]: Comprehension says (eliding some details) that $\{x\in A | \varphi(x)\}$ exists in $\mathfrak{M}$ for any formula $\phi$ (with parameters); presumably you intend $A$ here to be $P$.  But there is not generally going to be any such formula $\varphi$ to define a generic filter $G$, so comprehension won't tell you that such a $G \in \mathfrak{M}$.