P-generic filter

Question

Let $M$ be a countable transitive model of $ZF$ and let $P\in M$ be a partial order then how can we see

1. If $P$ is non atomic partial order and $G$ is a P-generic filter over $M$, then $G\notin M$.
2. If $\tau \in M$ is a P-name, and $G$ is P-generic filter over $M$ then $rank(\tau)>rank(\tau_G)$ .
3. What will be the forcing notion $P \in M$ and a P-name $\tau \in M$ such that for every $n<\omega$ there is a P-generic filter $G$ over $M$ such that $\tau_G=n$.

Answer 1

Again, look in Kunen's book.

1) Lemma VII 2.4

2) Lemma VII 2.15

3a) You might try the following: Suppose (in the ground model) you have a countable maximal antichain of $ P $ enumerated by $ \langle p_n : n < \omega \rangle $. For each $ n < \omega $, there is a generic filter $ G_n $ containing just $ p_n $. So define $$ \tau := \bigcup_{n < \omega} \{ (\check{m}, p_n) : m < n \}. $$ Now, think of well-known forcing notions and construct the maximal antichain! (Maybe there is a better solution, but this ad-hoc idea should work fine.)

3b) This is Asaf's idea: Just consider $ P := \omega + 1 $ and $$ {\leq} := \{ (\alpha, \alpha) : \alpha < \omega + 1 \} \cup \{ (n, \omega) : n < \omega \}. $$ Define $$ \tau := \{ (\check{n}, n) : n < \omega \}. $$ For each $ n < \omega $, $ G_n := \{ n, \omega \} $ is generic.