I have encountered the following exercise in kunen's forcing chapter and i have only partially solved it. Any hint or sketch would be very helpful.
Let $\mathbb{P} \in M$ be non-atomic. Let $$M = M_0 \subset M_1 \subset \dots \subset M_n \subset \dots \hspace{0.5cm} (n \in \omega)\\$$ such that $M_{n+1} = M_n[G_n]$ for some $G_n$ which is $\mathbb{P}\text{-generic}$ over $M_n$. Show that $\cup_nM_n$ cannot satisfy the Powerset Axiom. Furthermore, show that the $G_n$ may be chosen so that there is no c.t.m. $N$ for ZFC with $\langle G_n : n \in \omega \rangle \in N$ and $\text{o}(N) = \text{o}(M)$. Hint: $\{n : p \in G_n\}$ can code $\text{o}(M)$.
By non-atomic we mean: $$\forall p \in \mathbb{P} \hspace{0.1cm} \exists q,r( q \le p \wedge r \le p \wedge q \bot r ) \\$$ And $\text{o}(M) = \text{Ord} \cap M$.
Now we can easily see that $\mathbb{P}$ can have no power set in $\cup_nM_n$ because if it had a power set then it would be in some $M_n$ but then it couldn't have $G_n$ inside it because $G_n \not \in M_n$ and clearly $G_n \subset \mathbb{P}$, a contradiction.
Now for the second part i have so far chosen some natural $G_n$ by using the fact $\mathbb{P}$ is non-atomic like this: Choose some random $p\in \mathbb{P}$ and choose a generic $G_0$ such that $p \in G_0$. Now we know there are $q,r$ such that $( q \le p \wedge r \le p \wedge q \bot r )$ and hence both of them can't be in $G_0$, so choose one like $q$ now construct $G_1$ such that $q \in G_1$. Do this for all $n \in \omega$. At this stage i don't know how to code the ordinals of $M$ in sets like $F_p = \{n : p \in G_n\}$. Any kind of hint would be very helpful.$$$$Thanks for your patience.
You can code any subset, $X$, of $\omega$ as follows: take your incompatible $q$ and $r$. Now choose $G_n$ each time such that $q\in G_n$ if $n\in X$ and $r\in G_n$ if $n\notin X$. Then $X$ is definable from $q$ and the sequence $\langle G_n:n\in\omega\rangle$.