Forcing questions I have been looking at a proof of a technical forcing lemma, and I have a couple of questions.
Here is the setup: $(N, \epsilon) \prec (\mathbf{H}( \chi ), P, \epsilon ) $ is a countable submodel, $P$ is a notion of forcing.
Lemma: The following conditions are equivalent:
1. $q \in P$ is $N$-generic, and
2. $\forall \dot{\alpha} \in N, ( N \models$ "$\dot{\alpha}$ is a $P$-name for an ordinal" $\rightarrow q \Vdash \dot{\alpha} \in N)$
To prove $2 \rightarrow 1$, let $A \in N$, $N \models $"A is a maximal antichain."  In $N$, we enumerate $A = \{ a_{\xi} : \xi &lt \kappa \}$.  Then, we define $\dot{\alpha} = \{ ( \xi, a_\xi ) : \xi &lt \kappa \}$.
The argument then claims that $\dot{\alpha}$ is a $P$-name for an ordinal. But, isn't it actually a $P$-name for a set containing a single ordinal?  So how could we apply 2? 
 A: Your objection is correct, but the problem is easily fixed.  Define $\dot\alpha$ to be $\{(\check\beta,a_\xi):\beta<\xi\}$.  Then, because the $a_\xi$ form a maximal antichain, each $a_\xi$ will force $\dot\alpha=\check\xi$, and $\dot\alpha$ is forced (by all conditions) to be an ordinal. Furthermore, the name $\dot\alpha$ is in $N$ (because it's defined from the antichain $A$ and it enumeration, which are in $N$).  By statement 2, $q$ forces $\dot\alpha\in\check N$.  Some extension $r$ of $q$ forces a specific value, say $\xi\in N$, for $\dot\alpha$.  So $r$ can't be compatible with $a_\eta$ for any $\eta\neq\xi$ (because $a_\eta$ forces $\dot\alpha$ to equal $\check\eta$).  Therefore $r$ is an extension of $a_\xi$ (by maximality of the antichain and separativity of the forcing notion), which means $q$ is compatible with $a_\xi$.  Finally, since both the enumeration of $A$ and the ordinal $\xi$ are in $N$, $a_\xi$ is also in $N$, and that completes the proof of statement 1.
A few minor comments: In case you don't require separativity as part of the definition of "notion of forcing", either pass to the separative quotient first, or note that even though my argument won't make $r$ an extension of $a_\xi$, it will make them compatible, and this suffices.
One of the comments entertained the notion that $N$ might be transitive. It won't be, except in trivial cases.  Usually $\chi$ is large, and then every definable element of $H(\chi)$, for example $\omega_1$, will be in $N$, whereas a countable $N$ can't contain all the elements of $\omega_1$.  
Since a structure and an elementary substructure must have the same language, the setup at the start of the question should have $(N,P,\in)$ in place of $(N,\in)$.  By elementarity, $P$ will be a notion of forcing in $N$ iff it is one in $H(\chi)$, and that is equivalent to being a notion of forcing in $V$ if $\chi$ is big enough.  
