This is a part of exercise E4 from Chapter VII of Kunen's Set Theory. 


The hint (courtesy of A. Miller) goes like this: let ${\bf P} = Fn(I,2)$, $(I \geq \omega\_{1})^M$. Let G be ${\bf P}$-generic over M, N = ${\bf L}(\mathcal P \left({\omega}\right))^{M[G]}$, and assume AC in N such that $(\kappa = |\mathcal P \left({\omega}\right)|)^N$. Then $\mathbb{1} \Vdash (\check{\kappa} = |\mathcal P \left({\omega}\right)|)^{{\bf L}(\mathcal P \left({\omega}\right))}$.


At first glance it seems that this follows from the 'almost homogeneity' of $Fn(I, 2)$. But $\mathcal P \left({\omega}\right)^{M[G]}$ is not in M, and thus does not have a canonical name in M. I cannot determine if $\mathcal P \left({\omega}\right)^{M[G]}$ is independent of G. If not, does the transition from M[G] to ${\bf L}(\mathcal P \left({\omega}\right))^{M[G]}$ somehow manage to preserve the cardinality of $\mathcal P \left({\omega}\right)$? What am I missing here?



Cohen forcing: why is the cardinality of $\mathcal P \left({\omega}\right)$ in ${\bf L}(\mathcal P \left({\omega}\right))$ independent of G?
I'm a little confused by your question, since if we add uncountably many Cohen reals, which is what your forcing does, then the inner model $L(\mathbb{R})$, which is the same as your $L(P(\omega))$, does not actually satisfy the axiom of choice, so your assumption that $N\models\text{AC}$ is not warranted. In particular, $P(\omega)$ does not have a (well-ordered) cardinality there, and there is no such cardinal $\kappa$, if you mean cardinal in the sense of initial ordinal. To see why $L(\mathbb{R})$ doesn't satisfy AC, see Andres Caicedo's beautiful answer to [mbsq's question on elementary embeddings](https://mathoverflow.net/questions/81565/elementary-embeddings/81568#81568). 


But perhaps you are making that assumption towards contradiction, or perhaps you have another definable model in mind, where $P(\omega)$ does have a cardinality. In this case, let me answer that yes, homogeneity is all you need. You don't need to worry about having a name for $P(\omega)$, since this is a definable set in the extension. The point is that you don't need to use parameters explicitly when they are definable. If you define $\kappa$ as some cardinal obtained in a definable way from the new $P(\omega)$ in the extension, then if some condition forces that a particular $\kappa$ satisfies that definition, then by homogeniety all conditions must force that that cardinal satisfies the definition, since the only parameter in the assertion that $\kappa$ satisfied the definition was $\check\kappa$ itself, which is invariant under the induced automorphisms of the forcing. So in this case, such a $\kappa$ would indeed be independent of $G$. 


(I don't have Kunen's text here, so I'm not sure what the exercise was.)


