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?