Powers of infinite cardinals in the generic extension

Question

Let $\kappa, \lambda,\theta$ be infinite cardinals of $M$ where $M$ is a transitive model of ZFC. Let $\mathcal{P}=Fn(\kappa\times \omega,2)$ (finite functions from $\kappa\times \omega$ to 2) and G is a (M,P)-generic set. I want to show $(\lambda^\theta)^{M[G]}=(max(\kappa,\lambda)^\theta)^M$. One direction is easy $(\lambda^\theta)^M \leq (\lambda^\theta)^{M[G]}$ and $(\kappa^\theta)^M\leq (\kappa^\theta)^{M[G]}\leq ((2^\omega)^\theta)^{M[G]}=(2^\theta)^{M[G]}\leq (\lambda^\theta)^{M[G]}$ where $(\kappa \leq 2^\omega)^{M[G]}$ is by the forcing construction. But now I am not quite sure how to get the other direction. Any thoughts?

Answer 1

This is exercise (G1) in K.Kunen "Set Theory".

• If $\lambda \le \theta$ then $\lambda^\theta = 2^\theta$ and using lemma VII 5.13 we get the result.

• If $\lambda \gt \theta$ then using lemma 5.5 we can approximate this power in V ("up to $\omega$"), hence $(\lambda^\theta)^{M[g]} \le ( (\lambda^\omega))^\theta )^M = (\lambda^\theta)^M$