GCH is preserved when forcing with $Fn(\lambda,\kappa)$.

Question

Given a countable transitive model $M$ where $GCH$ holds it is an exercise from Kunen's book to show that GCH also holds in $M[G]$ when $G$ is a $P-$generic filter over $M$, and $P=Fn(\lambda,\kappa)$ ($\aleph_0\leq\kappa<\lambda$ in $M$). Recall that $Fn(\kappa,\lambda)$ is the set of partial functions from finite sets of $\kappa$ to finite sets of $\lambda$, ordered by reverse inclusion.

Does anybody have any idea to prove it? Thank you.

Answer 1

Regarding CH:

Working in $M$ we have $\mid Fn(\kappa,\lambda)\mid=\kappa^\lambda{=}\lambda^+$ and also $(\lambda^+)^{\aleph_0}=\lambda^+$ because $cf(\lambda^+)=\lambda^+>\aleph_0$ and $GCH$ holds in $M$. So using the well-known argument with nice names and $\lambda^+-cc$-ness, we conclude $(2^{\aleph_0}\leq (\lambda^+)^{M})^{M[G]}$. Since $Fn(\kappa,\lambda)$ collapses every cardinal in $M$ below $\lambda^+$, then $(\lambda^+)^M=\aleph_1^{M[G]}$ and finally $(2^{\aleph_0}\leq \aleph_1)^{M[G]}$

Regarding GCH:

Let us define

• $\kappa_0=(\lambda^+)^M$.
• $\kappa_{\alpha+1}=(\kappa_\alpha^+)^M$.
• $\kappa_\lambda=\sup_{\alpha<\lambda} \kappa_\alpha$ if $\lambda$ is a limit ordinal.

Now by induction it can be shown that $\kappa_\alpha=\aleph^{M[G]}_{\alpha + 1}$ and also $(2^{\aleph_\alpha}\leq \kappa_\alpha)^{M[G]}$. For the last one statement it is enough to apply the same argument as used in $CH$ noting that $cf^M(\kappa_\alpha)=cf^M(\aleph^{M[G]}_{\alpha +1})>\aleph_\alpha^{M[G]}$.