# Proof of PFA from Supercompact

In Jech's proof (in chapter 31) that the consistency of a supercompact cardinal implies the consistency of PFA, he needs the following fact:

Let $\mathbb{P}_\kappa$ be the countable support forcing constructed using a Laver function $f$ (as shown in Jech). Let $G$ be generic for $\mathbb{P}_\kappa$. In $V[G]$, suppose $\mathbb{P}$ is a proper forcing and $\mathcal{D} = \{D_\alpha : \alpha < \gamma\}$ with $\gamma < \kappa$ be a collection of dense subsets of $\mathbb{P}$. Let $(\dot{\mathbb{P}}, \dot{\mathcal{D}})$ be their names. There exists $\lambda > 2^{2^{|\mathbb{P}|}}$ sufficiently larges and elementary embedding $j : V \rightarrow M$ with $\text{crit}(j) = \kappa$, $j(\kappa) > \lambda$, $M^{\lambda} \subseteq M$, and $j(f)(\kappa) = (\dot{\mathbb{P}}, \dot{\mathcal{D}})$. Jech claims that $\mathbb{P}$ is proper in $M[G]$.

I do not understand his proof.

Jech states: Since $\mathbb{P}$ is proper in $V[G]$, there exists $2^{|\mathbb{P}|} < \eta < \lambda$ and a club set $C \subseteq [H_\eta]^{\omega}$ which witnesses the master condition definition of properness. Then he claims that since $M^\lambda \subseteq M$ and $\mathbb{P}_\kappa$ has the $\kappa$-chain condition imply that $M[G]^\lambda \subseteq M[G]$. This implies that $C \in M[G]$ and is club.

I do not see how those two facts can be used to show $M[G]^\lambda \subseteq M[G]$ or why this implies $C \in M[G]$ and $C$ is club there.

Thanks for any further details on this proof.

• It is not really Jech's proof, but rather Jech's presentation of Baumgartner's proof. – Andrés E. Caicedo Jul 1 '14 at 22:16
• How do you determine a $\lambda$ sequence of ordinals in $M[G]$? Fix a sequence of names. Each name can be interpreted in several ways, but not too many. – Andrés E. Caicedo Jul 1 '14 at 22:20