Normal measures on $P_{\kappa }(\lambda )$ extend the club filter

Question

This is (a variation on) exercise 20.4 in Jech's "Set Theory." Let $j : V \to M$ witness $\lambda$-supercompactness of $\kappa$, and consider the normal measure $U$ on $P_{\kappa}(\lambda)$ consisting of those $x$ such that $j[\lambda] \in j(x)$. (How do you make the left quotation mark symbol to denote 'j-image-of-lambda'?)

We want to show that this measure extends the club filter.

This hint is as follows: Suppose $C$ is club. Then define $D = j[C]$. Then:

1. $D$ is a directed subset of $j(C)$.
2. $D$ has size $|C| \leq \lambda ^{< \kappa} < j(\kappa )$.
3. Therefore $\bigcup D \in j(C)$.
4. $\bigcup D = j[\lambda ]$

I'm fine with 1. I'm not sure about 2 - where is the argument taking place, in $V$ or in $M$, or both? For 3, it appears the underlying argument is this:

$V \vDash \forall E \subset C\ (E$ directed and $|E| < \kappa \Rightarrow \bigcup E \in C)$

and so

$M \vDash \forall E \subset j(C)\ (E$ directed and $|E| < j(\kappa) \Rightarrow \bigcup \in j(C))$

I can accept this assuming that 2 means "$D \in M$ and $M \vDash |D| < j(\kappa )$." I'm having trouble with 4 as well - I believe that $j[\lambda] \subseteq \bigcup D$, but why does the reverse inclusion hold, i.e. why is it that $x \in C, \beta \in j(x) \Rightarrow \beta \in j[\lambda]$?

Answer 1

Suppose $\kappa$ is $\lambda$-supercompact for some $\lambda \geq \kappa$, and let $j: V \rightarrow M$ be an elementary embedding with critical point $\kappa$ such that $j(\kappa) > \lambda$ and $M^{\lambda} \subseteq M$ for some inner model $M$. First, observe that $V$ and $M$ agree on $P_{\kappa}\lambda$ because $M$ is closed under ${<}\kappa$ sequences. In particular, this means that $\lambda^{{<}\kappa} \leq (\lambda^{{<}\kappa})^M$ since $M \subseteq V$. But this then means that $j(\kappa) > (\lambda^{{<}\kappa})^M \geq \lambda^{{<}\kappa}$ because $j(\kappa)$ is inaccessible in $M$ and $j(\kappa)$ is greater than both $\lambda$ and $\kappa$. Next, note that any $x \in P_{\kappa}\lambda$ will be a subset of $\lambda$ having size less than the critical point $\kappa$ so that $j(x) = j''x \subseteq j''\lambda$.

[Specifically, if for some $\alpha < \kappa$, we have a bijection $f: \alpha \rightarrow x$, then $j(f)$ will be a bijection between $j(\alpha) = \alpha$ and $j(x)$. So every element of $j(x)$ is of the form $j(f)(\beta)$ for some $\beta < \alpha$, but $j(f)(\beta) = j(f(\beta)) \in j''x$ since $\beta$ is also below the critical point.]

Also, $M$ will contain $h = j \upharpoonright \lambda$ by its closure under $\lambda$ sequences. Therefore, $M$ will have $j''P_{\kappa}\lambda = \{j(x)| x \in P_{\kappa}\lambda\} = \{j''x| x \in P_{\kappa}\lambda\} = \{h''x| x \in P_{\kappa}\lambda\}$. Now letting $g: P_{\kappa}\lambda \rightarrow \lambda^{{<}\kappa}$ be a bijection in $V$, we will have a bijection $j(g) \upharpoonright j''P_{\kappa}\lambda: j''P_{\kappa}\lambda \rightarrow j''\lambda^{{<}\kappa}$ in $M$. Therefore, $M$ will have the range of $j(g)$, which is exactly $j''\lambda^{{<}\kappa}$. Now, since $C$ has size at most $\lambda^{{<}\kappa}$ (in $V$), we may let $e: \lambda^{{<}\kappa} \rightarrow C$ be a surjection. Then $j(e) \upharpoonright j''\lambda^{{<}\kappa}: j''\lambda^{{<}\kappa} \rightarrow j''C$ is a surjection in $M$ so similarly, its range, $D = j''C$, will be in $M$. But $M$ will also know that $j''\lambda^{{<}\kappa}$ has size $\lambda^{{<}\kappa} < j(\kappa)$ because $M$ can construct $j \upharpoonright \lambda^{{<}\kappa}$ from $j''\lambda^{{<}\kappa}$ by virtue of $j$ being order-preserving. Therefore $\bigcup D \in j(C)$ by the ${<}j(\kappa)$-directed closure of $j(C)$ in $M$, as you mention.

Also, if $x \in C$, then $|x| < \kappa$ and $x \subseteq \lambda$ so $j(x) = j''x \subseteq j''\lambda$. Therefore, $\bigcup D = \bigcup j''C \subseteq j''\lambda$.