Ineffable Cardinals and Critical Point of Elementary Embeddings

Question

A cardinal $\kappa$ is a ineffable if and only if for all sequences $\langle A_\alpha : \alpha < \kappa\rangle$ such that $A_\alpha \subseteq \alpha$ for all $\alpha < \kappa$, then there exists $A \subseteq \kappa$ such that $\{\alpha < \kappa : A \cap \alpha = A_\alpha\}$ is stationary in $\kappa$.

Now suppose $M$ and $N$ are transitive models of $\text{ZFC}$, $\mathcal{P}^M(\kappa) = \mathcal{P}^N(\kappa)$, and $j : M \rightarrow N$ is a nontrivial elementary embedding and $\kappa = \text{crit}(j)$. Lemma 17.32 of Jech claims that $\kappa$ is an ineffable cardinal in $M$.

Jech takes $\langle A_\alpha : \alpha < \kappa\rangle$ be any sequence as above. $j(\langle A_\alpha : \alpha < \kappa\rangle) = \langle A_\alpha : \alpha < j(\kappa)\rangle$ for some $A_\alpha \subseteq \alpha$ when $\kappa \leq \alpha < j(\kappa)$. $A_\kappa \in M$ by the assumption. He claims that $A_\kappa$ is such that $\{\alpha < \kappa : A_\kappa \cap \alpha = A_\alpha\}$ is stationary in $\kappa$. I can not see why this set should be stationary.

Answer 1

Let's let $B = \{ \alpha < \kappa : A_\kappa \cap \alpha = A_\alpha \}$.

Suppose $C \in M$ is a club subset of $\kappa$. We want to show that $C \cap B \neq \varnothing$. It follows that $j(C)$ is a club subset of $j(\kappa)$, and also that $j(B) = \{ \alpha < j(\kappa) : j(A_\kappa) \cap \alpha = A_\alpha \}$. Now note two things:

1. $\kappa \in j(C)$;
2. $j(A_\kappa) \cap \kappa = A_\kappa$.

Thus $\kappa \in j(C) \cap j(B)$, meaning $j(C) \cap j(B) \neq \varnothing$, and so by elementarity $C \cap B \neq \varnothing$.