In Kanamori's book on large cardinals, he defines a cardinal $\kappa$ to be extendible if for any $\eta$ there is some $j:V_{\kappa+\eta} \prec V_\zeta$ with crit($j$)=$\kappa$ an $j(\kappa)$> $\kappa+\eta$.
On page 323 he is proving 23.15, namely $\kappa$ is extendible iff for any $\eta > \kappa$ there is a $j:V_\eta \prec V_\zeta$ with crit($j$)=$\kappa$. For the proof, assume the latter statement and given $\eta \geq \kappa \cdot\omega$ we wish to prove that $\kappa$ is $\eta$-extendable. He lets $\gamma >\eta$ be such that
(i) if $\beta < \gamma$ and for some $\zeta$ there is $k:V_\eta \prec V_\zeta$ with crit($k$)=$\kappa$ and $k(\kappa$)=$\beta$, there is such a $k$ with a $\zeta < \gamma$.
(ii) cf($\gamma$)=$\omega_1$.
He says that such a $\gamma$ exists by a simple closure argument iterated $\omega_1$ times.
My question is what is the details of this closure argument?
In the paper 'Strong Axioms of Infinity and Elementary Embeddings' by Solovay, Reinhardt and Kanamori, on page 99 they are proving the same thing, and they say that we can get such a $\gamma$ as above by a reflection argument. I would also like to know what is the details of the reflection argument here.
Thanks!
The closure argument is actually simple: take $\gamma_0=\eta+1$, $\gamma_\delta=\sup_{\alpha<\delta}\gamma_\alpha$ if $\delta$ is a limit ordinal.
Now define $\gamma_{\alpha+1}$ as follows: For each $\beta < \gamma_\alpha$, choose $k_\beta\colon V_\eta\prec V_{\zeta_\beta}$ such that $\operatorname{crit}k_\beta=\kappa$ and $k_\beta(\kappa)=\beta$ if it exists. Let $X$ be the set of all $\beta<\gamma_\alpha$ such that $k_\beta$ exists, and let $\gamma_{\alpha+1}:=\sup \{\zeta_\beta\mid \beta\in X\}$.