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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!

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    $\begingroup$ The definition of extendibility should not say that the critical point is $>\kappa+\eta$ (which makes no sense) but that it equals $\kappa$ and is mapped above $\kappa+\eta$. $\endgroup$ 2 days ago
  • $\begingroup$ That is completely right. Thank you for pointing out the error, I have fixed the definition. Could you share the closure argument and the reflection argument in question? $\endgroup$
    – kaka
    2 days ago

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