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How can I prove the following fact? Let be a regular cardinal $\kappa$ such that $V_\kappa$ is a model of ZFC. Then $\kappa$ is inaccessible.

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  • $\begingroup$ Note that the proof that $\kappa$ is a strong limit doesn’t use the regularity of $\kappa$. So a more informative wording would be “if $V_{\kappa}$ is a model of ZFC then $\kappa$ is a strong limit.” (So of course if $\kappa$ also happens to be regular, then it is inaccessible.) $\endgroup$ – spaceisdarkgreen Sep 23 '19 at 16:40
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Since we assume $\kappa$ to be regular, we only need to prove that for any $\lambda < \kappa$ we have $2^\lambda < \kappa$. Note that for every ordinal $\alpha$ we have $\operatorname{rank}(\alpha) = \alpha$ (see, e.g. this answer). In particular, that means that the collection of ordinals in $V_\kappa$ is just $\kappa$. So for $\lambda < \kappa$, we have $\lambda \in V_\kappa$ and all of its subsets belong to $V_\kappa$. Then since $V_\kappa$ is a model of ZFC, the powerset $\mathcal{P}(\lambda)$ must belong to $V_\kappa$ again. Thus $2^\lambda$ (as an ordinal) is also in $V_\kappa$, and so we see $2^\lambda < \kappa$.

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  • $\begingroup$ The question asks about regular $\kappa$ and a regular wordly cardinal is inaccessible $\endgroup$ – Alessandro Codenotti Sep 23 '19 at 11:27
  • $\begingroup$ @AlessandroCodenotti Thanks for pointing that out, I did not read properly. Hopefully my answer now makes sense and actually answers the question. $\endgroup$ – Mark Kamsma Sep 23 '19 at 12:05
  • $\begingroup$ @Mark Kamsma Thank you. However, here I’m assuming that k is regular. So I’m not considering the case in which k is the least worldly cardinal. So the statement should be correct but I’ve no idea how to prove that. $\endgroup$ – SeTh Sep 23 '19 at 12:22
  • $\begingroup$ @SeTh Refresh the page ;) Alessandro pointed this out as well, and I edited. Hopefully it now actually answers your question. $\endgroup$ – Mark Kamsma Sep 23 '19 at 12:23
  • $\begingroup$ Thank you so much! Very elegant proof $\endgroup$ – SeTh Sep 23 '19 at 12:57

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