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I'm trying to prove that the following statements are equivalent:

1.$\forall\alpha\in\mathbb{ON}\ \exists\ \kappa>\alpha$ is an inaccessible cardinal.

2.$\forall x\ \exists\ U\ x\in U$ and $U$ is a Grothendieck universe.

(1) $\rightarrow$ (2) is pretty straightforward. For the other direction, I was thinking to use induction on the ordinals; the successor case slides right by, but I have no ideas for the limit ordinals. Any thoughts are appreciated! Please let me know if definitions are required.

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    $\begingroup$ I think the proof is more or less like Zermelo theorem that states that in second order set theory every model is of the form $V_{\kappa}$ for $\kappa$ strongly inaccessible. $\endgroup$
    – user40276
    Commented Dec 1, 2014 at 6:21
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    $\begingroup$ It suffices to show that $\left| U \right| = U \cap \mathbf{On}$ is an inaccessible cardinal for every Grothendieck universe $U$. $\endgroup$
    – Zhen Lin
    Commented Dec 1, 2014 at 8:27
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    $\begingroup$ As a reference, I would suggest N.H. Williams, On Grothendieck Universes, Compositio Mathematica, tome 21, no. 1 (1969). $\endgroup$
    – W. Rether
    Commented Jun 19, 2018 at 15:07

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