= Countable rank-initial segment cannot model ZFC
{c}
If $\alpha$ is a <countable ordinal> and $V_\alpha\models\mathsf{ZFC}$, then $\alpha$ must be a limit ordinal. Choose a countable cofinal sequence $(\alpha_n)$ in $\alpha$. The internal <axiom of choice> gives, for every $n$, a bijection between $V_{\alpha_n}$ and some ordinal below $\alpha$; that ordinal is externally countable, so every $V_{\alpha_n}$ is countable. Hence $V_\alpha=\bigcup_nV_{\alpha_n}$ is countable. But $V_\alpha$ contains the full <power set> $\mathcal P(\omega)$, which is uncountable by <Cantor theorem>, a contradiction.
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