Power-set failure at the first rank above V omega
= Power-set failure at the first rank above V omega
{title2=$(V_{\omega+1},\in)\not\models\mathrm{PowerSet}$}
Every <finite ordinal> belongs to $V_\omega$, so $\omega\in V_{\omega+1}$ but $\omega\notin V_\omega$. An internal <power set> $y$ of $\omega$ would contain $\omega$ itself. Yet $y\in V_{\omega+1}$ implies $y\subseteq V_\omega$, making that impossible. This example illustrates why one must check which subsets the structure sees when proving an axiom fails.