Power-set failure in hereditarily small sets
= Power-set failure in hereditarily small sets
{title2=$H_{\omega_2}\not\models\mathrm{PowerSet}$}
The <hereditarily small set> structure $H_{\omega_2}$ fails the <Axiom of power set> at $\omega_1$. Every subset of $\omega_1$ belongs to $H_{\omega_2}$, but their full <power set> has <cardinality> at least $\aleph_2$ by the <Cantor theorem>, so cannot itself be hereditarily smaller than $\aleph_2$.