Finite-power-set hereditary-small construction
= Finite-power-set hereditary-small construction
Let $z_0=\omega$ and $z_{n+1}=\mathcal P(z_n)$. Call $x$ small when it injects into some $z_n$, and let $\mathbf{HS}$ contain exactly the sets all of whose hereditary members are small. Then every $z_n$ and the set $y=\{z_n:n\in\omega\}$ belong to $\mathbf{HS}$, but $\bigcup y$ does not. Consequently $(\mathbf{HS},\in)$ fails the <Axiom of union>.