| definiendum | $V_\omega$ in it |
| postulate | $\emptyset\in V_\omega$ |
| for all | $x\in V_\omega$ |
| postulate | ${\mathcal P}(x)\in V_\omega $ |
| postulate | $x = \emptyset\ \lor\ \exists (y\in V_\omega).\ x = {\mathcal P}(y) $ |
This is the set of all finite sets constructable when starting with $\emptyset$. It's the smallest infinite Grothendieck universe, as well as a model of ZFC.
Wikipedia: Hereditarily finite set