Grothendieck universes and relative size

For a fixed universe $U$, “$U$-small” means being a member of $U$. It is relative to that specified set. Cardinal size alone does not determine it: a singleton can contain an ordinal of arbitrarily high rank, since $\mathrm{rank}(\{\alpha \})=\alpha +1$ follows immediately from the rank equation and $\mathrm{rank}(\alpha )=\alpha$.

Our closure convention does not impose $\omega \in U$. For example, $V_{\omega }$ satisfies the closure conditions: its members lie in finite stages; pairs and power sets raise the stage by only finitely much; and a family indexed by one of its finite members has a finite bound on the stages of its values, so its union again lies in a finite stage. It is nonempty and transitive, but $\omega \mathrm{\notin }{V}_{\omega }$ by the ordinal-intersection formula. This explains why an uncountable-inaccessible characterization needs an additional infinity convention. Universe existence axioms and large-cardinal characterizations are not asserted here.

Conventions and prerequisites: Grothendieck universe closure convention, Ranks of ordinals and hierarchy stages.