My intuition about Grothendieck universes says the following.
- Axiom of empty set $\leftrightarrow$ There exists a Grothendieck universe.
- Axiom of infinity $\leftrightarrow$ There exist at least two distinct Grothendieck universes.
- Tarski's axiom $\leftrightarrow$ There exist as many Grothendieck universes as there are entities in the domain of discourse.
Is this intuition correct? And if so, have people studied axioms that sit between 2 and 3? Say, asserting the existence of countably many distinct Grothendieck universes?