It is very near the bottom of Kanamori's chart. The very bottom of the chart is the level of a (strongly) inaccessible cardinals, which is the smallest large cardinal axiom. Right above the inaccessibles in the chart are the α-inaccessible cardinals. It turns out that the Universe Axiom (UA) is strictly weaker than the existence of a 2-inaccessible cardinal. In fact, κ is 2-inaccessible if and only if κ is regular and Vκ ⊧ ZFC + UA.
Specifically, a cardinal κ is:
- 0-inaccessible iff κ is regular,
- 1-inaccessible iff κ is a regular strong limit of 0-inaccessibles,
- 2-inaccessible iff κ is a regular strong limit of 1-inaccessibles,
- etc.
So an inaccessible cardinal is exactly the same as a 1-inaccessible cardinal, which are also precisely the regular cardinals κ such that Vκ ⊧ ZFC. If κ is 2-inaccessible, then there are unboundedly many inaccessibles λ < κ. These are inaccessible in Vκ too, which is why Vκ satisfies UA.