 There are two ways of stratifying the universe of sets. The first is in terms of ⟨Vα | α ∈ On ⟩, the second is in terms of ⟨H(κ) | κ ∈ Card⟩, where H(κ) is the set of all sets which have cardinality less than κ and whose members have cardinality less than κ, and whose members of members have cardinality less than κ, and so on. For example, H(ω) = Vω and the theories of the structures H(ω1) and Vω+1 are mutually interpretable. This latter structure is the structure of second-order arithmetic and, as mentioned above, large cardinal axioms give us an “effectively complete” understanding of this structure. We should like to be in the same position with regard to larger and larger fragments of the universe and the question is whether we should proceed in terms of the first or the second stratification. 