 The second stratification is potentially more fine-grained. Assuming CH one has that the theories of H(ω2) and Vω+2 are mutually interpretable and assuming larger and larger fragments of GCH this correspondence continues upward. But if CH is false then the structure H(ω2) is less rich than the structure Vω2. In this event the latter structure captures full third-order arithmetic, while the former captures only a small fragment of third-order arithmetic but is nevertheless rich enough to express CH. Given this, in attempting to understand the universe of sets by working up through it level by level, it is sensible to use the potentially more fine-grained stratification. 