This dissertation is a contribution to the project of second-order set
theory, which has seen a revival in recent years. The approach is to understand
second-order set theory by studying the structure of models of second-order set
theories. The main results are the following, organized by chapter. First, I
investigate the poset of T-realizations of a fixed countable model of
ZFC, where T is a reasonable second-order set theory such as
GBC or KM, showing that it has a rich structure. In
particular, every countable partial order embeds into this structure. Moreover,
we can arrange so that these embedding preserve the existence/nonexistence of
upper bounds, at least for finite partial orders. Second I generalize some
constructions of Marek and Mostowski from KM to weaker theories.
They showed that every model of KM plus the Class Collection schema
"unrolls" to a model of ZFCā with a largest cardinal. I calculate
the theories of the unrolling for a variety of second-order set theories, going
as weak as GBC+ETR. I also show that being T-realizable
goes down to submodels for a broad selection of second-order set theories T.
Third, I show that there is a hierarchy of transfinite recursion principles
ranging in strength from GBC to KM. This hierarchy is
ordered first by the complexity of the properties allowed in the recursions and
second by the allowed heights of the recursions. Fourth, I investigate the
question of which second-order set theories have least models. I show that
strong theories---such as KM or Ī 11ā-CA---do
not have least transitive models while weaker theories---from GBC to
GBC+ETROrdā---do have least transitive models.Comment: This is my PhD dissertatio