9 research outputs found
The Structure of Models of Second-order Set Theories
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 -realizations of a fixed countable model of
, where is a reasonable second-order set theory such as
or , 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 to weaker theories.
They showed that every model of plus the Class Collection schema
"unrolls" to a model of with a largest cardinal. I calculate
the theories of the unrolling for a variety of second-order set theories, going
as weak as . I also show that being -realizable
goes down to submodels for a broad selection of second-order set theories .
Third, I show that there is a hierarchy of transfinite recursion principles
ranging in strength from to . 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 or ---do
not have least transitive models while weaker theories---from to
---do have least transitive models.Comment: This is my PhD dissertatio
The Structure of Models of Second-order Set Theories
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 âunrollsâ to a model of ZFCâ 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 theoriesâsuch as KM or Î 11-CAâdo not have least transitive models while weaker theoriesâfrom GBC to GBC + ETROrd âdo have least transitive models
Non-tightness in class theory and second-order arithmetic
A theory T is tight if different deductively closed extensions of T (in the
same language) cannot be bi-interpretable. Many well-studied foundational
theories are tight, including PA [Visser2006], ZF, Z2, and KM [enayat2017]. In
this article we extend Enayat's investigations to subsystems of these latter
two theories. We prove that restricting the Comprehension schema of Z2 and KM
gives non-tight theories. Specifically, we show that GB and ACA0 each admit
different bi-interpretable extensions, and the same holds for their extensions
by adding Sigma^1_k-Comprehension, for k <= 1. These results provide evidence
that tightness characterizes Z2 and KM in a minimal way