23 research outputs found
Modern Class Forcing
We survey recent developments in the theory of class forcing for- malized in the second-order set-theoretic setting
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 āunrollsā 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 theoriesāsuch as KM or Ī 11-CAādo not have least transitive models while weaker theoriesāfrom GBC to GBC + ETROrd ādo have least transitive models
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
Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism
In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views