163 research outputs found
A topological interpretation of three Leibnizian principles within the functional extensions
Three philosophical principles are often quoted in connection with Leibniz:
"objects sharing the same properties are the same object" (Identity of
indiscernibles), "everything can possibly exist, unless it yields
contradiction" (Possibility as consistency), and "the ideal elements correctly
determine the real things" (Transfer). Here we give a precise
logico-mathematical formulation of these principles within the framework of the
Functional Extensions, mathematical structures that generalize at once
compactifications, completions, and elementary extensions of models. In this
context, the above Leibnizian principles appear as topological or algebraic
properties, namely: a property of separation, a property of compactness, and a
property of directeness, respectively. Abiding by this interpretation, we
obtain the somehow surprising conclusion that these Leibnizian principles may
be fulfilled in pairs, but not all three together.Comment: arXiv admin note: substantial text overlap with arXiv:1012.434
Forcing with Non-wellfounded Models
We develop the machinery for performing forcing over an arbitrary (possibly non-wellfounded) model of set theory. For consistency results, this machinery is unnecessary since such results can always be legitimately obtained by assuming that the ground model is (countable) transitive. However, for establishing properties of a given (possibly non-wellfounded) model, the fully developed machinery of forcing as a means to produce new related models can be useful. We develop forcing through iterated forcing, paralleling the standard steps of presentation found in [19] and [14]
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
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