334 research outputs found
The complexity of classification problems for models of arithmetic
We observe that the classification problem for countable models of arithmetic
is Borel complete. On the other hand, the classification problems for finitely
generated models of arithmetic and for recursively saturated models of
arithmetic are Borel; we investigate the precise complexity of each of these.
Finally, we show that the classification problem for pairs of recursively
saturated models and for automorphisms of a fixed recursively saturated model
are Borel complete.Comment: 15 page
Initial segments and end-extensions of models of arithmetic
This thesis is organized into two independent parts. In the first part, we extend the recent work on generic cuts by Kaye and the author. The focus here is the properties of the pairs (M, I) where I is a generic cut of a model M. Amongst other results, we characterize the theory of such pairs, and prove that they are existentially closed in a natural category. In the second part, we construct end-extensions of models of arithmetic that are at least as strong as ATR. Two new constructions are presented. The first one uses a variant of Fodorâs Lemma in ATR to build an internally rather classless model. The second one uses some weak versions of the GalvinâPrikry Theorem in adjoining an ideal set to a model of second-order arithmetic
Expansions, omitting types, and standard systems
Recursive saturation and resplendence are two important notions in models of
arithmetic. Kaye, Kossak, and Kotlarski introduced the notion of arithmetic
saturation and argued that recursive saturation might not be as rigid as first
assumed.
In this thesis we give further examples of variations of recursive
saturation, all of which are connected with expandability properties similar to
resplendence. However, the expandability properties are stronger than
resplendence and implies, in one way or another, that the expansion not only
satisfies a theory, but also omits a type. We conjecture that a special version
of this expandability is in fact equivalent to arithmetic saturation. We prove
that another of these properties is equivalent to \beta-saturation. We also
introduce a variant on recursive saturation which makes sense in the context of
a standard predicate, and which is equivalent to a certain amount of ordinary
saturation.
The theory of all models which omit a certain type p(x) is also investigated.
We define a proof system, which proves a sentence if and only if it is true in
all models omitting the type p(x). The complexity of such proof systems are
discussed and some explicit examples of theories and types with high
complexity, in a special sense, are given.
We end the thesis by a small comment on Scott's problem. We prove that, under
the assumption of Martin's axiom, every Scott set of cardinality <2^{\aleph_0}
closed under arithmetic comprehension which has the countable chain condition
is the standard system of some model of PA. However, we do not know if there
exists any such uncountable Scott sets.Comment: Doctoral thesi
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
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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