936 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
Representing Scott sets in algebraic settings
We prove that for every Scott set there are -saturated real closed
fields and models of Presburger arithmetic
Satisfaction classes in nonstandard models of first-order arithmetic
A satisfaction class is a set of nonstandard sentences respecting Tarski's
truth definition. We are mainly interested in full satisfaction classes, i.e.,
satisfaction classes which decides all nonstandard sentences. Kotlarski,
Krajewski and Lachlan proved in 1981 that a countable model of PA admits a
satisfaction class if and only if it is recursively saturated. A proof of this
fact is presented in detail in such a way that it is adaptable to a language
with function symbols. The idea that a satisfaction class can only see finitely
deep in a formula is extended to terms. The definition gives rise to new
notions of valuations of nonstandard terms; these are investigated. The notion
of a free satisfaction class is introduced, it is a satisfaction class free of
existential assumptions on nonstandard terms.
It is well known that pathologies arise in some satisfaction classes. Ideas
of how to remove those are presented in the last chapter. This is done mainly
by adding inference rules to M-logic. The consistency of many of these
extensions is left as an open question.Comment: Thesis for the degree of licentiate of philosophy, 74 pages, 4
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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 Fodor’s Lemma in ATR to build an internally rather classless model. The second one uses some weak versions of the Galvin–Prikry Theorem in adjoining an ideal set to a model of second-order arithmetic
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