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

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 figure

Representing Scott sets in algebraic settings

We prove that for every Scott set $S$ there are $S$-saturated real closed fields and models of Presburger arithmetic

Existentially Closed Models in the Framework of Arithmetic

We prove that the standard cut is definable in each existentially closed model of IΔ0 + exp by a (parameter free) П1–formula. This definition is optimal with respect to quantifier complexity and allows us to improve some previously known results on existentially closed models of fragments of arithmetic.Ministerio de Educación y Ciencia MTM2011–2684