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

Countable Short Recursively Saturated Models of Arithmetic

Short recursively saturated models of arithmetic are exactly the elementary initial segments of recursively saturated models of arithmetic. Since any countable recursively saturated model of arithmetic has continuum many elementary initial segments which are already recursively saturated, we turn our attention to the (countably many) initial segments which are not recursively saturated. We first look at properties of countable short recursively saturated models of arithmetic and show that although these models cannot be cofinally resplendent (an expandability property slightly weaker than resplendency), these models have non-definable expansions which are still short recursively saturated

Self-embeddings of models of arithmetic; fixed points, small submodels, and extendability

In this paper we will show that for every cut $I$ of any countable nonstandard model $\mathcal{M}$ of $\mathrm{I}\Sigma_{1}$, each $I$-small $\Sigma_{1}$-elementary submodel of $\mathcal{M}$ is of the form of the set of fixed points of some proper initial self-embedding of $\mathcal{M}$ iff $I$ is a strong cut of $\mathcal{M}$. Especially, this feature will provide us with some equivalent conditions with the strongness of the standard cut in a given countable model $\mathcal{M}$ of $\mathrm{I}\Sigma_{1}$. In addition, we will find some criteria for extendability of initial self-embeddings of countable nonstandard models of $\mathrm{I}\Sigma_{1}$ to larger models

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$_0$. Two new constructions are presented. The first one uses a variant of Fodor’s Lemma in ATR$_0$ 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