Interstructure Lattices and Types of Peano Arithmetic

The collection of elementary substructures of a model of PA forms a lattice, and is referred to as the substructure lattice of the model. In this thesis, we study substructure and interstructure lattices of models of PA. We apply techniques used in studying these lattices to other problems in the model theory of PA. In Chapter 2, we study a problem that had its origin in Simpson, who used arithmetic forcing to show that every countable model of PA has an expansion to PA∗ that is pointwise definable. Enayat later showed that there are 2ℵ0 models with the property that every expansion to PA∗ is pointwise definable. In this Chapter, we use techniques involved in representations of lattices to show that there is a model of PA with this property which contains an infinite descending chain of elementary cuts. In Chapter 3, we study the question of when subsets can be coded in elementary end extensions with prescribed interstructure lattices. This problem originated in Gaifman, who showed that every model of PA has a conservative, minimal elementary end extension. That is, every model of PA has a minimal elementary end extension which codes only definable sets. Kossak and Paris showed that if a model is countable and a subset X can be coded in any elementary end extension, then it can be coded in a minimal extension. Schmerl extended this work by considering which collections of sets can be the sets coded in a minimal elementary end extension. In this Chapter, we extend this work to other lattices. We study two questions: given a countable model M, which sets can be coded in an elementary end extension such that the interstructure lattice is some prescribed finite distributive lattice; and, given an arbitrary model M, which sets can be coded in an elementary end extension whose interstructure lattice is a finite Boolean algebra

Pure patterns of order 2

We provide mutual elementary recursive order isomorphisms between classical ordinal notations, based on Skolem hulling, and notations from pure elementary patterns of resemblance of order $2$, showing that the latter characterize the proof-theoretic ordinal of the fragment $\Pi^1_1$-$\mathrm{CA}_0$ of second order number theory, or equivalently the set theory $\mathrm{KPl}_0$. As a corollary, we prove that Carlson's result on the well-quasi orderedness of respecting forests of order $2$ implies transfinite induction up to the ordinal of $\mathrm{KPl}_0$. We expect that our approach will facilitate analysis of more powerful systems of patterns.Comment: corrected Theorem 4.2 with according changes in section 3 (mainly Definition 3.3), results unchanged. The manuscript was edited, aligned with reference [14] (moving former Lemma 3.5 there), and argumentation was revised, with minor corrections in (the proof of) Theorem 4.2; results unchanged. Updated revised preprint; to appear in the APAL (2017

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

Algebraic extensions in free groups

The aim of this paper is to unify the points of view of three recent and independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich and Miasnikov 2002), where similar modern versions of a 1951 theorem of Takahasi were given. We develop a theory of algebraic extensions for free groups, highlighting the analogies and differences with respect to the corresponding classical field-theoretic notions, and we discuss in detail the notion of algebraic closure. We apply that theory to the study and the computation of certain algebraic properties of subgroups (e.g. being malnormal, pure, inert or compressed, being closed in certain profinite topologies) and the corresponding closure operators. We also analyze the closure of a subgroup under the addition of solutions of certain sets of equations.Comment: 35 page