#### Date of Degree

9-2017

#### Document Type

Dissertation

#### Degree Name

Ph.D.

#### Program

Mathematics

#### Advisor

Roman Kossak

#### Committee Members

Alfred Dolich

Russell Miller

Philipp Rothmaler

#### Subject Categories

Logic and Foundations

#### Keywords

model theory, peano arithmetic, substructure lattices

#### Abstract

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 2^{ℵ0} 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?

#### Recommended Citation

Abdul-Quader, Athar, "Interstructure Lattices and Types of Peano Arithmetic" (2017). *CUNY Academic Works.*

https://academicworks.cuny.edu/gc_etds/2196