Topics in Model Theory

This book has two chapters. The first is a modern or contemporary account of stability theory. A focus is on the local (formula-by-formula) theory, treated a little differently from in the author's book Geometric Stability Theory. There is also a survey of general and geometric stability theory, as well as applications to combinatorics (stable regularity lemma) using pseudofinite methods.

The second is an introduction to 'continuous logic' or 'continuous model theory,' drawing on the main texts and papers, but with an independent point of view. This chapter includes some historical background, including some other formalisms for continuous logic and a discussion of hyperimaginaries in classical first order logic.

These chapters are based around notes, written by students, from a couple of advanced graduate courses in the University of Notre Dame, in Autumn 2018, and Spring 2021.

Contents:

• Preface

• Stability Theory:

  • Introduction
  • Preliminaries
  • Stability

• Continuous Logic:

  • Introduction
  • Background
  • 'Official' Continuous Logic
  • Stability in Continuous Logic

• Index

Readership: Graduate students and researchers in mathematics and related subjects interested in model theory and its applications.
Key Features:

• The book represents fairly up-to-date topics and points of view in model theory
• Regarding stability, local stability in a model is developed, pointing out connections with functional analysis and combinatorics
• The continuous logic chapter should serve as an introduction to the subject, also explaining the roots of the subject in classical first order model theory and also touching on stability and connections to combinatorics
• The author is an expert in stability theory, although the course on continuous logic was partly meant for the author and students to become acquainted with the subject