Topics In Model Theory

The book has two chapters. The first chapter is a modern or contemporary account of stability theory. After a preliminary section on some of the basic techniques of model theory, the focus is on local (formula-by-formula) stability theory, treated a little differently from in the author's Geometric Stability Theory book. There is also a survey of general and geometric stability theory, as well as detailed applications to combinatorics (regularity lemma) using pseudofinite methods. The second chapter is an introduction to 'continuous logic' or 'continuous model theory', where truth values are real numbers, drawing on the main texts and papers, but with an independent point of view. This chapter also includes some historical background, a discussion of hyperimaginaries in classical first order model theory, as well as an introduction to local stability in the continuous framework making use of some functional analysis results from Grothendieck's thesis. These chapters are based on notes, written by students, from a couple of advanced graduate courses in Notre Dame, in Autumn 2018, and Spring 2021.