Topology Atlas Document # iaai-10 | © 2000 Copyright by William Weiss and Cherie D'Mello.

## Fundamentals of Model Theory

### William Weiss and Cherie D'Mello

Department of Mathematics
University of Toronto

The textbook is available in several formats.

### Introduction

Model Theory is the part of mathematics which shows how to apply logic to the study of structures in pure mathematics. On the one hand it is the ultimate abstraction; on the other, it has immediate applications to every-day mathematics. The fundamental tenet of Model Theory is that mathematical truth, like all truth, is relative. A statement may be true or false, depending on how and where it is interpreted. This isn't necessarily due to mathematics itself, but is a consequence of the language that we use to express mathematical ideas.

What at first seems like a deficiency in our language, can actually be shaped into a powerful tool for understanding mathematics. This book provides an introduction to Model Theory which can be used as a text for a reading course or a summer project at the senior undergraduate or graduate level. It is also a primer which will give someone a self contained overview of the subject, before diving into one of the more encyclopedic standard graduate texts.

Any reader who is familiar with the cardinality of a set and the algebraic closure of a field can proceed without worry. Many readers will have some acquaintance with elementary logic, but this is not absolutely required, since all necessary concepts from logic are reviewed in Chapter 0. Chapter 1 gives the motivating examples and we recommend that you read it first, before diving into the more technical aspects of Chapter 0. Chapters 2 and 3 are selections of some of the most important techniques in Model Theory. The remaining chapters investigate the relationship between Model Theory and the algebra of the real and complex numbers. Thirty exercises develop familiarity with the definitions and consolidate understanding of the main proof techniques.

Throughout the book we present applications which cannot easily be found elsewhere in such detail. Some are chosen for their value in other areas of mathematics: Ramsey's Theorem, the Tarski-Seidenberg Theorem. Some are chosen for their immediate appeal to every mathematician: existence of infinitesimals for calculus, graph colouring on the plane. And some, like Hilbert's Seventeenth Problem, are chosen because of how amazing it is that logic can play an important role in the solution of a problem from high school algebra. In each case, the derivation is shorter than any which tries to avoid logic. More importantly, the methods of Model Theory display clearly the structure of the main ideas of the proofs, showing how theorems of logic combine with theorems from other areas of mathematics to produce stunning results.

The theorems here are all are more than thirty years old and due in great part to the cofounders of the subject, Abraham Robinson and Alfred Tarski. However, we have not attempted to give a history. When we attach a name to a theorem, it is simply because that is what mathematical logicians popularly call it.

The bibliography contains a number of texts that were helpful in the preparation of this manuscript. They could serve as avenues of further study and in addition, they contain many other references and historical notes. The more recent titles were added to show the reader where the subject is moving today. All are worth a look.

### Contents

Chapter 0.
Models, Truth and Satisfaction 4
Formulas, Sentences, Theories and Axioms 4
Prenex Normal Form 9
Chapter 1.
Notation and Examples 11
Chapter 2.
Compactness and Elementary Submodels 14
Compactness Theorem 14
Isomorphisms, elementary equivalence and complete theories 15
Elementary Chain Theorem 16
Löowenheim-Skolem Theorems 19
The Lós-Vaught Test 21
Every complex onetoone polynomial map is onto 23
Chapter 3.
Diagrams and Embeddings 24
Diagram Lemmas 25
Every planar graph can be four coloured 25
Ramsey's Theorem 26
The Leibniz Principle and infinitesimals 26
Robinson Consistency Theorem 27
Craig Interpolation Theorem 31
Chapter 4.
Model Completeness 32
Robinson's Theorem on existentially complete theories 32
Lindströom's Test 35
Hilbert's Nullstellensatz 38
Chapter 5.
The Seventeenth Problem 39
Positive definite rational functions are the sums of squares 39
Chapter 6. Submodel Completeness 45
Elimination of quantifiers 45
The Tarski-Seidenberg Theorem 49
Chapter 7.
Model Completions 50
Almost universal theories 52
Saturated models 54
Blum's Test 55
Bibliography 61
Index 62

### Distribution notice

This book began life as notes for William Weiss's graduate course at the University of Toronto. The notes were revised and expanded by Cherie D'Mello and William Weiss, based upon suggestions from several graduate students. The electronic version of this book may be downloaded and further modified by anyone for the purpose of learning, provided this paragraph is included in its entirety and so long as no part of this book is sold for profit.