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
The Compactness Theorem 14
Isomorphisms, elementary equivalence and complete theories 15...
William Weiss and Cherie D’Mello
Department of Mathematics
University of Toronto
c 2015 W.Weiss and C. D’Mello
,
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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 acquain-
tance 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 ex-
amples; it is short and we recommend that you peruse it first, before studying 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 mathe-
matics: Ramsey’s Theorem, the Tarski-Seidenberg Theorem. Some are chosen for
their immediate appeal to every mathematician: existence of infinitesimals for cal-
culus, 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 prepa-
ration 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.
This book began life as notes for William Weiss’s graduate course at the Uni-
versity of Toronto. The notes were revised and expanded by Cherie D’Mello and
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William Weiss, based upon suggestions from several graduate students. The elec-
tronic 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.
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