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Mathematical Logic and Model Theory: A Brief Introduction offers a streamlined yet easy-to-read introduction to mathematical logic and basic model theory. It presents, in a self-contained manner, the essential aspects of model theory needed to understand model theoretic algebra. As a profound application of model theory in algebra, the last part of this book develops a complete proof of Ax and Kochen's work on Artin's conjecture about Diophantine properties of p-adic number fields. The character of model theoretic constructions and results differ quite significantly from that commonly found in algebra, by the treatment of formulae as mathematical objects. It is therefore indispensable to first become familiar with the problems and methods of mathematical logic. Therefore, the text is divided into three parts: an introduction into mathematical logic (Chapter 1), model theory (Chapters 2 and 3), and the model theoretic treatment of several algebraic theories (Chapter 4). This book will be of interest to both advanced undergraduate and graduate students studying model theory and its applications to algebra. It may also be used for self-study.



0. Introduction

In the middle of the 1960s, several model theoretic arguments and methods of construction captured the attention of the mathematical world. J. Ax and S. Kochen succeeded, in joint work, in achieving a decisive contribution to the “Artin conjecture” on the solvability of homogeneous diophantine equations over p-adic number fields. This and other results led to an infiltration of certain model theoretic concepts and methods into algebra. Because of their strangeness, however, very few algebraists could become comfortable with them. This is not so astonishing once one traces the historical development of model theoretic concepts and methods from their origin up to their present-day applications: their applicability to algebra is not the result of a goal-directed development (more precisely, a development directed toward the goal of these applications), but is, rather, a by-product of an investigation directed toward a quite different goal, namely, the goal of getting to grips with the foundations of mathematics. The need to secure these foundations had become urgent after various contradictions in mathematics were discovered at the end of the nineteenth and the beginning of the twentieth centuries.
Alexander Prestel, Charles N. Delzell

Chapter 1. First-Order Logic

In this chapter we introduce a calculus of logical deduction, called first-order logic, that makes it possible to formalize mathematical proofs. The main theorem about this calculus that we shall prove is Gödel’s completeness theorem (1.5.2), which asserts that the unprovability of a sentence must be due to the existence of a counterexample. From the finitary character of a formalized proof we then immediately obtain the Finiteness Theorem (1.5.6), which is fundamental for model theory, and which asserts that an axiom system possesses a model provided that every finite subsystem of it possesses a model.
In (1.6) we shall axiomatize a series of mathematical (in particular, algebraic) theories. In order to show the extent of first-order logic, we shall also give within this framework the Zermelo–Fraenkel axiom system for set theory, a theory that allows us to represent all of ordinary mathematics in it.
Alexander Prestel, Charles N. Delzell

Chapter 2. Model Constructions

In this chapter we shall introduce various methods for the construction of models of an axiom system Σ of sentences in a formal language L. In Chapter 1 we have already encountered a method, in the form of the so-called term-models, for obtaining at least one model of Σ. Having presented that “absolute” construction, we shall now present a series of “relative” constructions. These relative methods allow us to start with one or more given models of Σ, and to produce a new model. The methods considered here do not (as often occurs in mathematics) depend on the particular axiom system Σ (e.g. the direct product of groups is again a group, while the analogue of this for fields does not hold); rather, our methods will work in every case. This will be guaranteed by the fact that an L-structure \(\mathfrak{A}'\) obtained by these methods from an L-structure \(\mathfrak{A}\) is elementarily equivalent to \(\mathfrak{A}\); i.e.
$$ \textit{ every $L$-sentence $\varphi$ that holds in $\mathfrak{A}$ also holds in $\mathfrak{A}'$, and conversely.}$$
Therefore, if \(\mathfrak{A}\) is a model of Σ, then so is \(\mathfrak{A}'\), independent of which axiom system Σ⊆Sent(L) we are working with.
Alexander Prestel, Charles N. Delzell

Chapter 3. Properties of Model Classes

In this chapter we wish to study properties of model classes. By a model class we mean the class of all models of an axiom system Σ.
First we shall furnish such a class with a topology whose compactness is exactly the content of the Finiteness Theorem 1.5.6. After that, we shall introduce several properties of the model class of an axiom system Σ, whose study can lead, among other things, to the proof of the completeness (recall §1.6) of Σ. We shall carry this out explicitly for a series of theories axiomatized in (1.6).
The properties of Σ or of its model class to be introduced are: categoricity in a fixed cardinality, model completeness and quantifier elimination. The study of these properties is based not only on the possible applicability to the proof of the completeness of a theory, but is, rather, also justified by its usefulness in concrete, mathematical (in particular, algebraic) theories. In this chapter we shall investigate such properties only for the theory of algebraically closed fields; in Chapter 4, other theories will follow.
Alexander Prestel, Charles N. Delzell

Chapter 4. Model Theory of Several Algebraic Theories

In this chapter we investigate a series of interesting algebraic theories for the properties of completeness, model completeness and quantifier elimination. Not only do we treat the standard examples that have already been frequently treated in the extant literature, but we shall place special value on the theory of valued fields. Since valuation theory does not belong in the standard repertoire of an algebra course, we shall first discuss the necessary concepts and theorems in detail, in Section 4.3. Thereafter we develop special cases (Sections 4.4 and 4.5), and finally the model theory of Henselian valued fields. The goal of this presentation is, among other things, a treatment of a purely number theoretic problem – Artin’s conjecture – in Theorem 4.6.5.
Alexander Prestel, Charles N. Delzell


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