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Publisher:

Springer

Publication Date:

2013

Number of Pages:

198

Format:

Paperback

Edition:

2

Series:

Universitext

Price:

69.95

ISBN:

9781461457459

Category:

Textbook

[Reviewed by , on ]

Annie Selden

06/2/2014

First, a disclaimer: I am not a logician. Indeed, I have never taken a course in mathematical logic, but I was interested in whether this book would provide some background information for teaching the logic portion of a traditional transition-to-proof course. The answer is no, but I will do my best to give some sense of what is in this book.

The goal of this paperback textbook, written for a course in logic, is to prove Gödel’s completeness and incompleteness theorems. Chapter 1 discusses first-order theories, which consist of a first-order language \(L\) and a set of formulas, basically expressions in \(L\). A number of examples of first order theories, including group theory and ring theory, are discussed. Chapter 2 discusses the semantics of first-order theories and is said by the author, to have the beginnings of model theory, which “can be thought of as the study of general mathematical structures.” More specifically, a *model *of a theory \(T\) is a structure in which all the nonlogical axioms of \(T\) are valid. A nonlogical axiom of group theory, for example, would be the existence of inverses.

Chapter 3 considers a simpler form of logic, namely propositional logic, defines what a proof is, and establishes the corresponding completeness theorem. Let \(\mathcal{A}\) be a set of nonlogical axioms. By a *proof* of a formula \(A\) is meant a finite sequence of formulas, each of which is either a logical axiom, a nonlogical axiom, or can be inferred from the previous formulas using a rule of inference, and the last of which is \(A\). If there is a proof of \(A\) from the set of axioms \(\mathcal{A}\), then one writes \(\mathcal{A}\vdash A\). If \(A\) is a true in *every * model of \(\mathcal{A}\), then one writes \(\mathcal{A}\models A\). With this terminology in place, the Completeness Theorem for Propositional Logic is then stated as follows:

Let \(\mathcal{A}\) be a set of formulas of \(L\). Then \(\mathcal{A}\vdash A \Leftrightarrow \mathcal{A}\models A\).

Chapter 4 defines proof for first-order theories and proves the corresponding completeness theorem. The author states that Chapters 1–4, plus Sections 5.1 and 5.2 would constitute an adequate course in mathematical logic for undergraduates. I leave that to those of you who would consider teaching such a course, perhaps to honors students or as a directed readings course.

There are three additional chapters on model theory; recursive functions and arithmetization of theories; and representability and incompleteness theorems, but I didn’t get that far. This book is replete with symbols such as \(\forall\), \(\exists\), \(\wedge\), \(\vee\), \(\neg\), \(\to\), \(\leftrightarrow\), that most mathematicians are used to, but there are also some symbols that seem specific to logic, such as \(\vdash\), \(\models\), and \(\nvDash\), whose definitions are not easy to find. It would have been helpful to have included a list of symbols and their meanings at the end of the book. There is an index, but not every technical word can be found there. However, for the most part, the book seems carefully written.

Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education. In 2003, she was elected a Fellow of the American Association for the Advancement of Science. She remains active in mathematics education research and curriculum development.

Preface

1 Syntax of First-Order Logic

2 Semantics of First-Order Languages

3 Propositional Logic

4 Completeness Theorem for First-Order Logic

5 Model Theory

6 Recursive Functions and Arithmetization of Theories

7 Incompleteness Theorems and Recursion Theory

References

Index.

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