Introduction to Mathematical Logic book cover

Introduction to Mathematical Logic book cover

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6th Edition

Introduction to Mathematical Logic

Copyright Year 2015

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Book Description

The new edition of this classic textbook, Introduction to Mathematical Logic, Sixth Edition explores the principal topics of mathematical logic. It covers propositional logic, first-order logic, first-order number theory, axiomatic set theory, and the theory of computability. The text also discusses the major results of Gödel, Church, Kleene, Rosser, and Turing.

The sixth edition incorporates recent work on Gödel's second incompleteness theorem as well as restoring an appendix on consistency proofs for first-order arithmetic. This appendix last appeared in the first edition. It is offered in the new edition for historical considerations. The text also offers historical perspectives and many new exercises of varying difficulty, which motivate and lead students to an in-depth, practical understanding of the material.

Table of Contents

Preface
Introduction

The Propositional Calculus

Propositional Connectives: Truth Tables
Tautologies
Adequate Sets of Connectives
An Axiom System for the Propositional Calculus
Independence: Many-Valued Logics
Other Axiomatizations

First-Order Logic and Model Theory
Quantifiers
First-Order Languages and Their Interpretations: Satisfiability and Truth Models
First-Order Theories
Properties of First-Order Theories
Additional Metatheorems and Derived Rules
Rule C
Completeness Theorems
First-Order Theories with Equality
Definitions of New Function Letters and Individual Constants
Prenex Normal Forms
Isomorphism of Interpretations: Categoricity of Theories
Generalized First-Order Theories: Completeness and Decidability
Elementary Equivalence: Elementary Extensions
Ultrapowers: Nonstandard Analysis
Semantic Trees
Quantification Theory Allowing Empty Domains

Formal Number Theory
An Axiom System
Number-Theoretic Functions and Relations
Primitive Recursive and Recursive Functions
Arithmetization: Gödel Numbers
The Fixed-Point Theorem: Gödel's Incompleteness Theorem
Recursive Undecidability: Church's Theorem
Nonstandard Models

Axiomatic Set Theory
An Axiom System
Ordinal Numbers
Equinumerosity: Finite and Denumerable Sets
Hartogs' Theorem: Initial Ordinals—Ordinal Arithmetic
The Axiom of Choice: The Axiom of Regularity
Other Axiomatizations of Set Theory

Computability
Algorithms: Turing Machines
Diagrams
Partial Recursive Functions: Unsolvable Problems
The Kleene–Mostowski Hierarchy: Recursively Enumerable Sets
Other Notions of Computability
Decision Problems

Appendix A: Second-Order Logic

Appendix B: First Steps in Modal Propositional Logic

Appendix C: A Consistency Proof for Formal Number Theory

Answers to Selected Exercises

Bibliography

Notations

Index

Author(s)

Biography

Elliott Mendelson is professor emeritus at Queens College in Flushing, New York, USA. Dr. Mendelson obtained his bachelor's degree at Columbia University and his master's and doctoral degrees at Cornell University, and was elected afterward to the Harvard Society of Fellows. In addition to his other writings, he is the author of another CRC Press book Introducing Game Theory and Its Applications.

Reviews

Praise for the Fifth Edition
"Since it first appeared in 1964, Mendelson's book has been recognized as an excellent textbook in the field. It is one of the most frequently mentioned texts in references and recommended reading lists … This book rightfully belongs in the small, elite set of superb books that every computer science graduate, graduate student, scientist, and teacher should be familiar with."
—Computing Reviews, May 2010

"The following are the significant changes in this edition: A new section (3.7) on the order type of a countable nonstandard model of arithmetic; a second appendix, Appendix B, on basic modal logic, in particular on the normal modal logics K, T, S4, and S5 and the relevant Kripke semantics for each; an expanded bibliography and additions to both the exercises and to the Answers to Selected Exercises, including corrections to the previous version of the latter."
—J.M. Plotkin, Zentralblatt MATH 1173

"Since its first edition, this fine book has been a text of choice for a beginner's course on mathematical logic. … There are many fine books on mathematical logic, but Mendelson's textbook remains a sure choice for a first course for its clear explanations and organization: definitions, examples and results fit together in a harmonic way, making the book a pleasure to read. The book is especially suitable for self-study, with a wealth of exercises to test the reader's understanding."
—MAA Reviews, December 2009