### Features

- Provides a compact introduction to the principal topics of mathematical logic
- Presents the fundamental assumptions and proof techniques that form the basis of mathematical logic
- Explores logic and computability theory—indispensable tools in theoretical computer science and artificial intelligence
- Includes many examples and exercises

### Summary

Retaining all the key features of the previous editions, **Introduction to Mathematical Logic, Fifth 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.

**New to the Fifth Edition**

- A new section covering basic ideas and results about nonstandard models of number theory
- A second appendix that introduces modal propositional logic
- An expanded bibliography
- Additional exercises and selected answers

This long-established text continues to expose students to natural proofs and set-theoretic methods. Only requiring some experience in abstract mathematical thinking, it offers enough material for either a one- or two-semester course on mathematical logic.

### Table of Contents

**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**

**Answers to Selected Exercises**

**Bibliography**

**Notation**

**Index**

### Reviews

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

"For the reviews of the previous editions see Zbl 192.01901, Zbl 498.03001, Zbl 681.03001 and Zbl 915.03002. 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

**Praise for the Fourth Edition**

"In my work as a math teacher, researcher, author, and journal editor, I often encounter problems with a logical component. When that need arises, my first choice of reference is always this book. It is the most concise and readable introductory text I have ever encountered and it is a rare occasion when I fail to find the background material needed to solve the problem. It is also an excellent source of problems and I have pulled the ideas for many test questions from it over the years."

—Charles Ashbacher