Introduction to Mathematical Logic, Fourth Edition: (Discrete Mathematics and Its Applications)

The Fourth Edition of this long-established text retains all the key features of the previous editions, covering the basic topics of a solid first course in mathematical logic. This edition includes an extensive appendix on second-order logic, a section on set theory with urlements, and a section on the logic that results when we allow models with empty domains. The text contains numerous exercises and an appendix furnishes answers to many of them.Introduction to Mathematical Logic includes:opropositional logicofirst-order logicofirst-order number theory and the incompleteness and undecidability theorems of G/del, Rosser, Church, and Tarskioaxiomatic set theoryotheory of computabilityThe study of mathematical logic, axiomatic set theory, and computability theory provides an understanding of the fundamental assumptions and proof techniques that form basis of mathematics. Logic and computability theory have also become indispensable tools in theoretical computer science, including artificial intelligence. Introduction to Mathematical Logic covers these topics in a clear, reader-friendly style that will be valued by anyone working in computer science as well as lecturers and researchers in mathematics, philosophy, and related fields.

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 Quantification Theory Quantifiers First-Order Languages and Their Interpretations 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. Non-Standard Analysis Semantic Trees Quantification Theory Allowing Empty Domains Formal Number Theory An Axiom System Number-Theoretic Functions and Relations Primitive Recursive and Recursive Functions Arithmatization. Gödel Numbers The Fixed Point Theorem. Gödel's Incompleteness Theorem Recursive Undecidability. Church's Theorem Axiomatic Set Theory An Axiom System Ordinal Numbers Equinumerousity. Finite and Denumerable Sets. Hartog's 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-Mosotovski Hierarchy. Recursively Enumerable Sets Other notions of Computability Decision Problems

Review :
"Nearly forty years after it was published (1964), Elliot Mendelson's Introduction to Mathematical Logic still remains the best textbook on the principle topics of this subjectI have used Mendelson's book to teach a one-semester course to advanced undergraduate and graduate students with great success." - Alan Berger "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 "I was sufficiently fortunate to have taken Professor Emeritus Mendelson's famous logic course at Queens College, the City University of New York, just two semesters before his retirement. I was, and continue to be, astonished by Dr. Mendelson's precise yet easy style, and the beautifully efficient organization of the subjects. Everything from the expository prose to the system of notational conventions has been carefully thought through so as to make the book both very substantive and very readable. In my opinion, it's the best introduction to serious mathematical logic currently on the market, and thanks to the genius of its author, it is likely to remain so for a long time. The buyer will not be disappointed." -Joseph Jay Stern