Introduction to Mathematical Logic, Fourth Edition

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: propositional logic first-order logic first-order number theory and the incompleteness and undecidability theorems of Gödel, Rosser, Church, and Tarski axiomatic set theory theory of computability The 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