A tour through mathematical logic

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     © 2005 by The Mathematical Association of America
     Complete Set ISBN 0-88385-000-1
     Vol. 30 ISBN 0-88385-036-2
     Library of Congress Catalog Card Number 200411354
A Tour through Mathematical Logic
Editorial Board
List of Published Monographs
Preface
     Acknowledgments
Contents

CHAPTER 1  Predicate Logic
     1.1 Introduction
          A very brief history of mathematical logic
     1.2 Propositional logic
     1.3 Quantifiers
          Uniqueness
          Proof methods based on quantifiers
          Translating statements into symbolic form
     1.4 First-order languages and theories
     1.5 Examples of first-order theories
     1.6 Normal forms and complexity
          Second-order logic and Skolem form
     1.7 Other logics
          Many-valued logic
          Fuzzy logic
          Modal logic
          Nonmonotonic logic
          Temporal logic

CHAPTER 2  Axiomatic Set Theory
     2.1 Introduction
     2.2 \"Naive\" set theory
     2.3 Zermelo-Fraenkel set theory
          Proper axioms of ZF set theory
          The regularity axiom
     2.4 Ordinals
     2.5 Cardinals and the cumulative hierarchy
          Von Neumann cardinals
          The cumulative hierarchy

CHAPTER 3  Recursion Theory and Computability
     3.1 Introduction
     3.2 Primitive recursive functions
     3.3 Turing machines and recursive functions
          The mu operator
     3.4 Undecidability and recursive enumerability
          Recursive enumerability
     3.5 Complexity theory
          Nondeterministic Turing machines, and P vs. NP

CHAPTER 4  Gödel\'s Incompleteness Theorems
     4.1 Introduction
     4.2 The arithmetization of formal theories
          The recursion theorem
     4.3 A potpourri of incompleteness theorems
          An historical perspective on Godel\'s work
          Godel\'s second incompleteness theorem
          Hilbert\'s formalist program, revisited
     4.4 Strengths and limitations of PA
          Ramsey\'s theorems and the Paris-Harrington results

CHAPTER 5  Model Theory
     5.1 Introduction
     5.2 Basic concepts of model theory
     5.3 The main theorems of model theory
          The Lowenheim-Skolem-Tarski theorem
     5.4 Preservation theorems
          Preservation under submodels and intersections
          Preservation under unions of chains
          Preservation under homomorphic images
          Preservation under direct products
     5.5 Saturation and complete theories
     5.6 Quantifier elimination
     5.7 Additional topics in model theory
          Axiomatizable and nonaxiomatizable classes
          Stone spaces
          Tarski\'s undefinability theorem
          Second-order model theory

CHAPTER 6  Contemporary Set Theory
     6.1 Introduction
          Some more history of set theory
     6.2 The relative consistency of AC and GCH
     6.3 Forcing and the independence results
     6.4 Modern set theory and large cardinals
          Large cardinals and the consistency of ZF
     6.5 Descriptive set theory
          Classical descriptive set theory
     6.6 The axiom of determinacy
          Infinite games
          Woodin\'s program

CHAPTER 7  Nonstandard Analysis
     7.1 Introduction
          \"Limits vs. infinitesimals\" through the ages
     7.2 Nonarchimedean fields
     7.3 Standard and nonstandard models
     7.4 Nonstandard methods in mathematics

CHAPTER 8  Constructive Mathematics
     8.1 Introduction
     8.2 Brouwer\'s intuitionism
     8.3 Bishop\'s constructive analysis
          Functions and continuity
          Differentiation
          Integration
APPENDIX A  A Deductive System for First-order Logic
     Logical axioms
     Rule of inference

APPENDIX B  Relations and Orderings
     Orderings
     Functions and equivalence relations
APPENDIX C  Cardinal Arithmetic
     Infinitary cardinal operations

APPENDIX D Groups, Rings, and Fields
     Groups
     Rings and fields
     Ordered algebraic structures
Bibliography
Symbols and Notation
Index
BackCover