The foundations of mathematics include mathematical logic, set theory, recursion theory, model theory, and Gödel's incompleteness theorems. Professor Wolf provides here a guide that any interested reader with some post-calculus experience in mathematics can read, enjoy, and learn from. It could also serve as a textbook for courses in the foundations of mathematics, at the undergraduate or graduate level. The book is deliberately less structured and more user-friendly than standard texts on foundations, so will also be attractive to those outside the classroom environment wanting to learn about the subject.
Author(s): Robert S. Wolf
Series: Carus Mathematical Monographs 30
Publisher: The Mathematical Association of America
Year: 2005
Language: English
Pages: xvi+397
Cover
S Title
Copyright
© 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
