A Tour Through Mathematical Logic

• Main Author: Wolf, Robert S.
• Format: eBook Book
• Language: English
• Published: Providence, Rhode Island American Mathematical Society 2005; Mathematical Association of America
• Edition: 1
• Series: Carus Mathematical Monographs
• Subjects: Algebra, Abstract; Logic, Symbolic and mathematical; Mathematics

A Tour through Mathematical Logic provides a tour through the main branches of the foundations of mathematics. It contains chapters covering elementary logic, basic set theory, recursion theory, Gödel’s (and others’) incompleteness theorems, model theory, independence results in set theory, nonstand...

Table of Contents:

• Predicate Logic -- Axiomatic Set Theory -- Recursion Theory and Computability -- Gödel’s Incompleteness Theorems -- Model Theory -- Contemporary Set Theory -- Nonstandard Analysis -- Constructive Mathematics
• Front Matter Preface Table of Contents CHAPTER 1: Predicate Logic CHAPTER 2: Axiomatic Set Theory CHAPTER 3: Recursion Theory and Computability CHAPTER 4: Gödel’s Incompleteness Theorems CHAPTER 5: Model Theory CHAPTER 6: Contemporary Set Theory CHAPTER 7: Nonstandard Analysis CHAPTER 8: Constructive Mathematics APPENDIX A APPENDIX B APPENDIX C APPENDIX D Bibliography Symbols and Notation Index
• cover -- copyright page -- title page -- Preface -- Contents -- 1 Predicate Logic -- 1.1 Introduction -- A very brief history of mathematical logic -- Short Bio: Bertrand Russell -- 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 -- Short Bio: Euclid -- 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 -- 2 Axiomatic Set Theory -- 2.1 Introduction -- 2.2 "Naive" set theory -- Short Bio: Georg Ferdinand Cantor -- 2.3 Zermelo-Fraenkel set theory -- Proper axioms of ZF set theory -- Short bio: John von Neumann -- The regularity axiom -- 2.4 Ordinals -- 2.5 Cardinals and the cumulative hierarchy -- Von Neumann cardinals -- The cumulative hierarchy -- 3 Recursion Theory and Computability -- 3.1 Introduction -- Short Bio: Emil Post -- 3.2 Primitive recursive functions -- 3.3 Turing machines and recursive functions -- Short Bio: Alan Turing -- The mu operator -- 3.4 Undecidability and recursive enumerability -- Recursive enumerability -- 3.5 Complexity theory -- Nondeterministic Turing machines, and P vs. NP -- 4 Gödel's Incompleteness Theorems -- 4.1 Introduction -- 4.2 The arithmetization of formal theories -- Short Bio: Kurt Gödel -- The recursion theorem -- 4.3 A potpourri of incompleteness theorems -- An historical perspective on Gödel's work -- Gödel's second incompleteness theorem -- Hilbert's formalist program, revisited -- 4.4 Strengths and limitations of PA -- Ramsey's theorems and the Paris-Harrington results -- Short Bio: Frank Ramsey -- 5 Model Theory -- 5.1 Introduction -- 5.2 Basic concepts of model theory
• 5.3 The main theorems of model theory -- The Löwenheim-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 -- Short Bio: Julia Robinson -- 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 -- 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 -- Short Bio: Nikolai Luzin -- 6.6 The axiom of determinacy -- Infinite games -- Woodin's program -- 7 Nonstandard Analysis -- 7.1 Introduction -- "Limits vs. infinitesimals" through the ages -- Short Bio: Archimedes -- Short Bio: Leonhard Euler -- 7.2 Nonarchimedean fields -- 7.3 Standard and nonstandard models -- 7.4 Nonstandard methods in mathematics -- 8 Constructive Mathematics -- 8.1 Introduction -- 8.2 Brouwer's intuitionism -- Short Bio: L. E. J. Brouwer -- 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