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.