Set Theory And Foundations Of Mathematics: An Introduction To Mathematical Logic - Volume Ii: Foundations Of Mathematics

A "This book provides an introduction to axiomatic set theory and descriptive set theory. It is written for the upper level undergraduate or beginning graduate students to help them prepare for advanced study in set theory and mathematical logic as well as other areas of mathematics, such as analysis, topology, and algebra. The book is designed as a flexible and accessible text for a one-semester introductory course in set theory, where the existing alternatives may be more demanding or specialized. Readers will learn the universally accepted basis of the field, with several popular topics added as an option. Pointers to more advanced study are scattered throughout the text"-- Provided by publisher Contents Preface About the Authors 1. Introduction 2. Propositional Logic 2.1. Basic Definitions 2.2. Truth Interpretations 2.3. The Disjunctive Normal Form Theorem 2.4. The Deductive Calculus 2.5. The Soundness Theorem 2.6. The Completeness Theorem 2.7. Completeness, Consistency, and Independence 2.8. Logical vs. Topological Compactness 3. Predicate Logic 3.1. The Language of Predicate Logic 3.2. Structures 3.3. The Deductive Calculus 3.4. Soundness Theorem for Predicate Logic 4. Models of Predicate Logic 4.1. The Completeness Theorem for Predicate Logic 4.2. Compactness 4.3. Isomorphism and Elementary Equivalence 4.4. Models, Theories, and Axioms 4.5. Categoricity and Quantifier Elimination 4.6. Examples of Theories and Structures 5. Boolean Algebras 5.1. Properties and Examples of Boolean Algebras 5.2. The Partial Ordering on a Boolean Algebra 5.3. Filters and Ideals 5.4. Ultraproducts 6. Computability 6.1. Finite State Machines 6.2. Turing Machines 6.3. Recursive Functions 6.4. The Halting Problem 7. Decidable and Undecidable Theories 7.1. Decidable vs. Undecidable Logical Systems 7.2. Decidable Theories 7.3. G ̈odel’s Incompleteness Theorems 8. Algorithmic Randomness 8.1. Kolmogorov Complexity 8.2. Incompressible Strings 8.3. Kolmogorov Complexity and Incompleteness 9. Nonstandard Numbers 9.1. Nonstandard Natural Numbers 9.2. Nonstandard Analysis 10. Foundations of Geometry 10.1. Axioms of Plane Geometry 10.2. Non-Euclidean Models 10.3. Finite Geometries Bibliography Index "This book provides an introduction to mathematical logic and the foundations of mathematics. It will help prepare students for advanced study in set theory and mathematical logic as well as other areas of mathematics, such as analysis, topology, and algebra. The presentation of finite state and Turing machines leads to the Halting Problem and Gödel's Incompleteness Theorem, which have broad academic interest, particularly in computer science and philosophy" Introduction -- Review of sets and logic -- Zermelo-Fraenkel set theory -- natural numbers and countable sets -- Ordinal numbers and the transfinite -- Cardinality and the axiom of choice -- Real numbers -- Models of set theory -- Ramsey theory