Basic Set Theory (eBook)

Part A. Pure Set Theory Chapter I. The Basic Notions 1. The Basic Language of Set Theory 2. The Axioms of Extensionality and Comprehension 3. Classes, Why and How 4. Classes, the formal Introduction 5. The Axioms of Set Theory 6. Relations and functions Chapter II. Order and Well-Foundedness 1. Order 2. Well-Order 3. Ordinals 4. Natural Numbers and finite Sequences 5. Well-Founded Relations 6. Well-Founded Sets 7. The Axiom of Foundation Chapter III. Cardinal Numbers 1. Finite Sets 2. The Partial Order of the Cardinals 3. The Finite Arithmetic of the Cardinals 4. The Infinite Arithmetic of the Well Orderd Cardinals Chapter IV. The Ordinals 1. Ordinal Addition and Multiplication 2. Ordinal Exponentiation 3. Cofinality and Regular Ordinals 4. Closed Unbounded Classes and Stationery Classes Chapter V. The Axiom of Choice and Some of Its Consequences 1. The Axiom of Choice and Equivalent Statements 2. Some Weaker Versions of the Axiom of Choice 3. Definable Sets 4. Set Theory with Global Choice 5. Cardinal ExponentiationPart B. Applications and Advanced Topics Chapter VI. A Review of Point Set Topology 1. Basic concepts 2. Useful Properties and Operations 3. Category, Baire and Borel Sets Chapter VII. The Real Spaces 1. The Real Numbers 2. The Separable Complete Metric Spaces 3. The Close Relationship Between the Real Numbers, the Cantor Space and the Baire Space Chapter VIII. Boolean Algebras 1. The Basic Theory 2. Prime Ideals and Representation 3. Complete Boolean Algebras 4. Martin's Axiom Chapter IX. Infinite Combinatorics and Large Cardinals 1. The Axiom of Constructibility 2. Trees 3. Partition Properties 4. Measurable CardinalsAppendix X. The Eliminability and Conservation Theorems Bibliography; Additional Bibliography; Index of Notation; IndexAppendix Corrections and Additions