Book:Kenneth Kunen/Set Theory: An Introduction to Independence Proofs

Subject Matter

 * Set Theory

Contents

 * Preface
 * Contents


 * Introduction
 * $\S$1. Consistency results
 * $\S$2. Prerequisites
 * $\S$3. Outline
 * $\S$4. How to use this book
 * $\S$5. What has been omitted
 * $\S$6. On references
 * $\S$7. The axioms


 * Chapter I. The foundations of set theory
 * $\S$1. Why axioms?
 * $\S$2. Why formal logic?
 * $\S$3. The philosophy of mathematics
 * $\S$4. What we are describing
 * $\S$5. Extensionality and Comprehension
 * $\S$6. Relations, functions, and well-ordering
 * $\S$7. Ordinals
 * $\S$8. Remarks on defined notions
 * $\S$9. Classes and recursion
 * $\S$10. Cardinals
 * $\S$11. The real numbers
 * $\S$12. Appendix 1: Other set theories
 * $\S$13. Appendix 2: Eliminating defined notions
 * $\S$14. Appendix 3: Formalizing the metatheory
 * Exercises for Chapter I


 * Chapter II. Infinitary combinatorics
 * $\S$1. Almost disjoint and quasi-disjoint sets
 * $\S$2. Martin's Axiom
 * $\S$3. Equivalents of MA
 * $\S$4. The Suslin problem
 * $\S$5. Trees
 * $\S$6. The c. u. b. filter
 * $\S$7. $\diamond$ and $\diamond^+$
 * Exercises for Chapter II


 * Chapter III. The well-founded sets
 * $\S$1. Introduction
 * $\S$2. Properties of the well-founded sets
 * $\S$3. Well-founded relations
 * $\S$4. The Axiom of Foundation
 * $\S$5. Induction and recursion on well-founded relations
 * Exercises for Chapter III


 * Chapter IV. Easy consistency proofs
 * $\S$1. Three informal proofs
 * $\S$2. Relativization
 * $\S$3. Absoluteness
 * $\S$4. The last word on Foundation
 * $\S$5. More absoluteness
 * $\S$6. The $H(\kappa)$
 * $\S$7. Reflection theorems
 * $\S$8. Appendix 1: More on relativization
 * $\S$9. Appendix 2: Model theory in the metatheory
 * $\S$10. Appendix 3: Model theory in the formal theory
 * Exercises for Chapter IV


 * Chapter V. Defining definability
 * $\S$1. Formalizing definability
 * $\S$2. Ordinal definable sets
 * Exercises for Chapter V


 * Chapter VI. The constructible sets
 * $\S$1. Basic properties of $\mathbf L$
 * $\S$2. ZF in $\mathbf L$
 * $\S$3. The Axiom of Constructibility
 * $\S$4. AC and GCH in $\mathbf L$
 * $\S$5. $\diamond$ and $\diamond^+$ in $\mathbf L$
 * Exercises for Chapter VI


 * Chapter VII. Forcing
 * $\S$l.General remarks
 * $\S$2.Generic extensions
 * $\S$3.Forcing
 * $\S$4. ZFC in $M[G]$
 * $\S$5. Forcing with finite partial functions
 * $\S$6. Forcing with partial functions of larger cardinality
 * $\S$7. Embeddings, isomorphisms, and Boolean-valued models
 * $\S$8. Further results
 * $\S$9. Appendix: Other approaches and historical remarks
 * Exercises for Chapter VII


 * Chapter VIII. Iterated forcing
 * $\S$1. Products
 * $\S$2. More on the Cohen model
 * $\S$3. The independence of Kurepa's Hypothesis
 * $\S$4. Easton forcing
 * $\S$5. General iterated forci ng
 * $\S$6. The consistency of MA + $\neg$CH
 * $\S$7. Countable iterations
 * Exercises for Chapter VIII


 * Bibliography
 * Index or special symbols
 * General Index