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

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Kenneth Kunen: Set Theory: An Introduction to Independence Proofs

Published $1980$, North-Holland

ISBN 0-444-85401-0.


Subject Matter


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