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

Kenneth Kunen: Set Theory: An Introduction to Independence Proofs

Published $\text {1980}$, North-Holland

ISBN 0-444-85401-0

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