Set Theory An Introduction To Independence Proofs (eBook)

K. Kunen (Autor)

eBook Download: PDF

2014 | 1. Auflage
330 Seiten
Elsevier Science (Verlag)
978-0-08-057058-7 (ISBN)

Infos

Studies in Logic and the Foundations of Mathematics, Volume 102: Set Theory: An Introduction to Independence Proofs offers an introduction to relative consistency proofs in axiomatic set theory, including combinatorics, sets, trees, and forcing. The book first tackles the foundations of set theory and infinitary combinatorics. Discussions focus on the Suslin problem, Martin's axiom, almost disjoint and quasi-disjoint sets, trees, extensionality and comprehension, relations, functions, and well-ordering, ordinals, cardinals, and real numbers. The manuscript then ponders on well-founded sets and easy consistency proofs, including relativization, absoluteness, reflection theorems, properties of well-founded sets, and induction and recursion on well-founded relations. The publication examines constructible sets, forcing, and iterated forcing. Topics include Easton forcing, general iterated forcing, Cohen model, forcing with partial functions of larger cardinality, forcing with finite partial functions, and general extensions. The manuscript is a dependable source of information for mathematicians and researchers interested in set theory.

Inhalt

Front Cover 1
Set Theory: An Introduction to Independence Proofs 4
Copyright Page 5
Table of Contents 9
Dedication 6
Preface 8
INTRODUCTION 12
1. Consistency results 12
2. Prerequisites 13
3. Outline 13
4. How to use this book 14
5. What has been omitted 15
6. On references 15
7. The axioms 16
CHAPTER I. THE FOUNDATIONS OF SET THEORY 18
1. Why axioms? 18
2. Why formal logic? 19
3. The philosophy of mathematics 23
4. What we are describing 25
5. Extensionality and Comprehension 27
6. Relations, functions, and well-ordering 29
7. Ordinals 33
8. Remarks on defined notions 39
9. Classes and recursion 40
10. Cardinals 44
11. The real numbers 52
12. Appendix 1: Other set theories 52
13. Appendix 2: Eliminating defined notions 53
14. Appendix 3: Formalizing the metatheory 55
EXERCISES 59
CHAPTER II. INFINITARY COMBINATORICS 64
1. Almost disjoint and quasi-disjoint sets 64
2. Martin's Axiom 68
3. Equivalents of MA 79
4. The Suslin problem 83
5. Trees 85
6. The c.u.b. filter 93
7. . and . + 97
EXERCISES 103
CHAPTER III. THE WELL-FOUNDED SETS 111
1. Introduction 111
2. Properties of the well-founded sets 112
3. Well-founded relations 115
4. The Axiom of Foundation 117
5. Induction and recursion on well-founded relations 119
EXERCISES 124
CHAPTER IV. EASY CONSISTENCY PROOFS 127
1. Three informal proofs 127
2. Relativization 129
3. Absoluteness 134
4. The last word on Foundation 141
5. More absoluteness 142
6. The H (k) 147
7. Reflection theorems 150
8. Appendix 1: More on relativization 158
9. Appendix 2: Model theory in the metatheory 159
10. Appendix 3: Model theory in the formal theory 160
EXERCISES 163
CHAPTER V. DEFINING DEFINABILITY 169
1. Formalizing definability 170
2. Ordinal definable sets 174
EXERCISES 180
CHAPTER VI. THE CONSTRUCTIBLE SETS 182
1. Basic properties of L 182
2. ZF in L 186
3. The Axiom of Constructibility 187
4. AC and GCH in L 190
5. . and . + in L 194
EXERCISES 197
CHAPTER VII. FORCING 201
1. General remarks 201
2. Generic extensions 203
3. Forcing 209
4. ZFC in M [G] 218
5. Forcing with finite partial functions 221
6. Forcing with partial functions of larger cardinality 228
7. Embeddings, isomorphisms, and Boolean-valued models 234
8. Further results 243
9. Appendix: Other approaches and historical remarks 249
EXERCISES 254
CHAPTER VIII. ITERATED FORCING 268
1. Products 269
2. More on the Cohen model 272
3. The independence of Kurepa's Hypothesis 276
4. Easton forcing 279
5. General iterated forcing 285
6. The consistency of MA + ¬ CH 295
7. Countable support iterations 298
EXERCISES 304
BIBLIOGRAPHY 322
INDEX OF SPECIAL SYMBOLS 326
GENERAL INDEX 328

Produkt-Details

• Erscheint lt. Verlag: 28.6.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Technik
• ISBN-10: 0-08-057058-5 / 0080570585
• ISBN-13: 978-0-08-057058-7 / 9780080570587