E-Book, Englisch, Band Volume 102, 330 Seiten, Web PDF
Kunen Set Theory An Introduction To Independence Proofs
1. Auflage 2014
ISBN: 978-0-08-057058-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark
E-Book, Englisch, Band Volume 102, 330 Seiten, Web PDF
Reihe: Studies in Logic and the Foundations of Mathematics
ISBN: 978-0-08-057058-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark
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.
Autoren/Hrsg.
Weitere Infos & Material
1;Front Cover;1
2;Set Theory: An Introduction to Independence Proofs;4
3;Copyright Page;5
4;Table of Contents;9
5;Dedication;6
6;Preface;8
7;INTRODUCTION;12
7.1;1. Consistency results;12
7.2;2. Prerequisites;13
7.3;3. Outline;13
7.4;4. How to use this book;14
7.5;5. What has been omitted;15
7.6;6. On references;15
7.7;7. The axioms;16
8;CHAPTER I. THE FOUNDATIONS OF SET THEORY;18
8.1;1. Why axioms?;18
8.2;2. Why formal logic?;19
8.3;3. The philosophy of mathematics;23
8.4;4. What we are describing;25
8.5;5. Extensionality and Comprehension;27
8.6;6. Relations, functions, and well-ordering;29
8.7;7. Ordinals;33
8.8;8. Remarks on defined notions;39
8.9;9. Classes and recursion;40
8.10;10. Cardinals;44
8.11;11. The real numbers;52
8.12;12. Appendix 1: Other set theories;52
8.13;13. Appendix 2: Eliminating defined notions;53
8.14;14. Appendix 3: Formalizing the metatheory;55
8.15;EXERCISES;59
9;CHAPTER II. INFINITARY COMBINATORICS;64
9.1;1. Almost disjoint and quasi-disjoint sets;64
9.2;2. Martin's Axiom;68
9.3;3. Equivalents of MA;79
9.4;4. The Suslin problem;83
9.5;5. Trees;85
9.6;6. The c.u.b. filter;93
9.7;7. . and . +;97
9.8;EXERCISES;103
10;CHAPTER III. THE WELL-FOUNDED SETS;111
10.1;1. Introduction;111
10.2;2. Properties of the well-founded sets;112
10.3;3. Well-founded relations;115
10.4;4. The Axiom of Foundation;117
10.5;5. Induction and recursion on well-founded relations;119
10.6;EXERCISES;124
11;CHAPTER IV. EASY CONSISTENCY PROOFS;127
11.1;1. Three informal proofs;127
11.2;2. Relativization;129
11.3;3. Absoluteness;134
11.4;4. The last word on Foundation;141
11.5;5. More absoluteness;142
11.6;6. The H (k);147
11.7;7. Reflection theorems;150
11.8;8. Appendix 1: More on relativization;158
11.9;9. Appendix 2: Model theory in the metatheory;159
11.10;10. Appendix 3: Model theory in the formal theory;160
11.11;EXERCISES;163
12;CHAPTER V. DEFINING DEFINABILITY;169
12.1;1. Formalizing definability;170
12.2;2. Ordinal definable sets;174
12.3;EXERCISES;180
13;CHAPTER VI. THE CONSTRUCTIBLE SETS;182
13.1;1. Basic properties of L;182
13.2;2. ZF in L;186
13.3;3. The Axiom of Constructibility;187
13.4;4. AC and GCH in L;190
13.5;5. . and . + in L;194
13.6;EXERCISES;197
14;CHAPTER VII. FORCING;201
14.1;1. General remarks;201
14.2;2. Generic extensions;203
14.3;3. Forcing;209
14.4;4. ZFC in M [G];218
14.5;5. Forcing with finite partial functions;221
14.6;6. Forcing with partial functions of larger cardinality;228
14.7;7. Embeddings, isomorphisms, and Boolean-valued models;234
14.8;8. Further results;243
14.9;9. Appendix: Other approaches and historical remarks;249
14.10;EXERCISES;254
15;CHAPTER VIII. ITERATED FORCING;268
15.1;1. Products;269
15.2;2. More on the Cohen model;272
15.3;3. The independence of Kurepa's Hypothesis;276
15.4;4. Easton forcing;279
15.5;5. General iterated forcing;285
15.6;6. The consistency of MA + ¬ CH;295
15.7;7. Countable support iterations;298
15.8;EXERCISES;304
16;BIBLIOGRAPHY;322
17;INDEX OF SPECIAL SYMBOLS;326
18;GENERAL INDEX;328
