Constructible Sets with Applications (eBook)

Lev D. Beklemishev (Autor)

eBook Download: PDF

2000 | 1. Auflage
268 Seiten
Elsevier Science (Verlag)
978-0-08-095776-0 (ISBN)

Infos

Provability, Computability and Reflection

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Constructible Sets with Applications

Inhalt

Front Cover 1
Constructible Sets With Applications 4
Copyright Page 5
CONTENTS 8
Chapter I. Axioms and auxiliary notions 12
1. Set theory ZF of Zermelo–Fraenkel 12
2. The meta-theory of classes 13
3. Definitions by transfinite induction. Ranks 19
4. Models, satisfaction 22
5. Derived semantical notions the Skolem–Löwenheim theorem
6. The contraction lemma 31
Chapter II. General principles of construction 33
1. Sufficient conditions for a class to be a model 33
2. The reflection theorem 34
3. Predicatively closed classes 38
4. The fundamental operations 40
Chapter III. Constructible sets 46
1. Enumeration 47
2. Constructible sets 50
3. Properties of constructible sets 51
4. Constructibility of ordinals 55
5. Models containing with each element its mappings into ordinals 57
6. Examples of functions satisfying (B.0)–(B.6) 59
7. Sets constructible in a class 59
Chapter IV. Functors and their definability 61
1. Strongly definable functors 62
2. Properties of strongly definable functors and relations 64
3. Examples of strongly definable functors 66
4. Definitions by transfinite induction 71
5. Why is all that necessary? 76
Chapter V. Constructible sets as values of a functor 80
1. Uniformly definable functions 80
2. Examples of uniformly definable functions 84
3. Uniform definability of the function C.B(a) 87
4. A generalization 92
5. Further properties of constructible sets 93
Chapter VI. Cp(A)(a) as a model 96
1. The reflection theorem again 96
2. Satisfiability of the power set axiom and of the axiom of substitution 99
3. Existence of models 104
4. Minimal models 107
Chapter VII. Consistency of the axiom of choice and of the continuum hypothesis 110
1. Axiom of choice 110
2. Auxiliary functors 113
3. Formulation of the generalized continuum hypothesis 116
4. A sufficient condition for the validity of GCH 118
5. Construction of models in which the continuum hypothesis is valid 119
6. Definability of the contracting function 121
7. A refinement of the Skolem–Löwenheim theorem 123
8. Consistency of GCH 126
9. Axioms of constructibility. Final remarks 129
Chapter VIII. Reduction of models 131
1. A reflection lemma 131
2. The validity of the power set axiom 135
Chapter IX. Generic points and forcing general theory
1. Auxiliary topological notions 139
2. Valuations 141
3. The forcing relation 143
4. A special valuation 145
5. Application of forcing to constructions of models 146
Chapter X. Polynomials 152
1. Polynomials 152
2. Reduction of Condition IV 156
3. Reduction of Condition V 158
Chapter XI. Explicit construction of polynomials for functions Bmin, Bo, BZ 164
1. The partial ordering < <
2. Auxiliary polynomials 166
3. Expressing EBaß and JBaß as polynomials of EBI(a,ß) and JBI(a,ß) 168
4. Generalization to the cases B = BZ and B = B0 173
5. Final reduction of Conditions IVand V 175
6. Appendix: list of the polynomials fj, j = 12 179
Chapter XII. Examples of models and of independence proofs 180
1. Examples of topological spaces 180
2. Examples of mappings p .a(p) 183
3. Proof of Condition VIII for sequences of the first and the second category 184
4. Examples of models 192
5. A theorem on generic points 195
6. Independence of the strong axiom of constructibility 198
Chapter XIII. The continuum hypothesis 200
1. Auxiliary notions concerning cardinals 201
2. The Souslin coefficient 203
3. The Souslin coefficient of product spaces 206
4. Relative cardinals, relative cofinality and relative Souslin coefficients 210
5. Determination of the relative Souslin coefficient 215
6. Absoluteness of cardinals and of the cofinality index 217
7. The function exp of a model 223
8. The independence of the continuum hypothesis 230
Chapter XIV. Independence of the axiom of choice 234
1. Action of homeomorphisms onto sets C.(a(p)) 234
2. Homeomorphisms and forcing 238
3. Invariance properties 240
4. Independence of the axiom of choice 240
5. The ordering of P(P(.)) 245
6. The existence of maximal ideals in P(.) 248
7. Cofinality of .1 252
Chapter XV. Problems of definability 259
1. Definable relations between ordinals 259
2. Non-definability of the well-orderings of P(.) 260
3. Definable well-ordered subsets of P(.) 262
Appendix 266
Bibliography 269
List of important symbols 271
Author index 277
Subject index 278

Produkt-Details

• Erscheint lt. Verlag: 1.4.2000
• Sprache: englisch
• Themenwelt: Informatik ► Software Entwicklung ► User Interfaces (HCI)
• Themenwelt: Informatik ► Theorie / Studium ► Algorithmen
• Themenwelt: Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-095776-5 / 0080957765
• ISBN-13: 978-0-08-095776-0 / 9780080957760