SET THEORY (eBook)

Lev D. Beklemishev (Autor)

eBook Download: PDF

2000 | 1. Auflage
416 Seiten
Elsevier Science (Verlag)
978-0-08-095772-2 (ISBN)

Infos

Provability, Computability and Reflection

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SET THEORY

Inhalt

Front Cover 1
Set Theory 4
Copyright Page 5
CONTENTS 10
Chapter I. Algebra of sets 14
1. Propositional calculus 14
2. Sets and operations on sets 17
3. Inclusion. Empty set 20
4. Laws of union, intersection. and subtraction 23
5. Properties of symmetric difference 26
6. The set 1, complement 31
7. Constituents 33
8. Applications of the algebra of sets to topology 39
9. Boolean algebras 45
10. Lattices 54
Chapter II. Axioms of set theory. Relations. Functions 58
1. Propositional functions. Quantifiers 58
2. Axioms of set theory 64
3. Some simple consequences of the axioms 71
4. Cartesian products. Relations 75
5. Equivalence relations 79
6. Functions 82
7. Images and inverse images 87
8. Functions consistent with a given equivalence relation. Factor Boolean algebras 91
9. Order relations 93
10. Relational systems. their isomorphisms and types 97
Chapter III. Natural numbers. Finite and infinite sets 101
1. Natural numbers 101
2. Definitions by induction 105
3. The mapping J of the set N × N onto N and related mappings 109
4. Finite and infinite sets 113
5. König's infinity lemma 117
6. Graphs. Ramsey's theorem 120
Chapter IV. Generalized union, intersection and cartesian product 124
1. Generalized union and intersection 124
2. Operations on infinite sequences of sets 133
3. Families of sets closed under given operations 139
4. s–additive and s–multiplicative families of sets 141
5. Generalized cartesian products 144
6. Cartesian products of topological spaces 148
7. The Tychonoff theorem 152
8. Reduced direct products 155
9. Inverse systems and their limits 160
10. Infinite operations in lattices and in Boolean algebras 162
11. Extensions of ordered sets to complete lattices 170
12. Representation theory for distributive lattices 175
Chapter V. Theory of cardinal numbers 182
1. Equipollence of Cardinal numbers 182
2. Countable sets 187
3. The hierarchy of cardinal numbers 192
4. The arithmetic of cardinal numbers 195
5. Inequalities between cardinal numbers. The Cantor-Bernstein theorem and its generalizations 198
6. Properties of the cardinals a and c 206
7. The generalized sum of cardinal numbers 209
8. The generalized product of cardinal numbers 214
Chapter VI. Linearly ordered sets 219
1. Introduction 219
2. Dense, scattered, and continuous sets 223
3. Order types ., . and . 229
4. Arithmetic of order types 234
5. Lexicographical ordering 237
Chapter VII. Well-ordered sets 241
1. Definitions. Principle of transfinite induction 241
2. Ordinal numbers 245
3. Transfinite sequences 248
4. Definitions by transfinite induction 252
5. Ordinal arithmetic 259
6. Ordinal exponentiation 265
7. Expansions of ordinal numbers for an arbitrary base 268
8. The well-ordering theorem 274
9. Von Neumann's method of elimination of ordinal numbers 282
Chapter VIII. Alephs and related topics 287
1. Ordinal numbers of power a 287
2. The cardinal N(m). Hartogs' aleph 290
3. Initial ordinals 292
4. Alephs and their arithmetic 295
5. The exponentiation of alephs 300
6. Equivalence of certain statements about cardinal numbers and the axiom of choice 304
7. The exponential hierarchy of cardinal numbers 310
8. Miscellaneous problems of power associated with Boolean algebras 316
Chapter IX. Inaccessible cardinals. The continuum hypothesis 321
1. Inaccessible cardinals 321
2. Classification of inaccessible cardinals 327
3. Measurability of cardinal numbers 328
4. The non-measurability of the first inaccessible aleph 333
5. The axiomatic introduction of inaccessible cardinals 339
6. The continuum hypothesis 341
7. ..-Sets 347
Chapter X. Introduction to the theory of analytic and projective sets 353
1. The operation (A) 353
2. The family A(R) 357
3. Hausdorff operations 361
4. Analytic sets 364
5. Projective sets 370
6. Universal functions 380
7. Sieves 388
8. Constituents 394
9. Universal sieve and the function t 398
10. The reduction theorem and the second separation theorem 403
11. The problem of projectivity for sets defined by transfinite induction 405
List of important symbols 412
Author index 418
Subject index 420

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: 0-08-095772-2 / 0080957722
• ISBN-13: 978-0-08-095772-2 / 9780080957722