SET THEORY (eBook)

Front Cover 1
Set Theory: With an Introduction to Descriptive Set Theory 4
Copyright Page 5
Contents 12
Preface to the first edition 6
Preface to the second edition 9
CHAPTER I. Algebra of sets 16
§ 1. Propositional calculus 16
§ 2. Sets and operations on sets 19
§ 3. Inclusion, Empty set 22
§ 4. Laws of union, intersection, and subtraction 25
§ 5. Properties of symmetric difference 28
§ 6. The set 1, complement 33
§ 7. Constituents 36
§ 8. Applications of the algebra of sets to topology 42
§ 9. Boolean algebras 48
10. Lattices 57
CHAPTER II. Axioms of set theory. Relations. Functions 61
§ 1. Set theoretical formulas. Quantifiers 61
§ 2. Axioms of set theory 67
§ 3. Some simple consequences of the axioms 73
§ 4. Cartesian products . Relations 77
§ 5. Equivalence relations. Partitions 81
§ 6. Functions 84
§ 7. Images and inverse images 89
§ 8. Functions consistent with a given equivalence relation. Factor Boolean algebras 93
§ 9. Order relations 95
§ 10. Relational systems, their isomorphisms and types 100
CHAPTER III. Natural numbers. Finite and infinite sets 104
§ 1. Natural numbers 104
§ 2. Definitions by induction 108
§ 3. The mapping J of the set N × N onto N and related mappings 113
§ 4. Finite and infinite sets 117
CHAPTER IV. Generalized union, intersection and Cartesian product 122
§ 1. Set-valued functions . Generalized union and intersection 122
§ 2. Operations on infinite sequences of sets 132
§ 3. Families of sets closed under given operations 136
§ 4. s-additive and d-multiplicative families of sets 139
§ 5. Reduction and separation properties 142
§ 6. Generalized Cartesian products 144
§ 7. Cartesian products of topological spaces 148
§ 8. The Tychonoff theorem 152
§ 9. Reduced direct products 155
§ 10. Infinite operations in lattices and in Boolean algebras 160
§ 11. Extensions of ordered sets to complete lattices 167
§ 12. Representation theory for distributive lattices 173
CHAPTER V. Theory of cardinal numbers 179
§ 1. Equipollence. Cardinal numbers 179
§ 2. Countable sets 184
§ 3. The hierarchy of cardinal numbers 189
§ 4. The arithmetic of cardinal numbers 193
§ 5. Inequalities between cardinal numbers. The Cantor–Bernstein theorem and its generalizations 196
§ 6. Properties of the cardinals a and c 203
§ 7. The generalized sum of cardinal numbers 206
§ 8. The generalized product of cardinal numbers 210
CHAPTER VI. Linearly ordered sets 216
§ 1. Introduction 216
§ 2. Dense, scattered, and continuous sets 220
§ 3. Order types ., ., and . 225
§ 4. Arithmetic of order types 232
§ 5. Lexicographical ordering 235
CHAPTER VII. Well-ordered sets 239
§ 1. Definitions. Principle of transfinite induction 239
§ 2. Ordinal numbers 243
§ 3. Transfinite sequences 245
§ 4. Definitions by transfinite induction 248
§ 5. Ordinal arithmetic 254
§ 6. Ordinal exponentiation 260
§ 7. Expansions of ordinal numbers for an arbitrary base 263
§ 8. The well-ordering theorem 269
§ 9. Von Neumann's method of elimination of ordinal numbers 277
CHAPTER VIII. Alephs and related topics 282
§ 1. Ordinal numbers of power a 282
§ 2. The cardinal K(m). Hartogs' aleph 285
§ 3. Initial ordinals 287
§ 4. Alephs and their arithmetic 290
§ 5. The exponentiation of alephs 295
§ 6. The exponential hierarchy of cardinal numbers 299
§ 7. The continuum hypothesis 305
§ 8. The number of prime ideals in the algebra P(A) 311
§ 9. m-disjoint sets 315
§ 10. Families of disjoint open sets 317
§ 11. Equivalence of certain statements about cardinal numbers with the axiom of choice 323
CHAPTER IX. Trees and partitions 330
§ 1. Trees 330
§ 2. The lexicographical ordering of zero-one sequences .. sets 334
§ 3. König's infinity lemma 341
§ 4. Arohszajn's trees 344
§ 5. Souslin trees 347
§ 6. Some partition theorems 351
CHAPTER X. Inaccessible cardinals 357
§ 1. Normal functions and stationary sets 357
§ 2. Weakly and strongly inaccessible cardinals 363
§ 3. A digression on models of S. [TR] 367
§ 4. Higher types of inaccessible numbers 371
§ 5. Weakly compact cardinals 375
§ 6. Measurable cardinals 381
§ 7. Measurable cardinals and reduced products 390
Incroduction to descriptive set theory 400
CHAPTER XI. Auxiliary notions 401
§ 1. The notion of a metric space. Various fundamental topological notions 401
§ 2. Exponential topology. Compact-open topology 407
§ 3. Complete and Polish spaces 411
§ 4. L-measurable mappings 415
§ 5. The operation A 424
§ 6. The Lusin sieve 427
CHAPTER XII. Borel sets. B-measurable functions. Baire property 430
§ 1. Elementary properties of Borel subsets of a metric space 430
§ 2. Ambiguous Borel sets 432
§ 3. Borel-measurable functions 434
§ 4. B-measurable complex and product functions 436
§ 5. Universal functions for Borel classes 438
§ 6. Borel subsets of Polish spaces 441
§ 7. Further properties of Borel sets 442
§ 8. Baire property 443
CHAPTER XIII. Souslin spaces. Projective sets 449
§ 1. Souslin spaces. Fundamental properties 449
§ 2. Applications of countable order types to Souslin spaces 459
§ 3. Coanalytic sets (CA-sets) 462
§ 4. The s-algebra S generated by Souslin sets and the S-measurablemappings 466
§ 5. The PCA-sets and sets of higher projective classes 470
CHAPTER XIV. Measurable selectors 473
§ 1. The general selector theorem 473
§ 2. Selectors for measurable partitions of Polish spaces 478
§ 3. Selectors for point-inverses of continuous mappings 481
Bibliography 491
List of important symbols 512
Subject index 517