NICOLAS BOURBAKI

ELEMENTS OF MATHEMATICS

Theory of Sets

Springe]

CONTENTS

TO THE READER V

CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES vn

INTRODUCTION 7

CHAPTER I. DESCRIPTION OF FORMAL MATHEMATICS 15

§ 1. Terms and relations 15 1. Signs and assemblies 15 2. Criteria of Substitution 18 3. Formative constructions 19 4. Formative criteria 21

§ 2. Theorems 24

1. The axioms 24 2. Proofs 25 3. Substitutions in a theory 26 4. Comparison of theories 26

§ 3. Logical theories 28

1. Axioms 28 2. First consequences 29 3. Methods of proof 30 4. Gonjunction 33 5. Equivalence 34

§ 4. Quantified theories 36

1. Definition of quantifiers 36 2. Axioms of quantified theories 37 3. Properties of quantifiers 38 4. Typical quantifiers 41

I

CONTENTS

§ 5. Equalitarian theories 44

1. The axioms 44 2. Properties of equality 45 3. Functional relations 47

Appendix. Gharacterization of terms and relations 50

1. Signs and words 50 2. Significant words 51 3. Gharacterization of significant words 51 4. Application to assemblies in a mathematical theory . . 53

Exercises for § 1 56

Exercises for § 2 56

Exercises for § 3 57

Exercises for § 4 58

Exercises for § 5 59

Exercises for the Appendix 60

CHAPTER II . THEORY OF SETS 65

§ 1. Gollectivizing relations 65

1. The theory of sets 65 2. Inclusion 66 3. The axiom of extent 67 4. Collectivizing relations 67 5. The axiom of the set of two elements 69 6. The scheme of selection and union 69 7. Gomplement of a set. The empty set 71

§ 2. Ordered pairs 72

1. The axiom of the ordered pair 72 2. Product of two sets 74

§ 3. Gorrespondences 75

1. Graphs and correspondences 75 2. Inverse of a correspondence 78 3. Composition of two correspondences 78 4. Functions 81 5. Restrictions and extensions of functions 82 6. Definition of a function by means of a term 83 7. Composition of two functions. Inverse function 84

2

CONTENTS

8. Retractions and sections 86 9. Functions of two argumenta 89

§ 4. Union and intersection of a family of sets 90

1. Definition of the union and the intersection of a family ofsets 90

2. Properties of union and intersection 93 3. Images of a union and an intersection 94 4. Complements of unions and intersections 96 5. Union and intersection of two sets 96 6. Coverings 98 7. Partitions 99 8. Sum of a family of sets 100

§ 5. Product of a family of sets 101

1. The axiom of the set of subsets 101 2. Set of mappings of one set into another 102 3. Definitions of the product of a family ofsets 103 4. Partial products 105 5. Associativity of products ofsets 106 6. Distributivity formulae 107 7. Extension of mappings to products 111

§ 6. Equivalence relations 112

1. Definition of an equivalence relation 113 2. Equivalence classes; quotient set 114 3. Relations compatible with an equivalence relation . 116 4. Saturated subsets 117 5. Mappings compatible with equivalence relations 118 6. Inverse image of an equivalence relation; induced

equivalence relation 119 7. Quotients of equivalence relations 119 8. Product of two equivalence relations 120 9. Classes of equivalent objects 121

Exercises for § 1 123

Exercises for § 2 123

Exercises for § 3 124

Exercises for § 4 125

Exercises for § 5 127

Exercises for § 6 127

3

CONTENTS

CHAPTER II I . ORDERED SETS, CARDINALS, INTEGERS 131

§ 1. Order relations. Ordered sets 131

1. Definition of an order relation 131 2. Preorder relations 133 3. Notation and tenninology 135 4. Ordered subsets. Product of ordered sets 136 5. Increasing mappings 138 6. Maximal and minimal elements 139 7. Greatest dement and least dement 140 8. Upper and lower bounds 141 9. Least upper bound and greatest lower bound 141

10. Directed sets 145 11. Lattices 145 12. Totally ordered sets 146 13. Intervals 147

§ 2. Well-ordered sets 148

1. Segments of a wdl-ordered set 148 2. The principle of transfinite induction 151 3. Zermelo's theorem 152 4. Inductive sets 154 5. Isomorphisms of well-ordered sets 155 6. Lexicographic products 157

§ 3. Equipotent sets. Cardinals 157

1. The cardinal of a set , 157 2. Order relation between cardinals 158 3. Operations on cardinals 160 4. Properties of the cardinals 0 and 1 162 5. Exponentiation of cardinals 163 6. Order relation and Operations on cardinals 164

§ 4. Natural integers. Finite sets 166

1. Definition of integers 166 2. Inequalities between integers 166 3. The principle of induction 168 4. Finite subsets of ordered sets 170 5. Properties of finite character 170

§ 5. Properties of integers 171

1. Operations on integers and finite sets 171 2. Strict inequalities between integers 173 3. Intervals in sets of integers 174

4

CONTENTS

4. Finite sequences 175 5. Characteristic functions of sets 175 6. Euclidean division 176 7. Expansion to base b 177 8. Combinatorial analysis 179

§ 6. Infinite sets 183

1. The set of natural integere 183 2. Definition of mappings by induction 184 3. Properties of infinite cardinals 186 4. Gountable sets 188 5. Stationary sequences 189

§ 7. Inverse limits and direct limits 191

1. Inverse limits 191 2. Inverse Systems of mappings 192 3. Double inverse limit 196 4. Conditions for an inverse limit to be non-empty 198 5. Direct limits 201 6. Direct Systems of mappings 204 7. Double direct limit. Product of direct limits 209

Exercises for § 1 212

Exercises for § 2 221

Exercises for § 3 229

Exercises for § 4 230

Exercises for § 5 235

Exercises for § 6 241

Exercises for § 7 251

Historical Note on § 5 255

Bibliography 257

CHAPTER IV. STRUCTURES 259

§ 1. Structures and isomorphisms 259 1. Echelons 259 2. Canonical extensions of mappings 260 3. Transportable relations 261 4. Species of structures 262 5. Isomorphisms and transport of structures 264

5

CONTENTS

6. Deduction of structures 266 7. Equivalent species of structures 268

§ 2. Morphisms and derived structures 271

1. Morphisms 271 2. Finer structures 273 3. Initial structures 274 4. Examples of initial structures 276 5. Final structures 280 6. Examples of final structures 281

§ 3. Universal mappings 283

1. Universal sets and mappings 283 2. Existence of universal mappings 284 3. Examples of universal mappings 286

Exercises for § 1 289

Exercises for § 2 290

Exercises for § 3 294

Historical Note on Chapters I-IV 296

Bibliography 343

SüMMARY OF RESULTS 3 4 7

Introduction 347 § 1. Elements and subsets of a set 347 § 2. Functions 351 § 3. Products of sets 357 § 4. Union, intersection, product of a family of sets 363 § 5. Equivalence relations and quotient sets 370 § 6. Ordered sets 373 § 7. Powers. Countable sets 380 § 8. Scales of sets. Structures 383

Index of notation 387

Index of terminology 392

Axioms and schemes of the theory of sets 414

•

6