Suitable for upper-level undergraduates, this accessible approach to set theory poses rigorous but simple arguments. Each definition is accompanied by commentary that motivates and explains new concepts. Starting with a repetition of the familiar arguments of elementary set theory, the level of abstract thinking gradually rises for a progressive increase in complexity.
A historical introduction presents a brief account of the growth of set theory, with special emphasis on problems that led to the development of the various systems of axiomatic set theory. Subsequent chapters explore classes and sets, functions, relations, partially ordered classes, and the axiom of choice. Other subjects include natural and cardinal numbers, finite and infinite sets, the arithmetic of ordinal numbers, transfinite recursion, and selected topics in the theory of ordinals and cardinals. This updated edition features new material by author Charles C. Pinter.
Author(s): Charles C Pinter
Series: Dover Books on Mathematics
Publisher: Dover Publications
Year: 2014
Language: English
Pages: 256
Tags: Set Theory;Pure Mathematics;Mathematics;Science & Math
Title Page ... 2
Copyright Page ... 3
Dedication ... 5
Contents ... 6
Preface ... 10
Chapter 0 Historical Introduction ... 12
1 THE BACKGROUND OF SET THEORY ... 12
2 THE PARADOXES ... 14
3 THE AXIOMATIC METHOD ... 16
4 AXIOMATIC SET THEORY ... 22
5 OBJECTIONS TO THE AXIOMATIC APPROACH. OTHER PROPOSALS ... 27
6 CONCLUDING REMARKS ... 34
Chapter 1 Classes and Sets ... 36
1 BUILDING SENTENCES ... 36
EXERCISES 1.1 ... 41
2 BUILDING CLASSES ... 42
3 THE ALGEBRA OF CLASSES ... 48
4 ORDERED PAIRS CARTESIAN PRODUCTS ... 54
5 GRAPHS ... 59
6 GENERALIZED UNION AND INTERSECTION ... 62
7 SETS ... 68
Chapter 2 Functions ... 73
1 INTRODUCTION ... 73
2 FUNDAMENTAL CONCEPTS AND DEFINITIONS ... 74
3 PROPERTIES OF COMPOSITE FUNCTIONS AND INVERSE FUNCTIONS ... 84
4 DIRECT IMAGES AND INVERSE IMAGES UNDER FUNCTIONS ... 90
5 PRODUCT OF A FAMILY OF CLASSES ... 95
6 THE AXIOM OF REPLACEMENT ... 100
Chapter 3 Relations ... 102
1 INTRODUCTION ... 102
2 FUNDAMENTAL CONCEPTS AND DEFINITIONS ... 102
3 EQUIVALENCE RELATIONS AND PARTITIONS ... 106
4 PRE-IMAGE, RESTRICTION AND QUOTIENT OF EQUIVALENCE RELATIONS ... 112
5 EQUIVALENCE RELATIONS AND FUNCTIONS ... 117
Chapter 4 Partially Ordered Classes ... 121
1 FUNDAMENTAL CONCEPTS AND DEFINITIONS ... 121
2 ORDER PRESERVING FUNCTIONS AND ISOMORPHISM ... 125
3 DISTINGUISHED ELEMENTS. DUALITY ... 131
4 LATTICES ... 137
5 FULLY ORDERED CLASSES. WELL-ORDERED CLASSES ... 143
6 ISOMORPHISM BETWEEN WELL-ORDERED CLASSES ... 147
Chapter 5 The Axiom of Choice and Related Principles ... 152
1 INTRODUCTION ... 152
2 THE AXIOM OF CHOICE ... 156
3 AN APPLICATION OF THE AXIOM OF CHOICE ... 159
4 MAXIMAL PRINCIPLES ... 161
5 THE WELL-ORDERING THEOREM ... 164
6 CONCLUSION ... 166
Chapter 6 The Natural Numbers ... 168
1 INTRODUCTION ... 168
2 ELEMENTARY PROPERTIES OF THE NATURAL NUMBERS ... 171
3 FINITE RECURSION ... 173
4 ARITHMETIC OF NATURAL NUMBERS ... 178
5 CONCLUDING REMARKS ... 185
Chapter 7 Finite and Infinite Sets ... 187
1 INTRODUCTION ... 187
2 EQUIPOTENCE OF SETS ... 192
3 PROPERTIES OF INFINITE SETS ... 195
4 PROPERTIES OF DENUMERABLE SETS ... 198
Chapter 8 Arithmetic of Cardinal Numbers ... 202
1 INTRODUCTION ... 202
2 OPERATIONS ON CARDINAL NUMBERS ... 204
3 ORDERING OF THE CARDINAL NUMBERS ... 210
4 SPECIAL PROPERTIES OF INFINITE CARDINAL NUMBERS ... 214
5 INFINITE SUMS AND PRODUCTS OF CARDINAL NUMBERS ... 218
5 INFINITE SUMS AND PRODUCTS OF CARDINAL NUMBERS ... 218
Chapter 9 Arithmetic of the Ordinal Numbers ... 223
1 INTRODUCTION ... 223
2 OPERATIONS ON ORDINAL NUMBERS ... 226
3 ORDERING OF THE ORDINAL NUMBERS ... 233
4 THE ALEPHS AND THE CONTINUUM HYPOTHESIS ... 241
5 CONSTRUCTION OF THE ORDINALS AND CARDINALS ... 243
Chapter 10 Transfinite Recursion. Selected Topics in the Theory of Ordinals ... 251
1 TRANSFINITE RECURSION ... 251
2 PROPERTIES OF ORDINAL EXPONENTIATION ... 256
3 NORMAL FORM ... 263
4 EPSILON NUMBERS ... 269
5 INACCESSIBLE ORDINALS AND CARDINALS ... 275
Chapter 11 Consistency and Independence in Set Theory ... 285
1 WHAT IS A SET? ... 285
2 MODELS ... 293
3 INDEPENDENCE RESULTS IN SET THEORY ... 295
4 THE QUESTION OF MODELS OF SET THEORY ... 296
5 PROPERTIES OF THE CONSTRUCTIBLE UNIVERSE ... 300
6 THE GÖDEL THEOREMS ... 309
Bibliography ... 312
Index ... 313
