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PART Ⅰ．HISTORICAL INTRODUCTION 1

1．INTRODUCTORY REMARKS 3

2．ZERMELO＇S SYSTEM．EQUALITY AND EXTENSIONALITY 5

3．＂CONSTRUCTIVE＂AXIOMS OF＂GENERAL＂SET THEORY 9

4．THE AXIOM OF CHOICE 15

5．AXIOMS OF INFINITY AND OF RESTRICTION 21

6．DEVELOPMENT OF SET-THEORY FROM THE AXIOMS OF Z 26

7．REMARKS ON THE AXIOM SYSTEMS OF VON NEUMANN，BERNAYS，GODEL 31

PART Ⅱ．AXIOMATIC SET THEORY 37

INTRODUCTION 39

CHAPTER Ⅰ．THE FRAME OF LOGIC AND CLASS THEORY 45

1．Predicate Calculus；Class Terms and Descriptions；Explicit Definitions 45

2．Equality and Extensionality．Application to Descriptions 52

3．Class Formalism．Class Operations 56

4．Functionality and Mappings 61

CHAPTER Ⅱ．THE START OF GENERAL SET THEORY 65

1．The Axicms of General Set Theory 65

2．Aussonderungstheorem．Intersection 69

3．Sum Theorem．Theorem of Replacement 72

4．Functional Sets．One-to-one Correspondences 76

CHAPTER Ⅲ．ORDINALS；NATURAL NUMBERS；FINITE SETS 80

1．Fundaments of the Theory of Ordinals 80

2．Existential Statements on Ordinals．Limit Numbers 86

3．Fundaments of Number Theory 89

4．Iteration．Primitive Recursion 92

5．Finite Sets and Classes 97

CHAPTER Ⅳ．TRANSFINITE RECURSION 100

1．The General Recursion Theorem 100

2．The Schema of Transfinite Recursion 104

3．Generated Numeration 109

CHAPTER Ⅴ．POWER；ORDER；WELLORDER 114

1．Comparison of Powers 114

2．Order and Partial Order 118

3．Wellorder 124

CHAPTER Ⅵ．THE COMPLETING AXIOMS 130

1．The Potency Axiom 130

2．The Axiom of Choice 133

3．The Numeration Theorem．First Concepts of Cardinal Arithmetic 138

4．Zorn＇s Lemma and Related Principles 142

5．Axiom of Infinity．Denumerability 147

CHAPTER Ⅶ．ANALYSIS；CARDINAL ARITHMETIC；ABSTRACT THEORIES 155

1．Theory of Real Numbers 155

2．Some Topics of Ordinal Arithmetic 164

3．Cardinal Operations 173

4．Formal Laws on Cardinals 179

5．Abstract Theories 188

CHAPTER Ⅷ．FURTHER STRENGTHENING OF THE AXIOM SYSTEM 195

1．A Strengthening of the Axiom of Choice 195

2．The Fundierungsaxiom 200

3．A one-to-one Correspondence between the Class of Ordinals and the Class of all Sets 203

INDEX OF AUTHORS(PART Ⅰ) 211

INDEX OF SYMBOLS(PART Ⅱ) 213

Predicates 213

Functors and Operators 214

Primitive Symbols 215

INDEX OF MATTERS(PART Ⅱ) 218

LIST OF AXIOMS(PART Ⅱ) 218

BIBLIOGRAPHY(PART Ⅰ AND Ⅱ) 219