《AXIOMATIC SET THEORY》

PART Ⅰ．HISTORICAL INTRODUCTION1

1．INTRODUCTORY REMARKS3

2．ZERMELO＇S SYSTEM．EQUALITY AND EXTENSIONALITY5

3．＂CONSTRUCTIVE＂AXIOMS OF＂GENERAL＂SET THEORY9

4．THE AXIOM OF CHOICE15

5．AXIOMS OF INFINITY AND OF RESTRICTION21

6．DEVELOPMENT OF SET-THEORY FROM THE AXIOMS OF Z26

7．REMARKS ON THE AXIOM SYSTEMS OF VON NEUMANN，BERNAYS，GODEL31

PART Ⅱ．AXIOMATIC SET THEORY37

INTRODUCTION39

CHAPTER Ⅰ．THE FRAME OF LOGIC AND CLASS THEORY45

1．Predicate Calculus；Class Terms and Descriptions；Explicit Definitions45

2．Equality and Extensionality．Application to Descriptions52

3．Class Formalism．Class Operations56

4．Functionality and Mappings61

CHAPTER Ⅱ．THE START OF GENERAL SET THEORY65

1．The Axicms of General Set Theory65

2．Aussonderungstheorem．Intersection69

3．Sum Theorem．Theorem of Replacement72

4．Functional Sets．One-to-one Correspondences76

CHAPTER Ⅲ．ORDINALS；NATURAL NUMBERS；FINITE SETS80

1．Fundaments of the Theory of Ordinals80

2．Existential Statements on Ordinals．Limit Numbers86

3．Fundaments of Number Theory89

4．Iteration．Primitive Recursion92

5．Finite Sets and Classes97

CHAPTER Ⅳ．TRANSFINITE RECURSION100

1．The General Recursion Theorem100

2．The Schema of Transfinite Recursion104

3．Generated Numeration109

CHAPTER Ⅴ．POWER；ORDER；WELLORDER114

1．Comparison of Powers114

2．Order and Partial Order118

3．Wellorder124

CHAPTER Ⅵ．THE COMPLETING AXIOMS130

1．The Potency Axiom130

2．The Axiom of Choice133

3．The Numeration Theorem．First Concepts of Cardinal Arithmetic138

4．Zorn＇s Lemma and Related Principles142

5．Axiom of Infinity．Denumerability147

CHAPTER Ⅶ．ANALYSIS；CARDINAL ARITHMETIC；ABSTRACT THEORIES155

1．Theory of Real Numbers155

2．Some Topics of Ordinal Arithmetic164

3．Cardinal Operations173

4．Formal Laws on Cardinals179

5．Abstract Theories188

CHAPTER Ⅷ．FURTHER STRENGTHENING OF THE AXIOM SYSTEM195

1．A Strengthening of the Axiom of Choice195

2．The Fundierungsaxiom200

3．A one-to-one Correspondence between the Class of Ordinals and the Class of all Sets203

INDEX OF AUTHORS(PART Ⅰ)211

INDEX OF SYMBOLS(PART Ⅱ)213

Predicates213

Functors and Operators214

Primitive Symbols215

INDEX OF MATTERS(PART Ⅱ)218

LIST OF AXIOMS(PART Ⅱ)218

BIBLIOGRAPHY(PART Ⅰ AND Ⅱ)219