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INTRODUCTION 1

Part One．The Elements 7

Ⅰ．LOGIC 9

1．Quantification and identity 9

2．Virtual classes 15

3．Virtual relations 21

Ⅱ．REAL CLASSES 28

4．Reality，extensionality，and the individual 28

5．The virtual amid the real 34

6．Identity and substitution 40

Ⅲ．CLASSES OF CLASSES 47

7．Unit classes 47

8．Unions，intersections，descriptions 53

9．Relations as classes of pairs 58

10．Functions 65

Ⅳ．NATURAL NUMBERS 74

11．Numbers unconstrued 74

12．Numbers construed 81

13．Induction 86

Ⅴ．ITERATION AND ARITHMETIC 95

14．Sequences and iterates 95

15．The ancestral 100

16．Sum，product，power 106

Part Two．Higher Forms of Number 117

Ⅵ．REAL NUMBERS 119

17．Program．Numerical pairs 119

18．Ratios and reals construed 124

19．Existential needs．Operations and extensions 130

Ⅶ．ORDER AND ORDINALS 139

20．Transfinite induction 139

21．Order 145

22．Ordinal numbers 150

23．Laws of ordinals 158

24．Their well-ordering and some consequences 165

Ⅷ．TRANSFINITE RECURSION 171

25．Transfinite recursion 171

26．Laws of transfinite recursion 177

27．Enumeration 184

Ⅸ．CARDINAL NUMBERS 193

28．Comparative size of classes 193

29．The Schroder-Bernstein theorem 203

30．Infinite cardinal numbers 208

Ⅹ．THE AXIOM OF CHOICE 217

31．Selections and selectors 217

32．Further equivalents of the axiom 224

33．The place of the axiom 231

Part Three．Axiomatic Theories 239

Ⅸ．RUSSELL＇S THEORY OF TYPES 241

34．The constructive part 241

35．Classes and the axiom of reducibility 249

36．The modern theory of types 259

Ⅻ．GENERAL VARIABLES AND ZERMELO 266

37．The theory of types with general variables 266

38．Cumulative types and Zermelo 272

39．Axioms of infinity and others 279

ⅩⅢ．STRATIFICATION AND ULTIMATE CLASSES 287

40．＂New foundations＂ 287

41．Non-Cantorian classes．Induction again 292

42．Ultimate classes added 299

ⅩⅣ．VON NEUMANN＇S SYSTEM AND OTHERS 310

43．The von Neumann-Bernays system 310

44．Departures and comparisons 315

45．Strength of systems 323

SYNOPSIS OF FIVE AXIOM SYSTEMS 331

LIST OF NUMBERED FORMULAS 333

BIBLIOGRAPHICAL REFERENCES 343

INDEX 351