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