INTRODUCTION

**PART ONE. THE ELEMENTS**

I. LOGIC

*Quantification and identity*

*Virtual classes*

*Virtual relations*

II. REAL CLASSES

*Reality, extensionality, and the individual*

*The virtual amid the real*

*Identity and substitution*

III. CLASSES OF CLASSES

*Unit classes*

*Unions, intersections, descriptions*

*Relations as classes of pairs*

*Functions*

IV. NATURAL NUMBERS

*Numbers unconstrued*

*Numbers construed*

*Induction*

V. ITERATION AND ARITHMETIC

*Sequences and iterates*

*The ancestral*

*Sum, product, power*

**PART TWO. HIGHER FORMS OF NUMBER**

VI. REAL NUMBERS

*Program. Numerical pairs*

*Ratios and reals construed*

*Existential needs. Operations and extensions*

VII. ORDER AND ORDINALS

*Transfinite induction*

*Order*

*Ordinal numbers*

*Laws of ordinals*

*The order of the ordinals*

VIII. TRANSFINITE RECURSION

*Transfinite recursion*

*Laws of transfinite recursion*

*Enumeration*

IX. CARDINAL NUMBERS

*Comparative size of classes*

*The SchrOder-Bernstein theorem*

*Infinite cardinal numbers*

X. THE AXIOM OF CHOICE

*Selections and selectors*

*Further equivalents of the axiom*

*The place of the axiom*

**PART THREE. AXIOM SYSTEMS**

XI. RUSSELL'S THEORY OF TYPES

*The constructive part*

*Classes and the axiom of reducibility*

*The modern theory of types*

XII. GENERAL VARIABLES AND ZERMELO

*The theory of types with general variables*

*Cumulative types and Zermelo*

*Axioms of infinity and others*

XIII. STRATIFICATION AND ULTIMATE CLASSES

*"New foundations"*

*Non-Cantorian classes. Induction again*

*Ultimate classes added*

XIV. VON NEUMANN'S SYSTEM AND OTHERS

*The von Neumann-Bernays system*

*Departures and comparisons*

*Strength of systems*

SYNOPSIS OF FIVE AXIOM SYSTEMS

LIST OF NUMBERED FORMULAS

BIBLIOGRAPHICAL REFERENCES

INDEX