Set Theory and Its Logic: Revised Edition

This is an extensively revised edition of W. V. Quine’s introduction to abstract set theory and to various axiomatic systematizations of the subject. The treatment of ordinal numbers has been strengthened and much simplified, especially in the theory of transfinite recursions, by adding an axiom and reworking the proofs. Infinite cardinals are treated anew in clearer and fuller terms than before. Improvements have been made all through the book; in various instances a proof has been shortened, a theorem strengthened, a space-saving lemma inserted, an obscurity clarified, an error corrected, a historical omission supplied, or a new event noted.

Table of Contents:
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

About the Author :
W. V. Quine was Edgar Pierce Professor of Philosophy, Harvard University. He wrote twenty-one books, thirteen of them published by Harvard University Press.

Review :
This is the masterpiece one would have expected it to be. For the expert it is a fresh and elegant treatise, brimming deliciously with new ideas and insights. For the beginner it is a brilliant gem of exposition, rendering a host of abstruse arguments crystal clear. For all it is a smooth and exciting journey to a vivid and comprehensive view of the alternative foundations of classical mathematics. Perhaps the most concise and readable general survey of axiomatic set theory at present available…Suitable reading not only for mathematics students…but also for philosophers with an interest in the foundations of mathematics. An excellent index and system of numbering formulae make it also a useful reference book. This revision of an important and lucid account of the various systems of axiomatic set theory preserves the basic format and essential ingredients of its highly regarded original…There have, however, been a number of important changes, generally in the interest of greater elegance and clarity…a generally improved version of an originally masterful and brilliant work. This book is most remarkable for its way of presenting the subject matter. A definite system of set theory is offered, but at the same time alternative ways are indicated and partly explored at every turn…The book is also remarkable for its style. Pithy, with never an unnecessary word (but with every necessary one), at times witty, the book is written in a way that is a great relief from ordinary textbooks. Quine's books always have style, but I consider this as one of the most successful from this point of view.