Set Theory and Its Logic: Revised Edition

by Willard Van Orman Quine

Overview

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.

Product Details

ISBN-13:	9780674802070
Publisher:	Harvard University Press
Publication date:	01/01/1971
Edition description:	Revised
Pages:	380
Product dimensions:	5.88(w) x 9.25(h) x 0.90(d)

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.

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

What People are Saying About This

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.

Jean van Heijenoort

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.

Editorial Reviews

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.
— A. A. Treherne

Proceedings of the Edinburgh Mathematical Society

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.
— Joseph S. Ullian

Philosophical Review

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.

Review of Metaphysics

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.

Jean van Heijenoort

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.

Proceedings of the Edinburgh Mathematical Society - A. A. Treherne