Set Theory and Its Logic

384 Pages·1969·7.59 MB·English

by  Quine, Willard Van Orman

SET THEORY AND ITS LOGIC

SET THEORY
AND ITS LOGIC
Revised Edition
W ILLARD VAN ORMAN QUINE
EDGAR PIERCE PROFESSOR OF PHILOSOPHY
HARVARD UNIVERSITY
THE BELKNAP PRESS
OF HARVARD UNIVERSITY PRESS
CAMBRIDGE, MASSACHUSETTS
LONDON. ENGLAND
& Copyright 1 963 and 1969 by the President
and Fellows of Harvard College
All rights reserved
1 .ibrary of Congress Catalog Card N umber 68*! 4271
ISBN 0*674-80207-1
Printed in the United States of America
lliis book has been digitally reprinted. The content
remains identical to that of previous printings.
To BERTRAND RUSSELL
whose ideas have long loomed large
in this subject and whose writings
inspired my interest in it
“How quaint the ways of paradox.”
W. S. GILBERT
PREFACE TO THE REVISED EDITION
The revisions have been extensive. Most of them made their
way meanwhile into the Japanese edition, thanks to the patience
of the Japanese translators and publishers. I am much relieved
that these improvements are now to go out to readers of English.
The latest major change is the adoption in §23 of an axiom
schema which provides in general that a class will exist if its
members can be mapped on the ordinals less than some ordinal.
In general my policy in the book has been to minimize existence
assumptions and to postpone them; but this much compromise
proved rewarding. While continuing to defer the assumption of
infinite classes, it served to sweep away most of the existence
premisses that had formerly cluttered the theorems of §§23-27 and
also to simplify a number of proofs. One notable effect of the
axiom schema is a proof that there is no last ordinal (new 24.9).
§§25—27 on transfinite recursion were already subject to another
major change, prompted by Charles Parsons; see bibliography.
He observed that some of my theorem schemata on this subject
carried existence premisses that would rule out some of the clearly
desirable interpretations of the schematic letters. He showed how
to remedy the situation, and I have adopted his central idea.
Parsons also pointed out a second and lesser limitation in my
treatment of transfinite recursion, and showed how to overcome
it. But it was a limitation that operates only if there is a last
ordinal; and this contingency is now excluded, as noted above.
So I am not following Parsons on this point
Another notable revision was instigated by Burton Dreben: the
reorganization of §30 on infinite cardinals. Extensions and cor-
viii PREFACE TO THE REVISED EDITION
rections relating to the axiom of Fundierung and the axiom schema
of replacement also have been prompted by Dreben and by
Kenneth Brown. Findings by Brown and by Hao Wang have led
me also to insert some further points on axioms in §13 and else­
where. §37 on variants of the theory of types has been improved
in the light mainly of findings by David Kaplan; a case of this h
the elimination of ‘(3x)Tas axiom at one stage.
Minor improvements have been made throughout the book.
Here and there a proof is shortened, a theorem strengthened, a
theoretical or clerical error corrected, a space-saving lemma in­
serted, an obscurity clarified, a historical error corrected or omis­
sion supplied, a new event noted. In these connections 1 am in­
debted variously to Burton Dreben, John Denton, Jean van
Heijenoort, Henry Hil, Saul Kripke, David K. Lewis, Donald
Martin, Akira Ohe, Charles Parsons, Dale R. Samson, Thomas
Scanlon, Leslie Tharp, Joseph S. IJIlian, and Natuhiko Yosida.
For supporting my work on this revision, as well as the proof­
reading and indexing of the first edition in 1963, 1 acknowledge
with thanks a grant GP-228 from the National Science Foun­
dation.
W.V.Q.
April 1967