The Axiom of Constructibility: A Guide for the Mathematician
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617 Keith J. Devlin
The Axiom of Constructibility: A Guide for the Mathematician
Springer-Verlag Berlin Heidelberg New York 1977
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
617 Keith J. Devlin
The Axiom of Constructibility: A Guide for the Mathematician
Springer-Verlag Berlin Heidelberg New York 1977
Author Keith J. Devlin Department of Mathematics University of Lancaster Lancaster/England
Library of Congress Cataloging in Publication Data Devlin, Keith J The axiom of constructibility. (Lecture notes in mathematics ; 617) Bibliography: p , Includes index.
1. Axiom of constructibility. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 617. QA3.L28 no. 617 [QA248] 510'.8s [511'.3] 7717119
AMS Subject Classifications (1970): 02 K15, 02 K25, 02 K05, 04-01, 04A30, 20A10, 20K35, 54015 ISBN 3-540-08520-3 ISBN 0-387-08520-3
Springer-Verlag Berlin Heidelberg New York Springer-Verlag New York Heidelberg Berlin
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PREFACE
Consider the following four theorems of pure mathematics. The Hahn-Banach Theorem 9f ;\nalysis: defined on a subspace
of a Banach space
functional G on !l such that
If F is a bounded linear functional there is an extension of F to a linear
\lG \I = II Fit.
The Nielsen-Schreier Theorem of Group Theory: is a subgroup of
Q,
If
is a free group and
then g is a free group.
The Tychonoff Product Theorem of General Topology:
The product of any
family of compact topological spaces is compact. The Zermelo liell-Ordering Theorem of Set Theory:
Every set can be well-
ordered.
The above theorems have
two things in common. Firstly they are all
fundamental results in contemporary mathematics. Secondly, none of them can be proved without the aid of some powerful set theoretical assumption: in this case the Axiom of Choice.
Now, there is nothing wrong about assuming the Axiom of Choice. But let us be sure about one thing: we are making an assumption here. lie are saying, in effect,. that when we speak of "set theory", the Axiom of Choice is one of the basic properties of sets which we intend to use. This is a perfectly reasonable assumption to make, as most pure mathematicians would agree. Moreover (and here we are at a distinct advantage over those who first advocated the use of the Axiom of Choice), we know for sure that assuming the Axiom of Choice does not lead to a contradiction with our other (more fundamental) assumptions about sets.
In Chapter I
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