A Formal Background to Mathematics Logic, Sets and Numbers
§1 Faced by the questions mentioned in the Preface I was prompted to write this book on the assumption that a typical reader will have certain characteristics. He will presumably be familiar with conventional accounts of certain portions of mathematics an
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R. E. Edwards
A Formal Background to Mathematics Ia Logic, Sets and Numbers
Springer Science+Business Media, LLC
Dr. Robert Edwards Institute of Advanced Studies The Australian National University Canberra, Australia
AMS Subject Classifications: 02-01, 04-01, 06-01, 08-01
Library of Congress Cataloging in Publication Data
Edwards, Robert E A formal background to mathematics. (Universitext) Bibliography: p. Inclu~es indexes. Contents: v. l. Logic, sets and numbers. l. Mathematics-1961I. Title. 510 79-15045 QA37.2.E38
ISBN 978-0-387-90431-3 ISBN 978-1-4612-9984-4 (eBook) DOI 10.1007/978-1-4612-9984-4
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag.
© 1979 by Springer Science+ Business Media New York Originally published by Springer-Verlag New York Inc 1979.
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Foreword
§1 Faced by the questions mentioned in the Preface I was prompted to write this book on the assumption that a typical reader will have certain characteristics.
He will presumably be familiar with conventional accounts of
certain portions of mathematics and with many so-called mathematical statements, some of which (the theorems) he will know (either because he has himself studied and digested a proof or because he accepts the authority of others) to be true, and others of which he will know (by the same token) to be false.
He will nevertheless
be conscious of and perturbed by a lack of clarity in his own mind concerning the concepts of proof and truth in mathematics, though he will almost certainly feel that in mathematics these concepts have special meanings broadly similar in outward features to, yet different from, those in everyday life; and also that they are based on criteria different from the experimental ones used in science.
He will be
aware of statements which are as yet not known to be either true or false (unsolved problems).
Quite possibly he will be surprised and dismayed by the possibility that
there are statements which are "definite"
(in the sense of involving no free
variables) and which nevertheless can never (strictly on the basis of an agreed collection of axioms and an agreed concept of proof) be either proved or disproved (refuted).
In spite of the aforesaid lack of clarity in detail, he will be firmly
convinced that mathematics is par excellence a deductive system : one commences from certain statements assumed to be true (axioms) and proceeds to deduce more true statements (theorems) by logical argument.
Yet the instances of axiomatics already
VI encountered (if any) will clearly fail to get close enough to the bottom of things, principally because they are founded on two things which are left too vague, namely, the concept of set and the (evidently pretty elaborate) logical apparatus for What he seeks is a clarification of all these matters, embracing in
deduction.
particular more satisfactory answers to the questions listed in the Preface. Many such questions have "surfaced" in the area of high
