Gothic Alphabet
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Dirk van Dalen
Logic and Structure
Springer-Verlag Berlin Heidelberg GmbH 1980
Dirk van Dalen Rijkuniversiteit Utrecht, Mathematisch Instituut Budapestlaan 6, Postbus 80.010 NL-3S08 TA Utrecht
AMS Subject Classification (1980): 03-01
ISBN 978-3-540-09893-5
Library of Congress Cataloging in Publication Data Dalen, Dirk van, 1932Logic and structure. (Universitext) Bibliography: p. Includes index. 1. Logic, Symbolic and mathematical. 1. Title. QA9.D14 1980 511 79-27728 ISBN 978-3-540-09893-5 ISBN 978-3-662-08402-1 (eBook) DOI 10.1007/978-3-662-08402-1
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of iIIustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the pUblisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Originally published by Springer-Verlag Berlin Heidelberg New York in 1980 2141/3140-543210
Preface
Logic appears in a 'sacred' and in a 'profane' form. The sacred form is dominant in proof theory, the profane form in model theory. The phenomenon is not unfamiliar, one observes this dichotomy also in other areas, e.g. set theory and recursion theory. For one reason or another, such as the discovery of the set theoretical paradoxes (Cantor, Russell), or the definability paradoxes (Richard, Berry), a subject is treated for some time with the utmost awe and diffidence. As a rule, however, sooner or later people start to treat the matter in a more free and easy way. Being raised in the 'sacred' tradition, I was greatly surprised (and somewhat shocked) when I observed Hartley Rogers teaching recursion theory to mathematicians as if it were just an ordinary course in, say, linear algebra or algebraic topology. In the course of time I have come to accept his viewpoint as the didactically sound one: before going into esoteric niceties one should develop a certain feeling for the subject and obtain a reasonable amount of plain working knowledge. For this reason I have adopted the profane attitude in this introductory text, reserving the more sacred approach for advanced courses. Readers who want to know more about the latter aspect of logic are referred to the immortal texts of Hilbert-Bernays or Kleene. The present book has developed out of courses given at the University at Utrecht in the mathematics department to undergraduates. The experience drawn from these courses and the reactions of the participants was that one should try to incorporate bits of real mathematics as soon as possible. For that reason the wellknown structures, such as groups, partially ordered sets, projective planes, are introduced at the earliest possible occasion. By now, it is generally agreed that a mathematician should know how to formalize his language and his semantics. One
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