Model Theory An Introduction
This book is a modern introduction to model theory which stresses applications to algebra throughout the text. The first half of the book includes classical material on model construction techniques, type spaces, prime models, saturated models, countable
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Editorial Board: S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
F.W. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA [email protected]
K.A. Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA [email protected]
Mathematics Subject Classification (2000): 03-01, 03Cxx Library of Congress Cataloging-in-Publication Data Marker, D. (David), 1958– Model theory : an introduction / David Marker p. cm. — (Graduate texts in mathematics ; 217) Includes bibliographical references and index. ISBN 0-387-98760-6 (hc : alk. paper) 1. Model theory. I. Title. II. Series. QA9.7 .M367 2002 511.3—dc21 2002024184 ISBN 0-387-98760-6
Printed on acid-free paper.
© 2002 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1
SPIN 10711679
Typesetting: Pages created by the author using a Springer TEX macro package. www.springer-ny.com Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH
In memory of Laura
Contents
Introduction
1
1 Structures and Theories 1.1 Languages and Structures . . . . . 1.2 Theories . . . . . . . . . . . . . . . 1.3 Definable Sets and Interpretability 1.4 Exercises and Remarks . . . . . . .
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2 Basic Techniques 2.1 The Compactness Theorem 2.2 Complete Theories . . . . . 2.3 Up and Down . . . . . . . . 2.4 Back and Forth . . . . . . . 2.5 Exercises and Remarks . . .
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3 Algebraic Examples 3.1 Quantifier Elimination . . . 3.2 Algebraically Closed Fields 3.3 Real Closed Fields . . . . . 3.4 Exercises and Remarks . . .
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71 . 71 . 84 . 93 . 104
4 Realizing and Omitting Types 115 4.1 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2 Omitting Types an
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