Model Theory.pdf

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David Marker Department of Mathematics University of Illinois 351 S. Morgan Street Chicago, IL 60607-7045 USA marker@math.uic.edu Editorial Board: S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA axler@sfsu.edu F.W. Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA fgehring@math.lsa.umich.edu K.A. Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA ribet@math.berkeley.edu 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. II. Series. QA9.7 .M367 2002 511.3—dc21 I. Title. 2002024184 Printed on acid-free paper. ISBN 0-387-98760-6 © 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 Structures and Theories 1.1 Languages and Structures . . . . . . . . . . . . . . . . . . . 1.2 Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Deﬁnable Sets and Interpretability . . . . . . . . . . . . . . 1.4 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . 1 7 7 14 19 29 33 33 40 44 48 60 3 Algebraic Examples 71 71 3.1 Quantiﬁer Elimination . . . . . . . . . . . . . . . . . . . . . 84 3.2 Algebraically Closed Fields . . . . . . . . . . . . . . . . . . 3.3 Real Closed Fields . . . . . . . . . . . . . . . . . . . . . . . 93 3.4 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 104 4 Realizing and Omitting Types 115 4.1 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2 Omitting Types and Prime Models . . . . . . . . . . . . . . 125

viii Contents 4.3 Saturated and Homogeneous Models . . . . . . . . . . . . . 138 4.4 The Number of Countable Models . . . . . . . . . . . . . . 155 4.5 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 163 5 Indiscernibles 175 5.1 Partition Theorems . . . . . . . . . . . . . . . . . . . . . . . 175 5.2 Order Indiscernibles . . . . . . . . . . . . . . . . . . . . . . 178 5.3 A Many-Models Theorem . . . . . . . . . . . . . . . . . . . 189 5.4 An Independence Result in Arithmetic . . . . . . . . . . . . 195 5.5 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 202 6 ω-Stable Theories 207 . . . . . . . . . . . . . . 207 6.1 Uncountably Categorical Theories 6.2 Morley Rank . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.3 Forking and Independence . . . . . . . . . . . . . . . . . . . 227 6.4 Uniqueness of Prime Model Extensions . . . . . . . . . . . . 236 6.5 Morley Sequences . . . . . . . . . . . . . . . . . . . . . . . . 240 6.6 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 243 7 ω-Stable Groups 251 7.1 The Descending Chain Condition . . . . . . . . . . . . . . . 251 7.2 Generic Types . . . . . . . . . . . . . . . . . . . . . . . . . 255 7.3 The Indecomposability Theorem . . . . . . . . . . . . . . . 261 7.4 Deﬁnable Groups in Algebraically Closed Fields . . . . . . . 267 7.5 Finding a Group . . . . . . . . . . . . . . . . . . . . . . . . 279 7.6 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 285 8 Geometry of Strongly Minimal Sets 289 8.1 Pregeometries . . . . . . . . . . . . . . . . . . . . . . . . . . 289 8.2 Canonical Bases and Families of Plane Curves . . . . . . . . 293 8.3 Geometry and Algebra . . . . . . . . . . . . . . . . . . . . . 300 8.4 Exercises and Remarks . . . . . . . . . . . . . . . . . . . . . 309 A Set Theory B Real Algebra References Index 315 323 329 337

Introduction Model theory is a branch of mathematical logic where we study mathemat- ical structures by considering the ﬁrst-order sentences true in those struc- tures and the sets deﬁnable by ﬁrst-order formulas. Traditionally there have been two principal themes in the subject: • starting with a concrete mathematical structure, such as the ﬁeld of real numbers, and using model-theoretic techniques to obtain new information about the structure and the sets deﬁnable in the structure; • looking at theories that have some interesting property and proving general structure theorems about their models. A good example of the ﬁrst theme is Tarski’s work on the ﬁeld of real numbers. Tarski showed that the theory of the real ﬁeld is decidable. This is a sharp contrast to G¨odel’s Incompleteness Theorem, which showed that the theory of the seemingly simpler ring of integers is undecidable. For his proof, Tarski developed the method of quantiﬁer elimination which can be used to show that all subsets of Rn deﬁnable in the real ﬁeld are geomet- rically well-behaved. More recently, Wilkie [103] extended these ideas to prove that sets deﬁnable in the real exponential ﬁeld are also well-behaved. The second theme is illustrated by Morley’s Categoricity Theorem, which says that if T is a theory in a countable language and there is an uncount- able cardinal κ such that, up to isomorphism, T has a unique model of cardinality κ, then T has a unique model of cardinality λ for every un- countable κ. This line has been extended by Shelah [92], who has developed deep general classiﬁcation results. For some time, these two themes seemed like opposing directions in the subject, but over the last decade or so we have come to realize that there

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