Course of Mathematical Logic Volume 2 Model Theory
This book is addressed primarily to researchers specializing in mathemat ical logic. It may also be of interest to students completing a Masters Degree in mathematics and desiring to embark on research in logic, as well as to teachers at universities and
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SYNTHESE LIBRARY MONOGRAPHS ON EPISTEMOLOGY, LOGIC, METHODOLOGY, PHILOSOPHY OF SCIENCE, SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE, AND ON THE MATHEMATICAL METHODS OF SOCIAL AND BEHAVIORAL SCIENCES
Editors: DONALD DAVIDSON,
J AAKKO
Rockefeller University and Princeton University
HINTIKKA,
Academy of Finland and Stanford University
GABRIEL NUCHELMANS, WESLEY
C.
SALMON,
University of Leyden
University of Arizona
VOLUME 69
ROLAND FRAlsSE
COURSE OF MATHEMATICAL LOGIC VOLUME 2
Model Theory
D. REIDEL PUBLISHING COMPANY DORDRECHT-HOLLAND / BOSTON-U.S.A.
COURS DE LOGIQUE MATHEMATIQUE, TOME 2 First published by Gauthier- Villars and E. Nauwelaerts, Paris, Louvain, 1967 Second revised and improved edition, Gauthier- Villars, Paris, 1972 Translated by David Louvish
Library of Congress Catalog Card Number 72-95893 ISBN-I3: 978-90-277-0510-5
e-ISBN-13: 978-94-010-2097-8
001: 10.1007/978-94-010-2097-8
Published by D. Reidel Publishing Company, P.O. Box 17, Dordrecht, Holland Sold and distributed in the U.S.A., Canada, and Mexico by D. Reidel Publishing Company, Inc. 306 Dartmouth Street, Boston, Mass. 02116, U.S.A.
All Rights Reserved Copyright © 1974 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover 1st edition 1974 No part of this book may be reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher
TABLE OF CONTENTS
PREFACE
IX
TRANSLATOR'S PREFACE
XI
INTRODUCTION
XIII
CHAPTER 1/ LOCAL ISOMORPHISM AND LOGICAL FORMULA; LOGICAL RESTRICTION THEOREM
1.1. (k, p)-Isomorphism 1.2. (k, p)-Equivalence 1.3. Characteristic of a Logical Formula Relations Between (k, p)-Isomorphism and Logical Formula 1.4. Logical Extension and Logical Restriction; Logical Restriction Theorem 1.5. Examples of Finitely-Axiomatizable and Non-FinitelyAxiomatizable Multirelations 1.6. (k, p)-Interpretability 1. 7. Homogeneous and Logically Homogeneous Multirelations 1.8. Rigid and Logically Rigid Multirelations Exercises
1 6 10 14 18 20 23 24 26
CHAPTER 2/ LOGICAL CONVERGENCE; COMPACTNESS, OMISSION AND INTERPRETABILITY THEOREMS
2.1. Logical Convergence 2.2. Compactness Theorem 2.3. Omission Theorem 2.4. Interpretability Theorem 2.5. Every Injective Logical Operator is Invertible Exercise
30 32 35 37 41 45
VI
COURSE OF MATHEMATICAL LOGIC CHAPTER 3/ ELIMINATION OF QUANTIFIERS
3.1. Absolute Eliminant 3.2. (k, p)-Eliminant
3.3. Elimination Algorithms for the Chain of Rational Numbers and the Chain of Natural Numbers 3.4. Positive Dense Sum; Elimination of Quantifiers over the Sum of Rational or Real Numbers 3.5. Positive Discrete Divisible Sum; Elimination of Quantifiers over the Sum of Natural Numbers 3.6. Real Field; Elimination of Quantifiers over the Sum and Product of Algebraic Numbers or Real Numbers Exercises
46 47 48 50 55 61 69
CHAPTER 4/ EXTENSION THEOREMS
4.1. Restrictive Sequence; (k, p)-Isomorphism and (k, p)- Identimorphism 4.2. Application to Logical Restriction 4.3. Projection Filter 4.4. Logical Extension T
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