ELEMENTS OF MATHEMATICS NICOLAS BOURBAKI ELEMENTS OF MATHEMATICS Theory of Sets ~ Springer Originally published as mements de Mathematique. Theorie des Ensembles Paris, 1970 eN.Bourbaki Mathematics Subject Classification (2000): 03-02, 03Bxx Library of Congress Control Number: 2004110815 Softcover edition of the 1st printing 1968 ISBN-13:978-3-540-22525-6 e-ISBN-13: 978-3-642-59309-3 DOl: 10.1007/978-3-642-59309-3 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved. whether the whole or part of the material is concerned. specifically the rights of translation. reprinting. reuse of illustrations. recitation. broad casting. reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9. 1965. in its current version. and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com o Springer-Verlag Berlin Heidelberg 2.004 Reprint of the original edition <2004> The use of general descriptive names,registered names, trademarks,etc.in this publication does not imply. even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the Translator Cover design: design+production GmbH. Heidelberg Printed on acid-free paper 41/3142/XT - 5 43 2. 1 0 TO THE READER I. This series of volumes, a list of which is given on pages vn and VIU, takes up mathematics at the beginning, and gives complete proofs. In prin ciple, it requires no particular knowledge of mathematics on the reader's part, but only a certain familiarity with mathematical reasoning and a certain capacity for abstract thought. Nevertheless, it is directed especially to those who have a good knowledge of at least the content of the first year or two of a university mathematics course. 2. The method of exposition we have chosen is axiomatic and abstract, and normally proceeds from the general to the particular. This choice has been dictated by the main purpose of the treatise, which is to provide a solid foundation for the whole body of modem mathematics. For this it is indispensable to become familiar with a rather large number of very general ideas and principles. Moreover, the demands of proof impose a rigorously fixed order on the subject matter. It follows that the utility of certain considerations will not be immediately apparent to the reader unless he has already a fairly extended knowledge of mathematics; other wise he must have the patience to suspend judgment until the occasion arises. 3. In order to mitigate this disadvantage we have frequendy inserted examples in the text which refer to facts the reader may already know but which have not yet been discussed in the series. Such examples * ... *. are always placed between two asterisks: Most readers will undoubtedly find that these examples will help them to understand the text, and will prefer not to leave them out, even at a first reading. Their omission would of course have no disadvantage, from a purely logical point of view. 4. This series is divided into volumes (here called " Books "). The first six Books are numbered and, in general, every statement in the text v TO THE READER assumes as known only those results which have already been discussed in the preceding volumes. This rule holds good within each Book, but for convenience of exposition these Books are no longer arranged in a consecutive order. At the beginning of each of these Books (or of these chapters), the reader will find a precise indication of its logical relation ship to the other Books and he will thus be able to satisfy himself of the absence of any vicious circle. 5. The logical framework of each chapter consists of the definitions, the axioms, and the theorems of the chapter. These are the parts that have mainly to be borne in mind for subsequent use. Less important results and those which can easily be deduced from the theorems are labelled as "propositions", "lemmas", "corollaries", "remarks", etc. Those which may be omitted at a first reading are printed in small type. A commentary on a particularly important theorem appears occasionally under the name of "scholium". To avoid tedious repetitions it is sometimes convenient to introduce notations or abbreviations which are in force only within a certain chapter or a certain section of a chapter (for example, in a chapter which is con cerned only with commutative rings, the word "ring" would always signify "commutative ring"). Such conventions are always explicidy mentioned, generally at the beginning of the chapter in which they occur. 6. Some passages in the text are designed to forewarn the reader against serious errors. These passages are signposted in the margin with the sign 2 ("dangerous bend"). 7. The Exercises are designed both to enable the reader to satisfY himself that he has digested the text and to bring to his notice results which have no place in the text but which are nonetheless of interest. The most difficult exercises bear the sign ~. 8. In general, we have adhered to the commonly accepted terminology, except where there appeared to be good reasons for deviating from it. 9. We have made a particular effort always to use rigorously correct language, without sacrificing simplicity. As far as possible we have drawn attention in the text to abuses of language, without which any mathe matical text runs the risk of pedantry, not to say unreadability. 10. Since in principle the text consists of the dogmatic exposition of a theory, it contains in general no references to the literature. Biblio graphical references are gathered together in Historical Notes, usually at the end of each chapter. These notes also contain indications, where appropriate, of the unsolved problems of the theory. VI TO THE READER The bibliography which follows each historical note contains in general only those books and original memoirs which have been of the greatest importance in the evolution of the theory under discussion. It makes no sort of pretence to completeness; in particular, references which serve only to determine questions of priority are almost always omitted. As to the exercices, we have not thought it worthwhile in general to indicate their origins, since they have been taken from many different sources (original papers, textbooks, collections of exercises). 11. References to a part of this series are given as follows : a) If reference is made to theorems, axioms, or definitions presented in the same section, they are quoted by their number. b) If they occur in another section oj the same chapter, this section is also quoted in the reference. c) If they occur in another chapter in the same Book, the chapter and section are quoted. d) If they occur in another Book, this Book is first quoted by its tide. The Summaries of Results are quoted by the letter R : thus Set Theory, R signifies "Summary oj Results oj the Theory oj Sets". CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES I. THEORY OF SETS 1. Description of formal mathematics. 2. Theory of sets. 3. Ordered sets; cardinals; natural numbers. 4. Structures. II. ALGEBRA 1. Algebraic structures. 2. Linear algebra. 3. Tensor algebras, exterior algebras, symmetric algebras. 4. Poylnomia1s and rational fractions. 5. Fields. 6. Ordered groups and fields. 7. Modules over principal ideal rings. 8. Semi-simple modules and rings. 9. Sesquilinear and quadratic forms. vn CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES III. GENERAL TOPOLOGY 1. Topological structures. 2. Uniform structures. 3. Topological groups. 4. Real numbers. 5. One-parameter groups. 6. Real number spaces, affine and projective spaces. 7. The additive groups Rli. 8. Complex numbers. 9. Use of real numbers in general topology. lO. Function spaces. IV. FUNCTIONS OF A REAL VARIABLE 1. Derivatives. 2. Primitives and integrals. 3. Elementary functions. 4. Differential equations. 5. Local study of functions. 6. Generalized Taylor expansions. The Euler-Maclaurin summation formula. 7. The gamma function. Dictonary. V. TOPOLOGICAL VECTOR SPACES 1. Topological vector spaces over a valued field. 2. Convex sets and locally convex spaces. 3. Spaces of continuous linear mappings. 4. Duality in topological vector spaces. 5. Hilbert spaces: elementary theroy. Dictionary. VI. INTEGRATION 1. Convexity inequalities. 2. Riesz spaces. 3. Measures on locally compact spaces. 4. Extension of a measure. LP spaces. 5. Integration of measures. 6. Vectorial integration. 7. Haar measure. 8. Convo lution and representation. Lm GROUPS AND Lm ALGEBRAS 1. Lie algebras. COMMUTATIVE ALGEBRA I. Flat modules. 2. Localization. 3. Graduations, filtrations and topo logies. 4. Associated prime ideals and primary decomposition. 5. In tegers. 6. Valuations. 7. Divisors. SPECTRAL THEORmS 1. Normed algebras. 2. Locally compact groups. vm CONTENTS TO THE READER ••..••.....•..•..•.••..••••...•.••••.••..••• V CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES . . . . • • • . . . . VII INTRODUCTION ...••.••.••.•..••..•.•.•••••••••.••.•.••••.••. 7 CHAPTER 1. DESCRIPTION OF FORMAL MATHEMATICS... ••••••••.• 15 § 1. Terms and relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1. Signs and assemblies ............................. 15 2. Criteria of substitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3. Formative constructions. . . . . . . . . . . . . . . . . . . . . . . . . . 19 4. Formative criteria ............................... 21 § 2. Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1. The axioms .................................... 24 2. Proofs ......................................... 25 3. Substitutions in a theory ......................... 26 4. Comparison of theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 § 3. Logical theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1. Axioms ........................................ 28 2. First consequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3. Met?ods .of proof. ............. .... ... ..... ... .. 30 4. ConjunctIon .................................... 33 5. Equivalence..................................... 34 § 4. Quantified theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1. Definition of quantifiers. . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2. Axioms of quantified theories ..................... 37 3. Properties of quantifiers. . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4. Typical quantifiers .............................. 41 CONTENTS § 5. Equalitarian theories ................................ 44 1. The axioms .................................... 44 2. Properties of equality ............................ 45 3. Functional relations ............................. 47 Appendix. Characterization of terms and relations . . . . . . . . . . 50 1. Signs and words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2. Significant words.. .... . ............ .... .... ..... 51 3. Characterization of significant words ............... 51 4. Application to assemblies in a mathematical theory .. 53 Exercises for § 1 ........................................ 56 Exercises for § 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Exercises for § 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Exercises for § 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Exercises for § 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Exercises for the Appendix 60 CHAPTER II. THEORY OF SETS 65 § 1. Collectivizing relations .. .. . . .. . .. .. . . . . . .. .. . .. .. . . . . 65 1. The theory of sets ............................... 65 2. Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3. The axiom of extent ............................. 67 4. Collectivizing relations ........................... 67 5. The axiom of the set of two elements. . . . . . . . . . . . . . . 69 6. The scheme of selection and union ................ 69 7. Complement ofa set. The empty set.............. 71 § 2. Ordered pairs...... . .. . ... . ........ .... .... .... . .. .. 72 1. The axiom of the ordered pair .................... 72 2. Product of two sets .............................. 74 § 3. Correspondences.................................... 75 1. Graphs and correspondences ...................... 75 2. Inverse of a correspondence . . . . . . . . . . . . . . . . . . . . . . . 78 3. Composition of two correspondences ............... 78 4. Functions ...................................... 81 5. Restrictions and extensions of functions . . . . . . . . . . . . 82 6. Definition of a function by means of a term. . . . . . . . 83 7. Composition of two functions. Inverse function ..... 84 2