# Book:Nicolas Bourbaki/Theory of Sets

## Nicolas Bourbaki: Theory of Sets

Published $\text {1968}$, Springer

ISBN 3-540-22525-0

Originally published as Éléments de Mathématique I: Théorie des Ensembles.

### Contents

Contents of the Elements of Mathematics Series
Introduction

#### Chapter I. Description of Formal Mathematics

$\S 1$. Terms and Relations
1. Signs and assemblies
2. Criteria of substitution
3. Formative constructions
4. Formative criteria
$\S 2$. Theorems
1. The Axioms
2. Proofs
3. Substitutions in a theory
4. Comparison of theories
$\S 3$. Logical theories
1. Axioms
2. First consequences
3. Methods of proof
4. Conjunction
5. Equivalence
$\S 4$. Quantified theories
1. Definition of quantifiers
2. Axioms of quantified theories
3. Properties of quantifiers
4. Typical quantifiers
$\S 5$. Equalitarian theories
1. The axioms
2. Properties of equality
3. Functional relations
Appendix. Characterization of terms and relations
1. Signs and words
2. Significant words
3. Characterization of significant words
4. Application to assemblies in a mathematical theory
Exercises for $\S 1$
Exercises for $\S 2$
Exercises for $\S 3$
Exercises for $\S 4$
Exercises for $\S 5$

#### Chapter II. Theory of Sets

$\S 1$. Collectivizing relations
1. The theory of sets
2. Inclusion
3. The axiom of extent
4. Collectivizing relations
5. The axiom of the set of two elements
6. The scheme of selection and union
7. Complement of a set. The empty set
$\S 2$. Ordered pairs
1. The axiom of the ordered pair
2. Product of two sets
$\S 3$. Correspondences
1. Graphs and correspondences
2. Inverse of a correspondence
3. Composition of two correspondences
4. Functions
5. Restrictions and extensions of functions
6. Definition of a function by means of a term
7. Composition of two functions. Inverse function
8. Retractions and sections
9. Functions of two arguments
$\S 4$. Union and intersection of a family of sets
1. Definition of the union and the intersection of a family of sets
2. Properties of union and intersection
3. Images of a union and an intersection
4. Complements of unions and intersections
5. Union and intersections of two sets
6. Coverings
7. Partitions
8. Sum of a family of sets
$\S 5$. Product of a family of sets
1. The axiom of the set of subsets
2. Set of mappings of one set into another
3. Definitions of the product of a family of sets
4. Partial products
5. Associativity of products of sets
6. Distributivity formulae
7. Extension of mappings to products
$\S 6$. Equivalence relations
1. Definition of an equivalence relation
2. Equivalence classes; quotient set
3. Relations compatible with an equivalence relation
4. Saturated subsets
5. Mappings compatible with equivalence relations
6. Inverse image of an equivalence relation; induced equivalence relation
7. Quotients of equivalence relations
8. Product of two equivalence relations
9. Classes of equivalent objects
Exercises for $\S 1$
Exercises for $\S 2$
Exercises for $\S 3$
Exercises for $\S 4$
Exercises for $\S 5$
Exercises for $\S 6$

#### Chapter III. Ordered Sets, Cardinals, Integers

$\S 1$. Order relations. Ordered sets
1. Definition of an order relation
2. Preorder relations
3. Notation and terminology
4. Ordered subsets. Product of ordered sets
5. Increasing mappings
6. Maximal and minimal elements
7. Greatest element and least element
8. Upper and lower bounds
9. Least upper bound and greatest lower bound
10. Directed sets
11. Lattices
12. Totally ordered sets
13. Intervals
$\S 2$. Well-ordered sets
1. Segments of a well-ordered set
2. The principle of transfinite induction
3. Zerrnelo's theorem
4. Inductive sets
5. Isomorphisms of well-ordered sets
6. Lexicographic products
$\S 3$. Equipotent sets. Cardinals
1. The cardinal of a set
2. Order relation between cardinals
3. Relations on cardinals
4. Properties of the cardinals 0 and 1
5. Exponentiation of cardinals
6. Order relation and operations on cardinals
$\S 4$. Natural integers. Finite sets
1. Definition of integers
2. Inequalities between integers
3. The principle of induction
4. Finite subsets of ordered sets
5. Properties of finite character
$\S 5$. Properties of integers
1. Operations on integers and finite sets
2. Strict inequalities between integers
3. Intervals in sets of integers
4. Finite sequences
5. Characteristic functions of sets
6. Euclidean division
7. Expansion to base $b$
8. Combinatorial analysis
$\S 6$. Infinite sets
1. The set of natural integers
2. Definition of mappings by induction
3. Properties of infinite cardinals
4. Countable sets
5. Stationary sequences
$\S 7$. Inverse limits and direct limits
1. Inverse limits
2. Inverse systems of mappings
3. Double inverse limit
4. Conditions for an inverse limit to be non-empty
5. Direct limits
6. Direct systems of mappings
7. Double direct limit. Product of direct limits
Exercises for $\S 1$
Exercises for $\S 2$
Exercises for $\S 3$
Exercises for $\S 4$
Exercises for $\S 5$
Exercises for $\S 6$
Exercises for $\S 7$
Historical note on $\S 5$
Bibliography

#### Chapter IV. Structures

$\S 1$. Structures and isomorphisms
1. Echelons
2. Canonical extensions of mappings
3. Transportable relations
4. Species of structures
5. Isomorphisms and transport of structures
6. Deduction of structures
7. Equivalent species of structures
$\S 2$. Morphisms and derived structures
1. Morphisms
2. Finer structures
3. Initial structures
4. Examples of initial structures
5. Final structures
6. Examples of final structures
$\S 3$. Universal mappings
1. Universal sets and mappings
2. Existence of universal mappings
3. Examples of universal mappings
Exercises for $\S 1$
Exercises for $\S 2$
Exercises for $\S 3$
Historical note on chapters I-IV
Bibliography

#### Summary of Results

Introduction
$\S 1$. Elements and subsets of a set
$\S 2$. Functions
$\S 3$. Products of sets
$\S 4$. Union, intersection, product of a family of sets
$\S 5$. Equivalence relations and quotient sets
$\S 6$. Ordered sets
$\S 7$. Powers. Countable sets
$\S 8$. Scales of sets. Structures
Index of notation
Index of terminology
Axioms and schemes of the theory of sets

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