Book:Nicolas Bourbaki/Theory of Sets

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

Subject Matter

 * Set Theory

Contents

 * To The Reader
 * 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