# Book:Nicolas Bourbaki/Theory of Sets

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## Nicolas Bourbaki:

## Nicolas Bourbaki: *Theory of Sets*

Published $\text {1968}$, **Springer**

- ISBN 3-540-22525-0.

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

### Subject Matter

### 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

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