Theory of sets : lements of mathematics / Nicolas Bourbaki.

Table of Contents

Chapter 1.
I. Description of Formal Mathematics
1. Terms and relations
2. Criteria of substitutions
3. Formative constructions
4. Formative Criteria

II. Theorems
1. The axioms
2. Proofs
3. Substitutions in a theory
4. Comparison of theories

III. Logical theories
1. Axioms
2. First Consequences
3. Methods of proof
4. Conjunction
5. Equivalence

IV. Quantified theories
1. Definition of quqntifiers
2. Axions of quqntified theories
3. Properties of Quantifiers
4. Typical Quantifiers

V. Equalitarian theories
1. The axioms
2. Properties of equality
3. Functional relations

Chapter 2.
II Theory of Sets
I. Collectivizing relations
1. The theory of sets
2. Inclusion
3. The axioms 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

II. Ordered pairs
1. The axiom of the ordered pair
2. Product of two sets

III. Correspondences
1. Graphs and Correspondences
2. Inverse of a correspondence
3. Composition of two correspondences
4. Functions
5. Restrictions and extensions of Functions
Etc.

IV. 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 union and intersection
4. Complements of unions and intersections
5. Union and intersection of two sets
Etc.

V. Product of a family of sets
1. The axiom of the set of subsets
2. Set of mappingsof one set into another
3. Definition of the product of a family of sets
4. Partial products
5. Associativity of products of sets
Etc.

VI. Equivalence relations
1. Definition of an equivalent relation
2. Equivalenceclasses ; quotient sets
3. Relations and compatible with an equivalence relation
4. Saturated subsets
5. Mappings compatible with equivalence relations
Etc.

Chapter 3 : Ordered Sets, Cardinals, Integers
I. Order relations. Ordered sets
1. Definition of an order relation
2. Preorder relations
3. Notation and Interminology
4. Ordered subsets. Product of ordered sets
5. Increasing mappings
Etc.

II. Well-ordered sets
1. Segments of a well-ordered set
2. The principle of a transfinite induction
3. Zermelo's theorem
4. Inductive sets
5. Isomorphisms of well-ordered sets
Etc.

III. Equipotent sets; Cardinals
1. The cardinal of a set
2. Order relation between ntegers
3. The principle of induction
4. Properties of the cardinals 0 and 1
5. Exponentiation of cardinals
Etc.

IV. 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

V. 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
Etc.

VI. 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

VII. 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
4. Direct limits
Etc.

Chapter 4 :Structures
I. Structures and isomorphisms
1. Echelons
2. Canonical extensions of mappings
3. Transportable relations
4. Species of structures
5. Isomorphisms and transport structures
Etc.

II. Morphisms and derived structures
1. Morphisms
2. Finer structures
3. Initial structures
4. Examples of initial structures
5. Final stuctures
6. Examples of final structures
Universal mappings.

Includes bibliographical references and indexes.