Theory of Sets

Inhaltsverzeichnis

• I. Description of Formal Mathematics.
• § 1. Terms and relations.
• 1. Signs and assemblies.
• 2. Criteria of substitution.
• 3. Formative constructions.
• 4. Formative criteria.
• § 2. Theorems.
• 1. The axioms.
• 2. Proofs.
• 3. Substitutions in a theory.
• 4. Comparison of theories.
• § 3. Logical theories.
• 1. Axioms.
• 2. First consequences.
• 3. Methods of proof.
• 4. Conjunction.
• 5. Equivalence.
• § 4. Quantified theories.
• 1. Definition of quantifiers.
• 2. Axioms of quantified theories.
• 3. Properties of quantifiers.
• 4. Typical quantifiers.
• § 5. Equalitarian theories.
• 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 § 1.
• Exercises for § 2.
• Exercises for § 3.
• Exercises for § 4.
• Exercises for § 5.
• Exercises for the Appendix.
• II. Theory of Sets.
• § 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.
• § 2. Ordered pairs.
• 1. The axiom of the ordered pair.
• 2. Product of two sets.
• § 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.
• § 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 intersection of two sets.
• 6. Coverings.
• 7. Partitions.
• 8. Sum of a family of sets.
• § 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.
• § 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 § 6.
• III. Ordered Sets, Cardinals, Integers.
• § 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.
• § 2. Well-ordered sets.
• 1. Segments of a well-ordered set.
• 2. The principle of transfinite induction.
• 3. Zermelo’s theorem.
• 4. Inductive sets.
• 5. Isomorphisms of well-ordered sets.
• 6. Lexicographic products.
• § 3. Equipotent sets. Cardinals.
• 1. The cardinal of a set.
• 2. Order relation between cardinals.
• 3. Operations on cardinals.
• 4. Properties of the cardinals 0 and 1.
• 5. Exponentiation of cardinals.
• 6. Order relation and operations on cardinals.
• § 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.
• § 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.
• § 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.
• § 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 § 7.
• Historical Note on § 5.
• IV. Structures.
• § 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.
• § 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.
• § 3. Universal mappings.
• 1. Universal sets and mappings.
• 2. Existence of universal mappings.
• 3. Examples of universal mappings.
• Historical Note on Chapters I-IV.
• Summary of Results.
• § 1. Elements and subsets of a set.
• § 2. Functions.
• § 3. Products of sets.
• § 4. Union, intersection, product of a family of sets.
• § 5. Equivalence relations and quotient sets.
• § 6. Ordered sets.
• § 7. Powers. Countable sets.
• § 8. Scales of sets. Structures.
• Index of notation.
• Index of terminology.
• Axioms and schemes of the theory of sets.