Theory of Sets [Paperback]

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.- 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 ? 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 lÃF