Elements of Mathematics: Theory of Sets

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.logy.- 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 in