Sets: Naïve, Axiomatic and Applied

PrefaceAcknowledgementsIntroductionChapter I. Naïve Set Theory 1. Some Important Sets and Notations Natural Numbers Integers Rationals Reals Singleton 2. Equality of Sets Extensionality Axiom 3. Subsets 4. The Naïve Comprehension Principle and the Empty Set 5. Union, Intersection and Relative Complement Complement De Morgan's Laws 6. Power Set 7. Unions and Intersections of Families Greatest and smallest Set with Property F Disjoint Pairwise Disjoint 8. Ordered Pairs Unordered Pairs Ordered n-Tuples 9. Cartesian Product 10. Relations Binary Relation Converse Composition Identity Relation Domain Range Reflexive Symmetric Transitive 11. Equivalence Relations Equivalence Class Representative Partition Quotient Set ¿, Q, 12. Real Numbers Chains of Segments Dedekind Cut Cauchy Sequence 13. Functions (Mappings) Injection Surjection Bijection Identity Mapping Characteristic Function Equality of Functions Composition Inverse Restriction Structure Isomorphism Cartesian Product of a Family 14. Orderings Lexicographic Ordering Partial Ordering Diagram Minimal Element Smallest Element Lower Bound Infimum Total(Linear) Ordering Well-Ordering Immediate Successor Principle of Transfinite Induction Initial Segment 15. Equivalence (Cardinality) =1 Diagonal Method Denumerable Countable Cantor-Bernstein Uncountable 16. Finite and Infinite Axiom of Choice Choice Function Dedekind-Infinite 17. Denumerable Sets Hilbert Hotel Closure Under Union and Product Ramsey's Theorem 18. UncountabZe Sets The Continuum P(¿) Continuum-Hypothesis 19. The Paradoxes Russell's Paradox The Set of All Sets The Separation Principle 20. The Set Theory of Zermelo-Fraenkel (ZF) Formal Language Axioms Natural Numbers ¿-Minimal 21. Peano's Arithmetic Axioms Definition by RecursionChapter II. Axiomatic Set Theory 1. The Axiom of Regularity Motivation Consistency 2. Induction and Recursion Well-Foundedness Transitive Closure Induction Principles Definition by Recursion Representation Theorem for Well-Founded Extensional Structures 3. Ordinal Numbers Von Neumann's Ordinals 4. The Cumulative Hierarchy Rank Partial universes 5. Ordinal Arithmetic Addition Multiplication Exponentiation Cantor's Normal Form 6. Normal Operations Cofinality Regularity Fixed Points Derivative 7. The Reflection Principle 8. Initial Numbers Hartogs' Function Weakly Inaccessible Well-Ordering of ORXOR and Consequences 9. The Axiom of Choice Motivation Equivalents Two Proofs of the Well-Ordering Theorem Zorn's Lemma GCH DC CC 10. Cardinal Numbers Motivation Possible Definitions Alephs Addition Multiplication Exponentiation Regularity Cofinality König's Inequality 11.