Set Theory and Logic

Chapter 1 SETS AND RELATIONS 1. Cantor's Concept of a Set 2. The Basis of Intuitive Set Theory 3. Inclusion 4. Operations for Sets 5. The Algebra of Sets 6. Relations 7. Equivalence Relations 8. Functions 9. Composition and Inversion for Functions 10. Operations for Collections of Sets 11. Ordering Relations Chapter 2 THE NATURAL NUMBER SEQUENCE AND ITS GENERALIZATIONS 1. The Natural Number Sequence 2. Proof and Definition by Induction 3. Cardinal Numbers 4. Countable Sets 5. Cardinal Arithmetic 6. Order Types 7. Well-ordered Sets and Ordinal Numbers 8. "The Axiom of Choice, the Well-ordering Theorem, and Zorn's Lemma" 9. Further Properties of Cardinal Numbers 10. Some Theorems Equivalent to the Axiom of Choice 11. The Paradoxes of Intuitive Set Theory Chapter 3 THE EXTENSION OF THE NATURAL NUMBERS TO THE REAL NUMBERS 1. The System of Natural Numbers 2. Differences 3. Integers 4. Rational Numbers 5. Cauchy Sequences of Rational Numbers 6. Real Numbers 7. Further Properties of the Real Number System Chapter 4 LOGIC 1. The Statement Calculus. Sentential Connectives 2. The Statement Calculus. Truth Tables 3. The Statement Calculus. Validity 4. The Statement Calculus. Consequence 5. The Statement Calculus. Applications 6. The Predicate Calculus. Symbolizing Everyday Language 7. The Predicate Calculus. A Formulation 8. The Predicate Calculus. Validity 9. The Predicate Calculus. Consequence Chapter 5 INFORMAL AXIOMATIC MATHEMATICS 1. The Concept of an Axiomatic Theory 2. Informal Theories 3. Definitions of Axiomatic Theories by Set-theoretical Predicates 4. Further Features of Informal Theories Chapter 6 BOOLEAN ALGEBRAS 1. A Definition of a Boolean Algebra 2. Some Basic Properties of a Boolean Algebra 3. Another Formulation of the Theory 4. Congruence Relations for a Boolean Algebra 5. Representations of Boolean Algebras 6. Statement Calculi as Boolean Algebras 7. Free Boolean Algebras 8. Applications of the Theory of Boolean Algebras to Statement Calculi 9. Further Interconnections between Boolean Algebras and Statement Calculi Chapter 7 INFORMAL AXIOMATIC SET THEORY 1. The Axioms of Extension and Set Formation 2. The Axiom of Pairing 3. The Axioms of Union and Power Set 4. The Axiom of Infinity 5. The Axiom of Choice 6. The Axiom Schemas of Replacement and Restriction 7. Ordinal Numbers 8. Ordinal Arithmetic 9. Cardinal Numbers and Their Arithmetic 10. The von Neumann-Bernays-Gödel Theory of Sets Chapter 8 SEVERAL ALGEBRAIC THEORIES 1. Features of Algebraic Theories 2. Definition of a Semigroup 3. Definition of a Group 4. Subgroups 5. Coset Decompositions and Congruence Relations for Groups 6. "Rings, Integral Domains, and Fields" 7. Subrings and Difference Rings 8. A Characterization of the System of Integers 9. A Characterization of the System of Rational Numbers 10. A Characterization of the Real Number System Chapter 9 FIRST-ORDER THEORIES 1. Formal Axiomatic Theories 2. The Statement Calculus as a Formal Axiomatic Theory 3. Predicate Calculi of First Order as Formal Axiomatic Theories 4. First-order Axiomatic Theories 5. Metamathematics 6. Consistency and Satisfiability of Sets of Formulas 7. "Consistency, Completeness, and Categoricity of First-order Theories" 8. Turing Machines and Recursive Functions 9. Some Undecidable and Some Decidable Theories 10. Gödel's Theorems 11. Some Further Remarks about Set Theory References Symbols and Notation Author Index Subject Index