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Beskrivning
"The best introductory text we have seen." — Cosmos. Lucidly and gradually explains sets and relations, the natural number sequence and its generalization, extension of natural numbers to real numbers, logic, informal axiomatic mathematics, Boolean algebras, informal axiomatic set theory, several algebraic theories, and 1st-order theories. Its clarity makes this book excellent for self-study.
Chapter 1 SETS AND RELATIONS1. Cantor's Concept of a Set2. The Basis of Intuitive Set Theory3. Inclusion4. Operations for Sets5. The Algebra of Sets6. Relations7. Equivalence Relations8. Functions9. Composition and Inversion for Functions10. Operations for Collections of Sets11. Ordering RelationsChapter 2 THE NATURAL NUMBER SEQUENCE AND ITS GENERALIZATIONS1. The Natural Number Sequence2. Proof and Definition by Induction3. Cardinal Numbers4. Countable Sets5. Cardinal Arithmetic6. Order Types7. Well-ordered Sets and Ordinal Numbers8. "The Axiom of Choice, the Well-ordering Theorem, and Zorn's Lemma"9. Further Properties of Cardinal Numbers10. Some Theorems Equivalent to the Axiom of Choice11. The Paradoxes of Intuitive Set TheoryChapter 3 THE EXTENSION OF THE NATURAL NUMBERS TO THE REAL NUMBERS1. The System of Natural Numbers2. Differences3. Integers4. Rational Numbers5. Cauchy Sequences of Rational Numbers6. Real Numbers7. Further Properties of the Real Number SystemChapter 4 LOGIC1. The Statement Calculus. Sentential Connectives2. The Statement Calculus. Truth Tables3. The Statement Calculus. Validity4. The Statement Calculus. Consequence5. The Statement Calculus. Applications6. The Predicate Calculus. Symbolizing Everyday Language7. The Predicate Calculus. A Formulation8. The Predicate Calculus. Validity9. The Predicate Calculus. ConsequenceChapter 5 INFORMAL AXIOMATIC MATHEMATICS1. The Concept of an Axiomatic Theory2. Informal Theories3. Definitions of Axiomatic Theories by Set-theoretical Predicates4. Further Features of Informal TheoriesChapter 6 BOOLEAN ALGEBRAS1. A Definition of a Boolean Algebra2. Some Basic Properties of a Boolean Algebra3. Another Formulation of the Theory4. Congruence Relations for a Boolean Algebra5. Representations of Boolean Algebras6. Statement Calculi as Boolean Algebras7. Free Boolean Algebras8. Applications of the Theory of Boolean Algebras to Statement Calculi9. Further Interconnections between Boolean Algebras and Statement CalculiChapter 7 INFORMAL AXIOMATIC SET THEORY1. The Axioms of Extension and Set Formation2. The Axiom of Pairing3. The Axioms of Union and Power Set4. The Axiom of Infinity5. The Axiom of Choice 6. The Axiom Schemas of Replacement and Restriction7. Ordinal Numbers8. Ordinal Arithmetic9. Cardinal Numbers and Their Arithmetic10. The von Neumann-Bernays-Gödel Theory of SetsChapter 8 SEVERAL ALGEBRAIC THEORIES1. Features of Algebraic Theories2. Definition of a Semigroup 3. Definition of a Group4. Subgroups5. Coset Decompositions and Congruence Relations for Groups6. "Rings, Integral Domains, and Fields"7. Subrings and Difference Rings8. A Characterization of the System of Integers9. A Characterization of the System of Rational Numbers10. A Characterization of the Real Number SystemChapter 9 FIRST-ORDER THEORIES1. Formal Axiomatic Theories2. The Statement Calculus as a Formal Axiomatic Theory3. Predicate Calculi of First Order as Formal Axiomatic Theories4. First-order Axiomatic Theories5. Metamathematics6. Consistency and Satisfiability of Sets of Formulas7. "Consistency, Completeness, and Categoricity of First-order Theories"8. Turing Machines and Recursive Functions9. Some Undecidable and Some Decidable Theories10. Gödel's Theorems11. Some Further Remarks about Set TheoryReferencesSymbols and NotationAuthor IndexSubject Index
