Description
Dover Number Systems and the Foundations of Analysis 2009 Edition by Elliott Mendelson
This study of basic number systems explores natural numbers, integers, rational numbers, real numbers, and complex numbers. Written by a noted expert on logic and set theory, it assumes no background in abstract mathematical thought. Undergraduates and beginning graduate students will find this treatment an ideal introduction to number systems, particularly in terms of its detailed proofs.Starting with the basic facts and notions of logic and set theory, the text offers an axiomatic presentation of the simplest structure, the system of natural numbers. It proceeds, by set-theoretic methods, to an examination of integers that covers rings and integral domains, ordered integral domains, and natural numbers and integers of an integral domain. A look at rational numbers and ordered fields follows, along with a survey of the real number system that includes considerations of least upper bounds and greatest lower bounds, convergent and Cauchy sequences, and elementary topology. Numerous exercises and several helpful appendixes supplement the text. Table of contents : - Chapter 1. Basic Facts and Notions of Logic and Set Theory1.1 Logical Connectives1.2 Conditionals1.3 Biconditionals1.4 Quantifiers1.5 Sets1.6 Membership. Equality and Inclusion of Sets1.7 The Empty Set1.8 Union and Intersection1.9 Difference and Complement1.10 Power Set1.11 Arbitrary Unions and Intersections1.12 Ordered Pairs1.13 Cartesian Product1.14 Relations1.15 Inverse and Composition of Relations1.16 Reflexivity, Symmetry, and Transitivity1.17 Equivalence Relations1.18 Functions1.19 Functions from A into (Onto) B1.20 One-One Functions1.21 Composition of Functions1.22 OperationsChapter 2. The Natural Numbers2.1 Peano Systems2.2 The Iteration Theorem2.3 Application of the Iteration Theorem: Addition2.4 The Order Relation2.5 Multiplication2.6 Exponentiation2.7 Isomorphism, Categoricity2.8 A Basic Existence AssumptionSupplementary ExercisesSuggestions for Further ReadingChapter 3. The Integers3.1 Definition of the Integers3.2 Addition and Multiplication of Integers3.3 Rings and Integral Domains3.4 Ordered Integral Domains3.5 Greatest Common Divisor, Primes3.6 Integers Modulo n3.7 Characteristic of an Integral Domain3.8 Natural Numbers and Integers of an Integral Domain3.9 Subdomains, Isomorphisms, Characterizations of the IntegersSupplementary ExercisesChapter 4. Rational Numbers and Ordered Fields4.1 Rational Numbers4.2 Fields4.3 Quotient field of an Integral Domain4.4 Ordered Fields4.5 Subfields. Rational Numbers of a Field.Chapter 5. The Real Number System5.1 Inadequacy of the Rationals5.2 Archimedean Ordered Fields5.3 Least Upper Bounds and Greatest Lower Bounds5.4 The Categoricity of the Theory of Complete Ordered Fields5.5 Convergent Sequences and Cauchy Sequences5.6 Cauchy Completion. The Real Number System5.7 Elementary Topology of the Real Number System5.8 Continuous Functions5.9 Infinite SeriesAppendix A. EqualityAppendix B. Finite Sums and the Sum NotationAppendix C. PolynomialsAppendix D. Finite, Infinite, and Denumerable Sets. Cardinal NumbersAppendix E. Axiomatic Set Theory and the Existence of a Peano SystemAppendix F. Construction of the Real Numbers via Dedekind CutsAppendix G.Complex NumbersBibliographyIndex of Special SymbolsIndex