Number Systems and the Foundations of Analysis

Chapter 1. Basic Facts and Notions of Logic and Set Theory 1.1 Logical Connectives 1.2 Conditionals 1.3 Biconditionals 1.4 Quantifiers 1.5 Sets 1.6 Membership. Equality and Inclusion of Sets 1.7 The Empty Set 1.8 Union and Intersection 1.9 Difference and Complement 1.10 Power Set 1.11 Arbitrary Unions and Intersections 1.12 Ordered Pairs 1.13 Cartesian Product 1.14 Relations 1.15 Inverse and Composition of Relations 1.16 Reflexivity, Symmetry, and Transitivity 1.17 Equivalence Relations 1.18 Functions 1.19 Functions from A into (Onto) B 1.20 One-One Functions 1.21 Composition of Functions 1.22 Operations Chapter 2. The Natural Numbers 2.1 Peano Systems 2.2 The Iteration Theorem 2.3 Application of the Iteration Theorem: Addition 2.4 The Order Relation 2.5 Multiplication 2.6 Exponentiation 2.7 Isomorphism, Categoricity 2.8 A Basic Existence Assumption Supplementary Exercises Suggestions for Further Reading Chapter 3. The Integers 3.1 Definition of the Integers 3.2 Addition and Multiplication of Integers 3.3 Rings and Integral Domains 3.4 Ordered Integral Domains 3.5 Greatest Common Divisor, Primes 3.6 Integers Modulo n 3.7 Characteristic of an Integral Domain 3.8 Natural Numbers and Integers of an Integral Domain 3.9 Subdomains, Isomorphisms, Characterizations of the Integers Supplementary Exercises Chapter 4. Rational Numbers and Ordered Fields 4.1 Rational Numbers 4.2 Fields 4.3 Quotient field of an Integral Domain 4.4 Ordered Fields 4.5 Subfields. Rational Numbers of a Field. Chapter 5. The Real Number System 5.1 Inadequacy of the Rationals 5.2 Archimedean Ordered Fields 5.3 Least Upper Bounds and Greatest Lower Bounds 5.4 The Categoricity of the Theory of Complete Ordered Fields 5.5 Convergent Sequences and Cauchy Sequences 5.6 Cauchy Completion. The Real Number System 5.7 Elementary Topology of the Real Number System 5.8 Continuous Functions 5.9 Infinite Series Appendix A. Equality Appendix B. Finite Sums and the Sum Notation Appendix C. Polynomials Appendix D. Finite, Infinite, and Denumerable Sets. Cardinal Numbers Appendix E. Axiomatic Set Theory and the Existence of a Peano System Appendix F. Construction of the Real Numbers via Dedekind Cuts Appendix G.Complex Numbers Bibliography Index of Special Symbols Index