Mendelson E. Number Systems and the Foundations of Analysis

New York: Dover Publications Inc., 2009. — 364 p.This is a textbook on the construction of the real numbers, aimed at students with no experience in proofs. It is a wide-ranging work, not only constructing the numbers but placing them in the context of more general algebraic structures and developing some topology of the real line. It has some discussion of continuous functions, convergence, and infinite series. It leads off with a lengthy section on logic, set theory, and functions.
The book has a clever approach to the negative numbers (p. 87), avoiding a lot of the tedious special-casing that these usually require: after having created the positive integers, the book defines the integers as equivalence classes of ordered pairs of positive integers, where all the pairs (a, b) with the same value of a – b form a class. Apart from this the book is fairly conventional in its approach. It uses Cauchy sequences to develop the reals, but gives Dedekind cuts in an appendix.
This book is nearly three times as long as Landau’s classic Foundations of Analysis, but it is not a flabby book. The added length comes partly because it covers a wider range of topics (Landau focuses single-mindedly on constructing the complex numbers from the Peano postulates) and because it has many exercises and examples (Landau has none).
One concern I have with the book is that it may be too wide-ranging; it’s hard to imagine any single college class or any reader who would want to study all these things at the same time. Another concern is that it takes a long time to get to the nominal subject of the book, number systems; by page 156 we have only developed the integers. That being said, it is well-written and a nice treatment of the subject, and has a bargain price.Basic Facts and Notions of Logic and Set Theory
Logical Connectives
Conditionals
Biconditionals
Quantifiers
Sets
Membership Equality and Inclusion of Sets
The Empty Set
Union and Intersection
Difference and Complement
Power Set
Arbitrary Unions and Intersections
Ordered Pairs
Cartesian Product
Relations
Inverse and Composition of Relations
Reflexivity, Symmetry, and Transitivity
Equivalence Relations
Functions
Functions from A into (Onto) B
One-One Functions
Composition of Functions
OperationsThe Natural Numbers
Peano Systems
The Iteration Theorem
Application of the Iteration Theorem: Addition
The Order Relation
Multiplication
Exponentiation
Isomorphism, Categoricity
A Basic Existence Assumption
Supplementary Exercises
Suggestions for Further ReadingThe Integers
Definition of the Integers
Addition and Multiplication of Integers
Rings and Integral Domains
Ordered Integral Domains
Greatest Common Divisor, Primes
Integers Modulo n
Characteristic of an Integral Domain
Natural Numbers and Integers of an Integral Domain
Subdomains, Isomorphisms, Characterizations of the Integers
Supplementary ExercisesRational Numbers and Ordered Fields
Rational Numbers
Fields
Quotient field of an Integral Domain
Ordered Fields
Subfields Rational Numbers of a FieldThe Real Number System
Inadequacy of the Rationals
Archimedean Ordered Fields
Least Upper Bounds and Greatest Lower Bounds
The Categoricity of the Theory of Complete Ordered Fields
Convergent Sequences and Cauchy Sequences
Cauchy Completion The Real Number System
Elementary Topology of the Real Number System
Continuous Functions
Infinite SeriesAppendix 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 SystemAppendix F Construction of the Real Numbers via DedekindAppendix GComplex NumbersIndex of Special Symbols