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Number Systems and the Foundations of Analysis

Elliott Mendelson
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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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.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical 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

Appendix G.Complex Numbers


Index of Special Symbols