Number Systems and the Foundations of Analysis

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.

Genres: Mathematics

358 pages, Paperback

Published December 18, 2008

Reviews

[Beau · Rating 5 out of 5]

I read the 1973 Academic Press edition (not the Dover edition). This book should be a model for how a math textbook is written. It is extremely clear and complete. The typesetting is immaculate. The book contains fewer errors and typos than any other textbook I've read. Everything is proven with the utmost rigor. University math departments usually have a course that serves as a bridge between the lower-level courses and upper-level courses like real analysis and abstract algebra. This book deserves to be assigned reading in all such courses. Every aspiring mathematician needs to know that a complete ordered field exists and that the theory of complete ordered fields is categorical. If you are a mathematics major, stop taking the real number system for granted and read this book.

[Soren Petersen · Rating 5 out of 5]

It’s a college textbook, but a very readable one. If you ever wanted a know how to rigorously derive real number arithmetic from first principles, boy is this ever the book for you! Starting with the Peano axioms (roughly speaking, they provide characterizations of “first”, “next”, and “all”) plus some basic set theory and logic, the book defines whole number addition and multiplication, and then negative numbers/subtraction, rational numbers/division, and finally real number arithmetic.

If this sort of thing sounds at all appealing, I highly recommend it. The strictly mathematical prerequisites are low: advanced techniques are developed as needed. In principle, there’s nothing here that a sufficiently motivated high school student couldn’t follow, although some familiarity with mathematical proofs would be very helpful.

[Giovanni Alberto Orlando]

Is this book necessary? ... Yes ... it is!