Everyone knows the real numbers, those fundamental quantities that make possible all of mathematics from high school algebra and Euclidean geometry through the calculus and beyond; and also serve as the basis for measurement in science, industry, and ordinary life. This book surveys alternative real number systems: systems that generalize and extend the real numbers yet stay close to these properties that make the reals central to mathematics.
Table of Contents
I. THE REALS
1. Axioms for the Reals
2. Construction of the Reals
II. MULTI-DIMENSIONAL NUMBERS
3. The Complex Numbers
4 The Quaternions
III. ALTERNATIVE LINES
5. The Constructive Reals
6. The Hyperreals
7. The Surreals
About the Author
Excerpt: Ch. 2 Construction of the Reals (p.35)
We will prove the existence of the real numbers two times–by twice constructing a complete, linearly ordered field. Afterwards, we prove that all complete, linearly ordered fields are isomorphic, meaning that our axiom system for the reals is categorical.
The reader might well object to this chapter in the following terms: “I’ve used the reals all my life; their properties are familiar; I know how to calculate with them; I know the calculus; I know everything. Why should I bother to construct the reals, when I know the result in advance?”
How do we know the reals exist? Just because we’ve used a few of them over the years and assumed some properties for them does not allow us to conclude that they are really out there. Physicists argued for years about properties of the æther before it was discovered that it doesn’t exist. To avoid a similar fate the reals’ existence must be proved. The simplest and most reliable way to do this is to construct them.
It was empirical evidence (the Michelson-Morley experiment, 1887) that did the æther in. The reader might argue that generations of mathematicians, scientists, and engineers have used the reals without encountering any problems: doesn’t that constitute experimental evidence for the existence of the reals? Certainly it does, but the goal of mathematics is to construct theories backed by stronger evidence than experiment. Experimental evidence only makes the existence of the reals plausible. It is possible, despite years of use, that the axioms of the reals contain a hidden inconsistency. By carrying out a construction, we prove that the axioms of the reals are consistent, or, more precisely, are at least as consistent as the axioms of set theory.
Furthermore, by constructing the reals we learn more about them and we learn methods that apply to the construction of other number systems. This the primary reason for constructing the reals here.
About the Author
Michael Henle was born in Washington, DC in 1944 and grew up in Arlington, in the Commonwealth of Virginia. He was educated at Washington-Lee High School and then at Swarthmore College (1961–1965, BA in Mathematics). He did his graduate work at Yale receiving three degrees (MA 1976, MPhil 1969, Ph.D. 1970).
He has taught at Oberlin College since fall 1970 as a Professor of Mathematics and Computer Science.
He has published a number of papers, including one in The College Mathematics Journal and the Mathematics Magazine. He has also published two books, A Combinatorial Introduction to Topology (San Francisco: W.H. Freeman and Co. 1978, reissued by Dover Publications 1994) and Modern Geometries: The Analytic Approach (Upper Saddle River: Prentice-Hall, Inc. 1996). The second edition of the latter (2001) has a slightly different title: Modern Geometries: Non-Euclidean, Projective and Discrete.
He is currently the editor of The College Mathematics Journal.
My first — and wrong — impression of the book was that it might be a modernized version of Edmund Landau’s classic Foundations of Analysis. After all, both books describe constructions of nothing else but several number systems. On closer inspection, though, I saw my error; the differences are significant. Landau constructs an expanding chain of number systems: natural numbers, integers, fractions, real and then complex numbers. The understanding of algebraic tools, say equivalence relations and classes, is taken for granted; the book is intended to be read (at least by a novice) from the beginning to the end. Henle, on the other hand, delves into the equivalence relations and equivalence classes at the beginning of the book. Integers and rationals are assumed to be known, but the constructions of rational numbers is well outlined. The number systems Henle describes are essentially different from Landau’s: real numbers, complex numbers, quarternions, constructive reals, hyperreals, and surreal numbers. Except for the first two chapters, which serve as the basis for the rest of the book, the remaining chapters are very much self-contained and could be read independently. Continued...