Essays on the Theory of Numbers

This book is a 1901 translation of two historically-important short books on the construction of the real numbers and of the integers. Both books are part of the “arithmetization of analysis” project that was underway in the late nineteenth century. The goal was to develop a rigorous theory of the real numbers that did not depend on an appeal to the geometry of the real line. The second essay was among the first works to use Cantor’s newly-developed theory of infinite sets. The “theory of numbers” in the title refers not to the general study of the integers, but to the construction of numbers.

The first book is a translation of the 1872 study Stetigkeit und irrationale Zahlen. In it Dedekind expounds a careful construction of the reals from the rationals that he developed in 1858, using what we today call the Dedekind cut. The construction is given in detail, but he only sketches the proofs of the properties of the reals. The Dedekind cut approach has fallen out of favor in recent years and the reals are usually constructed using Cauchy sequences, a method that generalizes easily to general metric spaces. Dedekind cuts are easier to conceptualize, and a few modern texts still use them, either in the main exposition or in an appendix. A complete and rigorous development is in Landau’s 1930 Foundations of Mathematics.

The second book is a translation of the 1888/1893 Was sind und was sollen die Zahlen?. This work starts from the beginning by developing some naive set theory and using this to construct what are essentially Peano’s axioms, then using these in a now-standard development to derive the properties of the integers. The book is hard to follow today because the notation and terminology have changed so much since Dedekind’s day.

The usual set-theoretic development today is to start with the empty set and recursively build up the positive integers as a sequence of sets, each containing the previous one as an element. Dedekind took a very different but equivalent approach, starting with any infinite set. Because the set is infinite, there is a one-to-one mapping from the set into a proper subset of itself, and Dedekind defines the successor operation as this mapping, and the number 1 as any element that is not in the range of this mapping. Part of the exposition here is to show that all such developments are isomorphic.

Most texts on set theory give an exposition of the modern development of the Peano axioms from set theory; an especially clear and concise exposition is in Halmos’s 1960 Naive Set Theory. The Landau text referenced above starts with the Peano axioms and presents a complete and rigorous development of the complex numbers.

Dedekind’s two books are historically important, but are they important for anything else? I say No. There are now better developments of the subjects covered here, that incorporate all the insights of Dedekind, and unless you are a historian or want to “learn from the masters,” there is no reason to read the books today.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.