An Examination of Richard Dedekind’s “Continuity and Irrational Numbers”

This paper explicates each of the seven sections of mathematician Richard Dedekind’s 1858 essay “Continuity and Irrational Numbers”, which he eventually published in 1872. In this essay, he provides a simple, completely arithmetic proof of the continuity of the set of real numbers, a property on which the validity of many mathematical theorems, especially those in calculus, depend. The intent of this paper is to familiarize the reader with the details of Dedekind’s argument, which is exceptionally easy to follow and self-contained. Although the real numbers were often imagined as points lying on an infinite line, as a calculus instructor in Zürich, Switzerland, Dedekind became deeply troubled by the need to reference geometry when teaching his students concepts such as functions and limits. This inspired him to develop a rigorous arithmetic foundation for the set of real numbers, in which, through the use of what are now called “Dedekind cuts,” he cleverly defines both rational and irrational numbers, and demonstrates how they fit together to form the continuum of real numbers. Alternative viewpoints and criticisms of his work exist, and one is briefly discussed at the conclusion of the paper, though it is noted that Dedekind’s essay accomplishes the goal he set for himself in its preface