A note on the history of the Cantor set and Cantor function

Summary

A search through the primary and secondary literature on Cantor yields little about the history of the Cantor set and Cantor function. In this note, we would like to give some of that history, a sketch of the ideas under consideration at the time of their discovery, and a hypothesis regarding how Cantor came upon them. In particular, Cantor was not the first to discover “Cantor sets”. Moreover, although the original discovery of Cantor sets had a decidedly geometric flavor, Cantor's discovery of the Cantor set and Cantor function was neither motivated by geometry nor did it involve geometry, even though this is how these objects are often introduced (see for example [24]). In fact, Cantor may have come upon them through a purely arithmetic program.

The systematic study of point set topology on the real line arose during the period 1870–1885 as mathematicians investigated two problems:

1. conditions under which a function could be integrated,

2. uniqueness of trigonometric series.

It was within the framework of these investigations that the two apparently independent discoveries of the Cantor set were made; each discovery was linked to one of these problems.

Bernhard Riemann (1826–1866) spent considerable time on the first question, and suggested conditions he thought might provide an answer. Although we will not discuss the two forms his conditions took ([16], pp. 17–18), we note that one of these conditions is important as it eventually led to the development of measure-theoretic integration (see [16], p. 28).