Bolzano's measurable numbers: are they real?

TL;DR

This paper explores Bolzano's early 19th-century work on measurable numbers, highlighting its historical significance in the development of real number theory and its foundational role predating Dedekind and Cantor.

Contribution

It provides a detailed analysis of Bolzano's concept of measurable numbers and their properties, emphasizing their precedence in real number construction.

Findings

Bolzano's measurable numbers predate Dedekind and Cantor.

Bolzano's work offers an early proof of the Cauchy criterion.

The reception of Bolzano's measurable numbers has been historically controversial.

Abstract

During the early 1830's Bernard Bolzano, working in Prague, wrote a manuscript giving a foundational account of numbers and their properties. In the final section of his work he described what he called `infinite number expressions' and `measurable numbers'. This work was evidently an attempt to provide an improved proof of the sufficiency of the criterion usually known as the `Cauchy criterion' for the convergence of an infinite sequence. Bolzano had in fact published this criterion four years earlier than Cauchy who, in his work of 1821, made no attempt at a proof. Any such proof required the construction or definition of real numbers and this, in essence, was what Bolzano achieved in his work on measurable numbers. It therefore pre-dates the well-known constructions of Dedekind, Cantor and many others by several decades. Bolzano's manuscript was partially published in 1962 and more fully published in 1976. We give an account of measurable numbers, the properties Bolzano proved about them, and the controversial reception they have prompted since their publication.