Bernard Bolzano: from Topological to Arithmetical Continuum and Back Again

TL;DR

This paper explores Bolzano's evolving concept of the continuum, his development of measurable numbers, and their role in understanding the properties and foundations of the real number system, linking topological and arithmetical perspectives.

Contribution

It clarifies Bolzano's concept of the continuum, analyzes his measurable numbers, and connects their properties to modern interpretations of the real numbers and continuum theory.

Findings

Bolzano's continuum is based on an infinite class of points with completeness.

Measurable numbers are key to understanding Bolzano's continuum and its properties.

Theorems on measurable numbers relate to their completeness and Bolzano's continuum concepts.

Abstract

Although Bolzano's concept of the continuum has gradually evolved, the basis remained the same: the continuum as an infinite class of points arranged in such a way that the so-called \emph{Bolzano completeness} holds. Bolzano realized over time that the central role of a general comprehension of continuum plays in its arithmetic description and constructed his measurable numbers. Their interpretations in the standard and non-standard models of real numbers clarify their relationship and also suggest why Bolzano did not base his theory of functions on infinitesimal numbers. The three main theorems on measurable numbers are various forms of their completeness. I argue why the second one is indeed the \emph{Supremum Theorem} and that an important corollary of the third one is a proof of the \emph{Bolzano completeness}. Only when the notion of continuum was supported by measurable numbers could Bolzano, in his last book, \emph{Paradoxes of the Infinite}, confidently defend the general properties of the continuum and reject the paradoxes associated with them.