Do Bolzano’s measurable numbers constitute a construction of the real numbers?

Determine whether Bolzano’s measurable numbers, as developed in the VIIth section "Infinite Quantity Concepts" of Reine Zahlenlehre, indeed provide a rigorous construction of the real numbers, resolving reported errors and inconsistencies and clarifying whether the system yields the real-number structure.

Background

The paper reviews Bolzano’s measurable numbers and notes historical and technical debate about their adequacy as a foundation for the real numbers. Various scholars have criticized inconsistencies or classified the system as a precursor; others argue that a small modification to the initial definition could repair the theory.

The author emphasizes that an affirmative resolution would place Bolzano’s construction decades ahead of Cantor’s and Dedekind’s work, underscoring the historical significance of the question. The paper also discusses interpretations via standard and non-standard models to clarify relationships and potential corrections, but concludes that consensus has not been reached.