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During the early 1830's Bernard Bolzano, working in Prague, wrote amanuscript giving a foundational account of numbers and their properties. Inthe final section of his work he described what he called `infinite numberexpressions' and `measurable numbers'. This work was evidently an attempt toprovide an improved proof of the sufficiency of the criterion usually known asthe `Cauchy criterion' for the convergence of an infinite sequence. Bolzano hadin fact published this criterion four years earlier than Cauchy who, in hiswork of 1821, made no attempt at a proof. Any such proof required theconstruction or definition of real numbers and this, in essence, was whatBolzano achieved in his work on measurable numbers. It therefore pre-dates thewell-known constructions of Dedekind, Cantor and many others by severaldecades. Bolzano's manuscript was partially published in 1962 and more fullypublished in 1976. We give an account of measurable numbers, the propertiesBolzano proved about them, and the controversial reception they have promptedsince their publication.

Bolzano's measurable numbers: are they real?