Abstract

A New Chance for Infinitesimals? This article discusses the connection between the Zenonian paradox of magnitude and probability on infinite sample spaces. Two important premises in the Zenonian argument are: the Archimedean axiom, which excludes infinitesimal magnitudes, and perfect additivity. Standard probability theory uses real numbers that satisfy the Archimedean axiom, but it rejects perfect additivity. The additivity requirement for real-valued probabilities is limited to countably infinite collections of mutually incompatible events. A consequence of this is that there exists no standard probability function that describes a fair lottery on the natural numbers. If we reject the Archimedean axiom, allowing infinitesimal probability values, we can retain perfect additivity and describe a fair, countable infinite lottery. The article gives a historical overview to understand how the first option has become the current standard, whereas the latter remains ‘non-standard’.