Less then Something, but More then Nothing - Zeno, Infinitesimals and the Continuum Paradox

Abstract

The hypothesis is that a line can be divided into an infinite number of parts. The question is whether extension is a property of these infinitely small parts of the line. If the infinitely small parts are extensional then they can be further divided. On the other hand, if they have no extension how can they compose the line? The sum of non-extensional units can not be extensional. None of the answers explains the problem which is thus further considered to be the continuum paradox. Although the infinitesimal calculus does not solve the continuum paradox, it finds a way to deal with it by introducing infinitesimals as infinitely small parts that are neither extensional nor non-extensional. Even if infinitesimals have no extension, they can compose a line.