Is a Space Interval a Set of Infinite Points? A Very Old Question

In this paper we will address the question whether a space interval is a set of infinite points . It is a very old problem, but despite its age it is still a live issue, and one we have to confront. We will analyze some topics regarding this question using the most influential objections against it, i.e. The Large and the Small paradox (in particular its Small Horn). We will consider classical contemporary reformulations of the argument (Grünbaum in Philosophy of Science 19:280–306, 1952; Grünbaum in Modern science and Zeno’s paradoxes. Allen and Unwin, London, 1968) and the possible ‘solutions’ to it. Finally, we will propose a new formulation of the paradox and analyze its consequences. In particular, we will bring further arguments supporting the standard thesis that it is possible that a segment of space is composed of a non-denumerable set of indivisible 0-length points.