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

Part of the Studies in Applied Philosophy, Epistemology and Rational Ethics book series (SAPERE, volume 39)


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.


Zeno’s paradoxes Argument from complete divisibility Large and small paradox Paradox of measure Metrical paradox 



We would like to thanks our colleagues from the Department of Mathematics at the Polytechnic University of Milan, from the Department of Philosophy at the University of Bologna, from the Department of Philosophy at the University of Cagliari, and from the Department in Pure and Applied Sciences at the University of Urbino for their helpful comments during the presentations of earlier versions of this paper. In particular we would like to thank Claudio Calosi for his helpful suggestions.


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Authors and Affiliations

  1. 1.Department of Pure and Applied SciencesUniversity of UrbinoUrbinoItaly
  2. 2.Department of Philosophical, Pedagogical and Economic-Quantitative SciencesUniversity of Chieti-PescaraChietiItaly

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