On the Onto-Epistemological Status of the Empty Set and the Pure Singleton


BİRGÜL O. G.

Axiomathes, 2021 (Journal Indexed in AHCI) identifier identifier

  • Publication Type: Article / Article
  • Volume:
  • Publication Date: 2021
  • Doi Number: 10.1007/s10516-021-09571-6
  • Title of Journal : Axiomathes
  • Keywords: Absence, Nihil privativum, Nihil negativum, The empty set axiom, The pure singleton, Zero

Abstract

© 2021, The Author(s), under exclusive licence to Springer Nature B.V.This article discusses the quiddity of the empty set from its epistemological and linguistic aspects. It consists of four parts. The first part compares the concept of nihil privativum and the empty set in terms of representability, arguing the empty set can be treated as a negative and formal concept. It is argued that, unlike Frege’s definition of zero, the quantitative negation with a full scope is what enables us to represent the empty set conceptually without committing to an antinomy. The second part examines the type and scope of the negation in the concept of nihil privativum and the empty set. In the third part the empty set is interpreted as a rigid abstract general term. The uniqueness of the empty set is explained via a widened version of Kripke’s notion of rigidity. The fourth part proposes a construction for the pure singleton, comparing it with Zermelo’s conception of singletons with the Ur-elements. It is argued that the proposed construction does not face the criterion and ontological inflation problems. The first conclusion of the article is that the empty set can be construed as a negative, formal and unique abstract general term, with quantitative negation full in scope. The second conclusion is that the pure singleton constructed out of the empty set construed in this way overcomes the criterion and ontological inflation problems.