Regions-based two dimensional continua: The euclidean case

Geoffrey Hellman, Stewart Shapiro

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

We extend the work presented in [7, 8] to a regions-based, two-dimensional, Euclidean theory. The goal is to recover the classical contin-uum on a point-free basis. We first derive the Archimedean property for a class of readily postulated orientations of certain special regions, "gener-alized quadrilaterals" (intended as parallelograms), by which we cover the entire space. Then we generalize this to arbitrary orientations, and then establishing an isomorphism between the space and the usual point-based R × R. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit "extremal clause" (to the effect that "these are the only ways of generating regions"), and we have no axiom of induction other than ordinary numerical (mathematical) induction. Finally, having explicitly defined 'point' and 'line', we will derive the characteristic Paral-lel's Postulate (Playfair axiom) from regions-based axioms, and point the way toward deriving key Euclidean metrical properties.

Original languageEnglish (US)
Pages (from-to)499-534
Number of pages36
JournalLogic and Logical Philosophy
Volume24
Issue number4
DOIs
StatePublished - Dec 2015

Keywords

  • Archimedean Property
  • Euclidean Geometry
  • Gunk
  • Mereology
  • Point-Free Geometry
  • Points
  • Tarski

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