Why did Euclid not need the Pasch axiom?

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Abstract

An axiom system for Euclidean geometry with Euclid’s version of the parallel postulate, in which the order axioms are introduced in terms of the separation a line introduces in the plane, as pioneered by Sperner (Math Ann 121:107–130, 1949), in which the compass can be used only to transport segments, which lacks the Pasch axiom, is shown to imply the Pasch axiom due to the very form in which Euclid chose to express his fifth postulate. This shows, as first noted without proof by Salvatore di Noi in that same year 1949, that Euclid did not need the Pasch axiom.

Original languageEnglish (US)
Article number13
JournalJournal of Geometry
Volume115
Issue number1
DOIs
StatePublished - Apr 2024

Keywords

  • 51G05
  • Axiom system
  • Euclidean geometry
  • Euclid’s postulate V
  • Primary 03B30
  • Secondary 51M05
  • The Pasch axiom

ASJC Scopus subject areas

  • Geometry and Topology

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