On the simplicity of ordered geometry

Research output: Contribution to journalArticlepeer-review


The Pasch axiom P is shown to be equivalent, given the linear order axioms, to the conjunction of Pasch's Theorem PT with the Weak Crossbar Theorem WCBT. Replacing P with PT and WCBT one gets a simplest axiom system for ordered planes. A likely missing link between PT and the outer form of the Pasch axiom OP, as well as an axiom system for the fragment of the geometry of ordered planes that can be expressed by at most 5 variables, when written in prenex form, are also determined. It is an open problem whether P can be proved inside the 5-variable fragment of plane ordered geometry.

Original languageEnglish (US)
Pages (from-to)153-163
Number of pages11
JournalJournal of Geometry
Issue number1
StatePublished - Apr 2013


  • Pasch's theorem
  • The Pasch axiom
  • ordered planes
  • simplicity
  • splitting an axiom
  • the inner and outer form of the Pasch axiom

ASJC Scopus subject areas

  • Geometry and Topology


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