Axiomatizations of hyperbolic geometry: A comparison based on language and quantifier type complexity

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3 Scopus citations

Abstract

Hyperbolic geometry can be axiomatized using the notions of order and congruence (as in Euclidean geometry) or using the notion of incidence alone (as in projective geometry). Although the incidence-based axiomatization may be considered simpler because it uses the single binary point-line relation of incidence as a primitive notion, we show that it is syntactically more complex. The incidence-based formulation requires some axioms of the quantifier-type ∀∃∀, while the axiom system based on congruence and order can be formulated using only ∀∃-axioms.

Original languageEnglish (US)
Pages (from-to)331-341
Number of pages11
JournalSynthese
Volume133
Issue number3
DOIs
StatePublished - Dec 1 2002

ASJC Scopus subject areas

  • Philosophy
  • Social Sciences(all)

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