Axiomatizations of Hyperbolic Geometry: A Comparison Based on Language and Quantifier Type Complexity

Author: Pambuccian V.

Source: Synthese, Volume 133, Number 3, December 2002 , pp. 331-341(11)

Publisher: Springer

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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 \forall\exists\forall, while the axiom system based on congruence and order can be formulated using only \forall\exists-axioms.

Language: English

Document Type: Research article

Affiliations: 1: Department of Integrative Studies Arizona State University West P.O. Box 37100 Phoenix AZ 85069-7100 U.S.A.

Publication date: 2002-12-01