Abstract
Hyperbolic geometry can be axiomatized using the notions of order andcongruence (as in Euclidean geometry) or using the notion of incidencealone (as in projective geometry). Although the incidence-based axiomatizationmay be considered simpler because it uses the single binary point-linerelation of incidence as a primitive notion, we show that it issyntactically 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 beformulated using only \forall\exists-axioms.
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Pambuccian, V. Axiomatizations of Hyperbolic Geometry: A Comparison Based on Language and Quantifier Type Complexity. Synthese 133, 331–341 (2002). https://doi.org/10.1023/A:1021294808742
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DOI: https://doi.org/10.1023/A:1021294808742