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(2002) Synthese 133 (3).
Axiomatizations of hyperbolic geometry
a comparison based on language and quantifier type complexity
Victor Pambuccian
pp. 331-341
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 forallexistsforall, while the axiom system based on congruence and order can beformulated using only forallexists-axioms.
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Pambuccian, V. (2002). Axiomatizations of hyperbolic geometry: a comparison based on language and quantifier type complexity. Synthese 133 (3), pp. 331-341.
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