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

Synthese 133 (3):331 - 341 (2002)
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 forallexistsforall, while the axiom system based on congruence and order can beformulated using only forallexists-axioms.
Keywords Philosophy   Philosophy   Epistemology   Logic   Metaphysics   Philosophy of Language
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Reprint years 2004
DOI 10.1023/A:1021294808742
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