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.
ASJC Scopus subject areas
- Social Sciences(all)