Axiomatizations of hyperbolic geometry: A comparison based on language and quantifier type complexity (Q1868171)

The author compares systems of axioms for the (plane) hyperbolic geometry, one, \(L(B,\equiv)\), that uses the notations of order ``\(B\)'' and congruence ``\(\equiv\)'' (given by a ternary betweenness and a quaternary congruence relation) and another one that uses the notion of incidence. For \(L(B,\equiv)\) collinearity is expressed through the betweenness relation. The author considers eleven axioms (T1--T11) for \(L(B,\equiv)\) using only \(\forall \exists\)-axioms. (T1--T10) resp. (T1--T9,T11) characterize the Euclidean resp. hyperbolic planes over Pythagorean resp. Euclidean fields. The incidence-based formulation requires some axioms of the quantifier-type \(\forall\exists\forall\). Here the author claims two axioms, (MS4) and (MS5), which allow him to introduce ``rays'' and ``rimpoints (ends)''. Then he claims that two rimpoints can be joined by a line (MS6), that Pascal's theorem for rimpoints (MS7), and that the axioms of Pappus (MS8) and Desargues (MS9) are satisfied.