The complexity of plane hyperbolic incidence geometry is (forall)(exist)(forall)(exist)

We show that plane hyperbolic geometry, expressed in terms of points and the ternary relation of collinearity alone, cannot be expressed by means of axioms of complexity at most ∀∃∀, but that there is an axiom system, all of whose axioms are ∀∃∀∃ sentences. This remains true for Klingenberg's generalized hyperbolic planes, with arbitrary ordered fields as coordinate fields. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)