The complexity of plane hyperbolic incidence geometry is ∀∃∀∃

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 for all there exists for all, but that there is an axiom system, all of whose axioms are for all there exists for all there exists sentences. This remains true for Klingenberg's generalized hyperbolic planes, with arbitrary ordered fields as coordinate fields. (c) 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.