The complexity of plane hyperbolic incidence geometry is ヨヨ

摘要

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 ヨヨ, but that there is an axiom system, all of whose axioms are ヨヨ sentences. This remains true for Klingenberg's generalized hyperbolic planes, with arbitrary ordered fields as coordinate fields.