Consistency and Categoricalness of the Hyperbolic Axioms; The Classical Models

Abstract

We take as primary model of the hyperbolic plane an abstract surface S in the sense of Section 5.3, whose geometry is determined by the methods of differential geometry in such a way that all the axioms of the hyperbolic plane are satisfied. We consider several different coordinate systems, each of which covers the entire surface S, some of which are more useful than others for certain purposes. Each coordinate system leads to one of the classical models of the hyperbolic plane based on Euclidean geometry. Differential geometry is based on analysis, which is based on the real number system ℝ. It follows that the hyperbolic axioms are consistent, if the axioms of ℝ are consistent. It is proved that the axiom system is categorical, in the that any model of the hyperbolic plane is isomorphic to any other model. Lastly, as an amusement, we describe a hyperbolic model of the Euclidean plane.