Dehn plane

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The Dehn plane consists of all points (x,y), where x and y are finite hyperreal numbers. The parallel postulate fails in the Dehn plane.

All pairs (x, y) where x and y are any members of a hyperreal field F with the metric $||(x,y)|| = \sqrt{x^2+y^2}$ taking values in F gives a non-standard model of Euclidean geometry. The parallel postulate is true in this model, but if the deviation from the perpendicular is infinitesimal, the intersecting lines intersect at a point which is not in the finite part of the plane. Hence, if the model is restricted to the finite part of the plane, a geometry is obtained in which the parallel postulate fails. Modulo caveats about the use of proper classes which can be got round by using Grothendieck universes, the surreal numbers (for some inaccessible cardinal) define a suitable hyperreal field, and therefore a model of the Dehn plane.