Simple Foundations for the Hyperbolic Plane

TL;DR

This paper simplifies the foundational axioms of hyperbolic geometry by replacing complex theorems with two more straightforward axioms, advancing the understanding of its incidence structure.

Contribution

It introduces a more streamlined axiom system for hyperbolic geometry that eliminates the need for Pappus and Desargues theorems, simplifying the foundational framework.

Findings

Hyperbolic geometry can be axiomatized with fewer axioms.

The new axioms are simpler and more direct.

Hyperbolic geometry remains incidence geometry under the new system.

Abstract

H. L. Skala (1992) gave the first elegant first-order axiom system for hyperbolic geometry by replacing Menger's axiom involving projectivities with the theorems of Pappus and Desargues for the hyperbolic plane. In so doing, Skala showed that hyperbolic geometry is incidence geometry. We improve upon Skala's formulation by doing away with Pappus and Desargues altogether, by substituting for them two simpler axioms.