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The Simplest Axiom System for Plane Hyperbolic Geometry

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Abstract

We provide a quantifier-free axiom system for plane hyperbolic geometry in a language containing only absolute geometrically meaningful ternary operations (in the sense that they have the same interpretation in Euclidean geometry as well). Each axiom contains at most 4 variables. It is known that there is no axiom system for plane hyperbolic consisting of only prenex 3-variable axioms. Changing one of the axioms, one obtains an axiom system for plane Euclidean geometry, expressed in the same language, all of whose axioms are also at most 4-variable universal sentences. We also provide an axiom system for plane hyperbolic geometry in Tarski's language L B≡ which might be the simplest possible one in that language.

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Authors and Affiliations

  1. Department of Integrative Studies, Arizona State University West, P. O. Box, 37100, Phoenix, AZ, 85069-7100, U.S.A.

    Victor Pambuccian

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Pambuccian, V. The Simplest Axiom System for Plane Hyperbolic Geometry. Studia Logica 77, 385–411 (2004). https://doi.org/10.1023/B:STUD.0000039031.11852.66

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