The Simplest Axiom System for Plane Hyperbolic Geometry |
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Authors: | Pambuccian Victor |
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Institution: | (1) Department of Integrative Studies, Arizona State University West, P. O. Box, 37100, Phoenix, AZ, 85069-7100, U.S.A. |
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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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Keywords: | Hyperbolic geometry constructive axiomatization Euclidean geometry simplicity |
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