David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Studia Logica 77 (3):385 - 411 (2004)
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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Marco Aiello, Guram Bezhanishvili, Isabelle Bloch & Valentin Goranko (2012). Logic for Physical Space. Synthese 186 (3):619-632.
Victor Pambuccian (2011). The Simplest Axiom System for Plane Hyperbolic Geometry Revisited. Studia Logica 97 (3):347 - 349.
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