Abstract
We give a non-Paschian plane based on the property of betweenness which cannot be derived from an ordering of the points of a line. In this model there is no possibility to define the congruence of segments but we can define angle, triangle and angle measure, respectively. With respect to our definitions the plane has an elliptic character, meaning that the sum of the angles of a triangle is greater than \({\pi}\).
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References
Adler A.: Determinateness and Pasch axiom. Canad. Math. Bull. 16(2), 159–160 (1973)
Berger M.: Geometry I–II. Springer-Verlag, Berlin (1994)
Bonola R.: Non-euclidean geometry. Dover Publication, USA (1955)
Coxeter H.S.M.: Non-Euclidean geometry. Mathematical Association of America, USA (1998)
Horváth Á.G.: Wonderful geometry. Typotex, Hungary (2013)
Horváth Á.G.: Malfatti’s problem on the hyperbolic plane. Studia Sci. Math. 51(2), 201–212 (2014)
Horváth Á.G.: Hyperbolic plane geometry revisited. J. Geom. 106(2), 341–362 (2015)
Horváth Á.G.: On the hyperbolic triangle centers. Stud. Univ. Zilina 27(1), 11–34 (2015)
Hilbert, D.: The Foundations of Geometry, 2nd edn. Open Court 1980, Chicago (1899)
Kreuzer A.: Klassifizierung von Halbordnungen. J. Geom. 33(1), 73–82 (1988)
Moussong, G.: The models of the hyperbolic geometry (in hungarian). In: Bolyai emlékkönyv Bolyai János születésének 200. évfordulójára, Vince Kiadó, pp. 143–165 (2004)
Givant S., Tarski A.: Tarski’s system of geometry. Bull. Symb. Logic 5(2), 175–214 (1999)
Szczerba W.: Independence of Paschs Axiom. Bul. Acad. Polon. Sci. Ser. Sci. Tech. 11(18), 659–666 (1970)
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Horváth, Á.G. On the non-Paschian ordered planes. J. Geom. 108, 255–263 (2017). https://doi.org/10.1007/s00022-016-0337-z
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DOI: https://doi.org/10.1007/s00022-016-0337-z