Geodesic
Axioms for Absolute Geometry. III
Canadian journal of mathematics, Tome 22 (1970) no. 1, pp. 185-190

Voir la notice de l'article provenant de la source Cambridge University Press

This paper is a continuation of [1; 2]. In [2], I stated that I had been unable to construct examples of planes satisfying various conditions. Some of the examples that I have since constructed are given below. A discussion of one-dimensional absolute geometries, with examples, will be given in a separate paper. The relevant parts of [1] and [2] are [1, § 1, § 2 up to 2.4; 2, § 2]. We shall use the notation and terminology of [1; 2]; the axioms Cl*-C4* and C4** (referred to below) can all be found in [1].We shall show here that spaces of dimension greater than 1 exist, both Archimedean and non-Archimedean, satisfying Cl*-C4*, in which not all points are isometric, and that C4** does not follow from Cl*-C4* in non- Archimedean geometries of dimension greater than 1. 
Rigby, J. F. Axioms for Absolute Geometry. III. Canadian journal of mathematics, Tome 22 (1970) no. 1, pp. 185-190. doi: 10.4153/CJM-1970-022-7
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[1] 1. Rigby, J. F., Axioms for absolute geometry, Can. J. Math. 20 (1968), 158–181. Google Scholar

[2] 2. Rigby, J. F., Axioms for absolute geometry. II, Can. J. Math. 21 (1969), 876–883. Google Scholar

[3] 3. Sierpinski, W., Cardinal and ordinal numbers, Polska Akademia Nauk, Monographie Matematyczne, Tom 34 (Panstwowe Wydawnictwo Naukowe, Warsaw, 1958). Google Scholar

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