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Axioms for Absolute Geometry. II

Published online by Cambridge University Press:  20 November 2018

J. F. Rigby
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In this paper I continue the process, begun in (2), of reducing and weakening the axioms of congruence needed for absolute geometry. The congruence axioms Cl*–C4*, C4**, and C5a–C5c (frequently referred to below) can all be found in (2) and will not be quoted again here. (This paper should be read in conjunction with (2); any attempt to make it self-contained would result in the repetition of large parts of (2).) The notation of (2) will be used throughout the paper.

The main result here is that axiom C5c is unnecessary. This is shown in § 1. In § 2 I discuss three other points arising from (2).

Note added in proof.Since writing this paper, I have constructed examples of (a) Archimedean planes satisfying Cl*-C4* in which not all points are isometric, (b) non-Archimedean planes satisfying Cl*-C4*but not C4**,and (c) one-dimensional geometries in which 2.1 (with “plane” replaced by “line“) is false.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 876 - 883
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Forder, H. G., The foundations of Euclidean geometry (Macmillan, London, 1927; Dover, New York, 1956).Google Scholar
2. Rigby, J. F., Axioms for absolute geometry, Can. J. Math. 20 (1968), 158–181.Google Scholar
3. Rigby, J. F., Axioms for absolute geometry. III , Can. J. Math, (to appear).Google Scholar
4. Sierpinski, W., Cardinal and ordinal numbers, Polska Akademia Nauk, Monographie Matematyczne, Tom 34 (Państwowe Wydawnictwo Naukowe, Warsaw, 1958).Google Scholar