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OM DEN PLANA GEOMETRINS AXIOMSYSTEM

GUNNAR AF HÄLLSTRÖM
Nordisk Matematisk Tidskrift
Vol. 9, No. 4 (1961), pp. 145-166
Published by: Mathematica Scandinavica
Stable URL: http://www.jstor.org/stable/24524741
Page Count: 22
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OM DEN PLANA GEOMETRINS AXIOMSYSTEM
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Abstract

The article contains an axiomatic development of plane absolute geometry, up to and including the theorem of parallels, but not introducing the circle. In addition to the completeness and consistency of the system of axioms, the following conditions are satisfied: 1° The axioms are formulated on the basis of the first concepts and notions only; later definitions like "angle" do not appear explicitly. Congruency of angles is based on the following axiom: If B is between A and C, B' between A' and C', and if AB ≡ A'B', BC ≡ B'C', AD ≡ A'D', BD ≡ B'D', then CD ≡ C'D'. 2° It is tried to avoid "overlapping" and so to make the axioms completely independent. It is a well known — and often used — fact, that a certain redundancy in the system of axioms greatly facilitates the later development. In the article, much of the complication is contributed to the lack of an "auxiliary" axiom of the type: If A'B' ≡ AB, there are exactly two triangles A'B'C' and A'B'C'', congruent to the given triangle ABC.

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