Hilbert's Statement of Pasch's Axiom
David Hilbert uses Pasch's axiom in his book Foundations of Geometry which provides an axiomatic basis for Euclidean geometry. Depending upon the edition, it is numbered either II.4 or II.5 and is stated as:
- Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A, B,C. If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of segment BC.
The fact that both segments AC and BC are not intersected by the line a is proved in Supplement I,1, which was written by P. Bernays.
In Hilbert's treatment, this axiom appears in the section concerning axioms of order and is referred to as a plane axiom of order. Since he does not phrase the axiom in terms of the sides of a triangle there is no need to talk about internal and external intersections of the line a with the sides of the triangle ABC. However, if one applies the triangle terminology to Hilbert's version of the axiom, the fact that it is equivalent to Pasch's version is apparent.
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