Abstract
While the Euclidean parallel postulate P can be replaced with the conjunction of the two axioms, “Given three parallel lines, there is a line that intersects all three of them” (ML) and “Given a line a and a point P on a, as well as two intersecting lines m and n, both parallel to a, there exists a line g through P which intersects m but not n” (S) to obtain plane Euclidean geometry based on Hilbert’s plane absolute geometry \({{\mathcal {A}}}\), it is shown if \({{\mathcal {A}}}\) is slightly weakened, in the sense that either the order axioms are weakened or a congruence axiom is weakened, then the conjunction of ML and S is no longer equivalent to P.
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Acknowledgements
I thank Andrew Bremner for having promptly answered a question that was instrumental in proving that \({\mathfrak {M}}\) satisfies ML and the referee for a crucial remark regarding ordered metric planes.
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Pambuccian, V. On a splitting of the parallel postulate. J. Geom. 113, 12 (2022). https://doi.org/10.1007/s00022-022-00626-6
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DOI: https://doi.org/10.1007/s00022-022-00626-6