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 A, it is shown if 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.
Original language | English (US) |
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Article number | 12 |
Journal | Journal of Geometry |
Volume | 113 |
Issue number | 1 |
DOIs | |
State | Published - Apr 2022 |
Keywords
- Lotschnittaxiom
- metric-Euclidean planes
- ordered geometry
- Parallel postulate
ASJC Scopus subject areas
- Geometry and Topology