Sets of metrical hypotheses for geometry

Robert L. Moore
1908 Transactions of the American Mathematical Society  
In this paper is given a set of assumptions, C, concerning point, order f and congruence, which, together with a certain set of order assumptions 0,% a continuity assumption F,% and a very weak parallel assumption P0,% are sufficient for the establishment of ordinary Euclidean geometry. Each assumption of this set is independent of all the remaining ones. Moreover congruence is here an independent symbol in the sense that it would be impossible § to prove that if the order assumptions 0, TTand
more » ... 0 are true, then there must exist a relation satisfying the congruence assumptions C with reference to the points and order in terms of which O, F and P0 are stated.
doi:10.1090/s0002-9947-1908-1500823-x fatcat:pdpyhwfbmja5jghy2srwi7od3i