In order to possibly gain one common axiomatic framework from which to derive synthetically as many of the important geometries as possible, my idea is to base as many geometries as possible on Hilbert's axioms for 3D Euclidean geometry, by either changing some of the axioms, by omitting some of them or by adding new ones. (Maybe this is even possible for higher dimensioninal geometries.)
What about affine geometry in 3D (affine space)? Affine geometry is usually characterized by "omitting from Euclidean geometry the notions of distance and angle". I am still not sure whether I understand that properly. What does this mean with respect to Hilbert's axioms? Can we simply omit the primitive of congruence, the third group of the (congruence) axioms and the Archimedean property of the fifth group?
Are there any redundances by the first (incidence) and second (order) group of axioms with respect to affine geometry?
Surely the parallel axiom of the fourth group has to stay.
I think the first (the Archimedean) axiom of the fifth group has to be omitted because of the lack of the primitive of congruence. But I am not sure about the second one on completeness.