Elliptic geometry is not a subset of absolute geometry (as absolute geometry is typically formalized). The problem isn't with Euclid's other postulates, it's wi

Elliptic geometry is not a subset of absolute geometry (as absolute geometry is typically formalized). The problem isn't with Euclid's other postulates, it's with things that Euclid didn't formalize, specifically, betweenness. In absolute geometry, given any line containing three distinct points, exactly one of the points will be between the other two. There's no sensible way to define this in elliptic geometry. Intuitively, this makes sense; when you have three points on a line in elliptic geometry, you can travel from any of the three points to any other without passing by the third point. It's been a while since I've actively studied this, but my recollection is that all you have to do is replace the axioms for betweenness with axioms for separation, and you get a theory which is suitable for elliptic geometry. (Separation is an arity-4 relationship on points. Intuitively, given any four distinct and collinear points, A and C separate B and D if the order of the points on the line (traveling in either direction) is ABCD.) https://en.wikipedia.org/wiki/Foundations_of_geometry#Elliptic_geometry https://en.wikipedia.org/wiki/Foundations_of_geometry#Ellipt... appears to agree