I make reference to Coxeter, _Introduction to Geometry_, Wiley, Second edition.
In section 12.2 _Intermediacy_ , Coxeter discusses the relation [ABC]--informally "point B is between points A and C"--and then he defines segment, interval, ray and line as sets using that relation.
Not much set theory seems to be needed, but how much? If one were to formalize this set theory should it be a theory with points as Urelemente?
Would one do better to formalize it using many-sorted first order logic with points, segments, intervals, rays and lines as sorts?
In either case, what then about triangles and other plane figures?
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