Variations on a theme of Euclid

Abstract

Moritz Pasch's axiomatic definition of a Euclidean line interval [1, 2] is generalized to allow arbitrarily many multiple points and to apply especially to the point sets associated with unclosed geodesics having two endpoints. Many other ‘uniquely extensible’ examples of the theory exist (see section 2). The main primitive ideas are a set of objects S(points), a set of subsets of S(G strings, generalizing intervals) and end points of G strings. It is a theory in the large, described with elementary set theory. To construct a theory of linear order we (1) define (in either of two ways) a relation on the set of Gstrings having endpoint A,(2) show that the relation is an equivalence relation, assuming several generalizations of Pasch's order properties of intervals, (3) define a ray emanating from Aas an equivalence class of Gstrings with endpoint Aand show that a ray is linearly ordered. A G string is ‘oriented’ with theorem 1.4. To specialize this order theory to the Euclidean or projective lines read theorems 3.2, 3.3 and definition. This treatment may be a possible first step towards a synthetic theory of geodesics in the large. (Synthetic proofs are sometimes simplest.)