Do somewhere full axiomatic system axioms of one-dimensional geometry presented in Hilbert's "Foundations of Geometry" style? (With incidence, order, congruence and continuity axioms, as described in existing systems for three- and two-dimensional cases.)
2026-02-23 17:00:58.1771866058
Axioms of one-dimension geometry in Hilbert's style
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If you remove the lower dim axiom (saying that there are three non collinear points) from Hilbert's or Tarski's axioms and add an upper dim axiom saying that all points are collinear, I guess you obtain an axiom system for one-dimentional geometry.
You can find axiomatization of the order relation on a line in Section 2 of the following paper by Victor Pambuccian: The axiomatics of ordered geometry: I. Ordered incidence spaces.