Ein Axiomensystem für den Anschauungsraum. (An axiom system for the Euclidean space) (Q578586)

The author displays an interesting axiom system, consisting of only seven axioms, for ``complete'' Euclidean space of dimension three. The conceptually convincing axioms are, briefly: axioms of non-degeneracy, co-planarity of three points, planarity of lines, parallelism, and existence of a parallelogram; a ``Euclidean'' axiom which states that a line through an inner point of a sphere cuts the sphere, and a ``completeness'' axiom. A classical Euclidean space of dimension 3 satisfies all of the above except the last, while a normal or fanonian Eucliden space of dimension 3 satisfies all but the last two axioms. The rest of the paper discusses these three types of spaces, and answers questions concerning co-ordinatization, isomorphism, etc.. The paper ends by showing that the axioms are ``consecutively'' independent, i.e. that each axiom is independent of the previous axioms introduced in the above order.