On a problem of H. N. Gupta
The problem mentioned in the title is as follows. \textit{H. N. Gupta} [Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 13, 551-552 (1965; Zbl 0139.14404)] provided an elementary axiomatization of finite-dimensional Cartesian spaces coordinatized by arbitrary ordered fields. In Can. Math. Bull. 12, 831-836 (1969; Zbl 0192.57302) he introduced axiom \(B\), stating that for any points \(x,y,z\) such that \(y\) is between \(x\) and \(z\) there is a right triangle having \(x\) and \(z\) as endpoints of the hypotenuse and \(y\) as the foot of the altitude of the hypotenuse, and axiom \(E\), which implies that the coordinate field is Euclidean. He asked the question: are axioms \(E\) and \(B\) equivalent when added to Euclidean geometry of arbitrary dimension \(n\geq 2\) over arbitrary ordered fields? It is easily seen that the answer is positive for \(n=2\), and \textit{W. Schwabhäuser} [Proc. Am. Math. Soc. 22, 233-234 (1969; Zbl 0177.23901)] showed that it is negative for \(n\geq 5\). The author of the present article shows that the answer is negative for \(n\geq 3\).
scientific article; zbMATH DE number 913000