On the stepwise construction of the parallel postulate
In an absolute plane given by the axioms I 1-3, II, III of D. Hilbert the author shows that Euclid's parallel postulate is equivalent to the following two weaker axioms: \textbf{(A)} (Bachmann's ``Lotschnittaxiom''): For any three distinct points \(a\), \(b\), \(c\) with \(\overline{a,b} \perp \overline{b,c}: (a \perp \overline{a,b}) \cap (c \perp \overline{c,b})\neq \emptyset\). \textbf{(Ar)} (Aristotle's Axiom, cf. \textit{M. J. Greenberg}, J. Geom. 33, No. 1/2, 53-57 (1988; Zbl 0655.51010)): Given any three distinct points \(a\), \(b\), \(c\) with \(\overline{a,b} \perp \overline{b,c}\) and two further points \(d\), \(e\). Then there exists a point \(r \in \vec{a,c}\) such that if \(s := \overline{a,b} \cap (r \perp \overline{a,b})\), then \((r,s) > (d,e)\). The proof is based on the algebraic description of absolute planes given by \textit{W. Pejas} [Math. Ann. 143, 212-235 (1961; Zbl 0109.390)]. To such a plane there corresponds an ordered field \(K\) and a ``metric constant'' \(k \in K\). By \textit{M. J. Greenberg} [loc. cit.] one has \(k \leq 0\) if \textbf{(Ar)} is valid and the author proves that in a plane with \textbf{(A)} and \(k < 0\) the axiom \textbf{(Ar)} is not valid. Furthermore the author presents a statement equivalent to \textbf{(A)}.
scientific article; zbMATH DE number 700400