The simplest axiom system for plane hyperbolic geometry revisited
This note solves a problem left open in the author's paper [Stud. Log. 77, No. 3, 385--411 (2004; Zbl 1069.03007)]. There, the author presented an axiom system \(\Sigma\) consisting of 22 axioms in Tarski's language \(L_{B\equiv}\) --- with points as the only individual variables, the ternary relation \(B\), with \(B(abc)\) standing for `point \(b\) lies between \(a\) and \(c\)', and the quaternary relation \(\equiv\), with \(ab\equiv cd\) standing for `the segment \(ab\) is congruent to the segment \(cd\)' --- for plane hyperbolic geometry, all of whose axioms --- with one exception, axiom A20, which requires 6 variables --- being statements which, when written in prenex form, contain at most 5 variables. \(\Sigma\) was based on an axiom system in terms of collinearity (a ternary relation \(L\), with \(L(abc)\) standing for `points \(a, b, c\) are collinear') and \(\equiv\) stands for non-elliptic metric planes from \textit{K. Sörensen} [J. Geom. 22, 15--30 (1984; Zbl 0537.51019)], A20 being one of the axioms of Sörensen's axiom system --- it states that \[ \neg L(xyz)\wedge B(xay) \wedge ax\equiv ay \wedge B(ybz)\wedge by\equiv bz \wedge B(zcx)\wedge cz\equiv cx \rightarrow \neg L(abc). \] The author shows in this note, by using an axiom system for non-elliptic metric planes in which every point-pair has a midpoint from \textit{C. Augat} [Ein Axiomensystem für die hyperbolischen Ebenen über euklidischen Körpern. Stuttgart: Univ. Stuttgart, Fakultät Mathematik und Physik (Diss.) (2008; Zbl 1195.51001)], that A20 is superfluous in \(\Sigma\), and thus that \(\Sigma\setminus \{\text{A20}\}\) is an axiom system in \(L_{B\equiv}\) for plane hyperbolic geometry, all of whose axioms are statements in prenex form regarding at most 5 variables. Since, as shown in [Stud. Log. 77, No. 3, 385--411 (2004; Zbl 1069.03007)], there is no axiom system in \(L_{B\equiv}\) consisting entirely of statements in prenex form regarding at most 4 variables, \(\Sigma\setminus \{\text{A20}\}\) can be said to be the simplest axiom system for plane hyperbolic geometry in \(L_{B\equiv}\).
scientific article; zbMATH DE number 5900600