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The simplest axiom system for plane hyperbolic geometry revisited. (English) Zbl 1234.51011
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 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 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 \(L_{B\equiv}\).

MSC:
51M10 Hyperbolic and elliptic geometries (general) and generalizations
51M30 Line geometries and their generalizations
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[1] Augat, C., Ein Axiomensystem für die hyperbolischen Ebenen über euklidischen Körpern, Dissertation, Universität Stuttgart, 2008. · Zbl 1195.51001
[2] Pambuccian V.: ’Simple axiom systems for Euclidean geometry’. Mathematical Chronicle 18, 63–74 (1989) · Zbl 0699.51010
[3] Pambuccian V.: ’The simplest axiom system for plane hyperbolic geometry’. Studia Logica 77, 385–411 (2004) · Zbl 1069.03007 · doi:10.1023/B:STUD.0000039031.11852.66
[4] Rigby J.F.: ’Axioms for absolute geometry’. Canadian Journal of Mathematics 20, 158–181 (1968) · Zbl 0159.21703 · doi:10.4153/CJM-1968-017-6
[5] Sörensen K.: ’Ebenen mit Kongruenz’. Journal of Geometry 22, 15–30 (1984) · Zbl 0537.51019 · doi:10.1007/BF01230121
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