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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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References:
 [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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