Universal-existential axiom systems for geometries expressed with Pieri's isosceles triangle as single primitive notion

Weprove that, building uponthe universal-existential orthogonality-based axiom system for metric planes presented in [28], one can provide universal-existential axiom systems – expressed solely in terms of the ternary predicate I , with I(abc) standing for ‘ab is congruent to ac’ , which Pieri has introduced 100 years ago –for metric planes, for absolute geometry with the circle axiom, for Euclidean planes, for Euclidean geometry with the circle axiom, for Klingenberg’s generalized hyperbolic planes, for plane elementary hyperbolic geometry, as well as for all thefinite-dimensional versions of these geometries.

[1]  Aufbau der hyperbolischen Geometrie aus dem Geradenorthogonalitätsbegriff , 2003 .

[2]  Richard Kramer.  The undefinability of intersection from perpendicularity in the three-dimensional Euclidean geometry of lines , 1993 .

[3]  Raphael M. Robinson.  Binary Relations as Primitive Notions in Elementary Geometry , 1959 .

[4]  L. W. Szczerba,et al.  Relations on lines as primitive notions for Euclidean geometry , 1975 .

[5]  Kay Srensen,et al.  Ebenen mit Kongruenz , 1984 .

[6]  Dana Scott,et al.  A Symmetric Primitive Notion for Euclidean Geometry , 1956 .

[7]  A. Tarski,et al.  Über die Beschränktheit der Ausdrucksmittel deduktiver Theorien , 1936 .

[8]  J. Ahrens.  Begründung der absoluten Geometrie des Raumes aus dem Spiegelungsbegriff , 1959 .

[9]  F. J. Pelletier,et al.  316 Notre Dame Journal of Formal Logic , 1982 .

[10]  W. Pejas,et al.  Die Modelle des Hilbertschen Axiomensystems der absoluten Geometrie , 1961 .

[11]  Alfred Tarski,et al.  Tarski's System of Geometry , 1999, Bulletin of Symbolic Logic.

[12]  Victor Pambuccian.  FRAGMENTS OF EUCLIDEAN AND HYPERBOLIC GEOMETRY , 2001 .

[13]  Rafal Gruszczynski,et al.  Pieri's Structures , 2007, Fundam. Informaticae.

[14]  Victor Pambuccian,et al.  The Simplest Axiom System for Plane Hyperbolic Geometry , 2004, Stud Logica.

[15]  Victor Pambuccian.  Sphere tangency as single primitive notion for hyperbolic and Euclidean geometry , 2003 .

[16]  J. Diller.  Eine algebraische Beschreibung der metrischen Ebenen mit ineinander beweglichen Geraden , 1970 .

[17]  Elena Anne Marchisotto,et al.  The legacy of Mario Pieri in geometry and arithmetic , 2007 .

[18]  Friedrich Bachmann,et al.  Aufbau der Geometrie aus dem Spiegelungsbegriff , 1959 .

[19]  Victor Pambuccian.  ON THE SIMPLICITY OF AN AXIOM SYSTEM FOR PLANE EUCLIDEAN GEOMETRY , 1997 .

[20]  Malgorzata Grochowska.  EUCLIDEAN TWO DIMENSIONAL, EQUIDISTANCE THEORY , 1984 .

[21]  V. Pambuccian.  Orthogonality as single primitive notion for metric planes , 2007 .

[22]  Discrete versions of the Beckman—Quarles theorem , 1999, math/9904047.

[23]  Victor Pambuccian,et al.  Groups and Plane Geometry , 2005, Stud Logica.

[24]  A. Tarski,et al.  Metamathematische Methoden in der Geometrie , 1983 .