The Axiomatics of Ordered Geometry I
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Expositiones Mathematicae 29 (2011) 24–66 Contents lists available at ScienceDirect Expositiones Mathematicae journal homepage: www.elsevier.de/exmath The axiomatics of ordered geometry I. Ordered incidence spaces Victor Pambuccian Division of Mathematical and Natural Sciences, Arizona State University - West Campus, P. O. Box 37100, Phoenix, AZ 85069-7100, USA article info a b s t r a c t Article history: We present a survey of the rich theory of betweenness and Received 7 September 2009 separation, from its beginning with Pasch's 1882 Vorlesungen über Received in revised form neuere Geometrie to the present. 17 July 2010 ' 2010 Elsevier GmbH. All rights reserved. Keywords: Ordered geometry Axiom system Half-ordered geometry Somewhere there is some knowledge; under the stone A message perhaps; or hidden by strange signs In places only the stupid visit. Malcolm Lowry. Ein Zeichen sind wir, deutungslos, Schmerzlos sind wir und haben fast Die Sprache in der Fremde verloren. Friedrich Hölderlin, Mnemosyne. 1. Introduction Unlike most concepts of elementary geometry, whose origin is shrouded in the ineluctable fog of all cultural beginnings, the history of the notion of betweenness can be traced back to one person, one book, and one year: Moritz Pasch,1 Vorlesungen über neuere Geometrie, and 1882. The aim of this pioneering book was to provide a solid foundation, very much in the spirit of synthetic geometry, for 1 For a view of one of Pasch's contemporaries on his life and his work's significance for the axiomatics of geometry, see [325].
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