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The Ubiquitous Axiom

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Abstract

This paper starts with a survey of what is known regarding an axiom, referred to as the Lotschnittaxiom, stating that the perpendiculars to the sides of a right angle intersect. Several statements are presented that turn out to rather unexpectedly be equivalent, with plane absolute geometry without the Archimedean axiom as a background, to the Lotschnittaxiom. One natural statement is shown to be strictly weaker than the Lotschnittaxiom, creating a chain of four statements, starting with the Euclidean parallel postulate, each weaker than the previous one. It then moves on to provide surprising equivalents, expressed as pure incidence statements, for both the Lotschnittaxiom and Aristotle’s axiom, whose conjunction is equivalent to the Euclidean parallel postulate. The new incidence-geometric axioms are shown to be syntactically simplest.

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Notes

  1. wenn man beweisen könnte dass ein geradlinigtes Dreieck möglich sei, dessen Inhalt grösser wäre als eine jede gegebne Fläche so bin ich im Stande die ganze Geometrie völlig streng zu beweisen. [58, pp. 36f]

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Acknowledgements

Thanks are due to Franz Kalhoff for the proof of Theorem 4.2, to Julien Narboux for having brought to our attention Misha Lavrov’s question on math.stackexchange.com regarding the strength of ML, and to the referee for a very close reading of the manuscript.

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Authors and Affiliations

  1. School of Mathematical and Natural Sciences (MC 2352), Arizona State University - West Campus, P. O. Box 37100, Phoenix, AZ 85069-7100, USA

    Victor Pambuccian

  2. Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA

    Celia Schacht

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  1. Victor Pambuccian
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  2. Celia Schacht
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Correspondence to Victor Pambuccian.

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Heinrich Wefelscheid in memoriam.

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Pambuccian, V., Schacht, C. The Ubiquitous Axiom. Results Math 76, 114 (2021). https://doi.org/10.1007/s00025-021-01424-3

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