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The principle of duality in Euclidean and in absolute geometry

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Abstract

In Euclidean geometry and in absolute geometry fragments of the principle of duality hold. Bachmann (Aufbau der Geometrie aus dem Spiegelungsbegriff, 1973, §3.9) posed the problem to find a general theorem which describes the extent of an allowed dualization. It is the aim of this paper to solve this problem. To this end a first-order axiomatization of Euclidean (resp. absolute) geometry is provided which allows the application of Gödel’s Completeness Theorem for first-order logic and the solution of Bachmann’s problem.

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References

  1. Bachmann F.: Aufbau der Geometrie aus dem Spiegelungsbegriff, 2nd edn. Springer, Heidelberg (1973)

    Book  MATH  Google Scholar 

  2. Bachmann F.: Ebene Spiegelungsgeometrie. BI-Verlag, Mannheim (1989)

    MATH  Google Scholar 

  3. Gödel K.: Über die Vollständigkeit der Axiome des logischen Funktionenkalküls. Monatsh. f. Math. u. Phys. 36, 349–360 (1930)

    Article  MATH  Google Scholar 

  4. Gray J.: Worlds Out of Nothing. A Course in the History of Geometry in the 19th Century. Springer, London (2007)

    MATH  Google Scholar 

  5. Hughes D.R., Piper F.C.: Projective Planes. Springer, Heidelberg (1973)

    MATH  Google Scholar 

  6. Pambuccian, V.: Review of: David Hilbert’s lectures on the foundations of geometry, 1891–1902. M. Hallett, U. Majer, eds., Springer, Berlin (2004). Philosophia Mathematica 21, 255–277 (2013)

  7. Schmidt A.: Die Dualität von Inzidenz und Senkrechtstehen in der absoluten Geometrie. Math. Ann. 118, 609–625 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  8. Schwabhäuser W., Szmielew W., Tarski A.: Metamathematische Methoden in der Geometrie. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  9. Struve H., Struve R.: Non-euclidean geometries: the Cayley–Klein approach. J. Geom. 98, 151–170 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Struve, R.: An axiomatic foundation of Cayley–Klein geometries. J. Geom. doi:10.1007/s00022-015-0293-z

  11. Yaglom I. M.: A Simple Non-Euclidean Geometry and Its Physical Basis. Springer, Heidelberg (1979)

    MATH  Google Scholar 

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  1. SIGNAL IDUNA Gruppe, Joseph-Scherer-Strasse 3, 44139, Dortmund, Germany

    Rolf Struve

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  1. Rolf Struve
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Correspondence to Rolf Struve.

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Struve, R. The principle of duality in Euclidean and in absolute geometry. J. Geom. 107, 707–717 (2016). https://doi.org/10.1007/s00022-016-0314-6

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  • DOI: https://doi.org/10.1007/s00022-016-0314-6

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