Geometry as an extension of the group theory
DOI:
https://doi.org/10.12775/LLP.2002.008Abstract
Klein’s Erlangen program contains the postulate to study the group of automorphisms instead of a structure itself. This postulate, taken literally, sometimes means a substantial loss of information. For example, the group of automorphisms of the field of rational numbers is trivial. However in the case of Euclidean plane geometry the situation is different. We shall prove that the plane Euclidean geometry is mutually interpretable with the elementary theory of the group of authomorphisms of its standard model. Thus both theories differ practically in the language only.References
Bachmann, F., “Aufbau der Geometrie aus dem Spiegelungsbegriff”, in Zweite ergänzte Auflage. Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin–New York, 1973.
Szczerba, L.W., “Interpretations of elementary theories”, in Logic Foundations of Mathematics and Computability Theory, Reidel 1977, Dordrecht, Boston, pp. 129–145.
Szmielew, W., From the Affine to Euclidean Geometry, PSP, Warsaw, 1983.
Tarski, A., “What is elementary geometry?”, pages 16–29 in Studies in Logic and the Foundation of Math., Proc. Internat. Sympos. (Univ. of Calif., Berkeley 1957/1958), North-Holland, Amsterdam 1959.
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2004-01-19
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PRUSIŃSKA, A. and SZCZERBA, L. Geometry as an extension of the group theory. Logic and Logical Philosophy. Online. 19 January 2004. Vol. 10, no. 10, p. 131–135. [Accessed 2 July 2025]. DOI 10.12775/LLP.2002.008.
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