Abstract
The fruitless attempts to prove Euclid’s parallel postulate, in particular the theory of limit parallels, lead eventually the mathematicians of the nineteenth century to consider that the negation of this postulate could possibly be taken as an axiom. The discovery of the models of non-Euclidean geometry—like the Beltrami–Klein and the Poincaré disk—give evidence that the negation of the parallel postulate is “as consistent as the parallel postulate”.
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Bibliography
F. Borceux, A Differential Approach to Geometry. Geometric Trilogy III (Springer, Berlin, 2014)
K. Gödel, Über formal Unentscheidbare Sätze der Principia Mathematica und Verwandter Systeme, I. Monatshefte Math. Phys. 38, 173–198 (1931)
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Borceux, F. (2014). Non-Euclidean Geometry. In: An Axiomatic Approach to Geometry. Springer, Cham. https://doi.org/10.1007/978-3-319-01730-3_7
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DOI: https://doi.org/10.1007/978-3-319-01730-3_7
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-01729-7
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