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    Reid BarnesMay 6, 2018 at 7:50 am

    Hawking’s new assumption that the universe has an overall geometry that is saddle-shaped cannot be valid. This geometry is a type of non-Euclidean geometry which proved to be self-contradicting. The validity of geometry and math with coordinates was demonstrated by David Hilbert based on his Theorem 8 [5 in earlier editions of his book] about a line dividing a plane in two. This theorem required for its proof his Axiom I. 2 and Pasch’s triangle axiom, which Hilbert believed was an independent foundational axiom common to Euclidean and non-Euclidean geometry. However, contrary to what Hilbert believed, the triangle axiom was not an independent foundational axiom. It was a proposition that combined a more elementary triangle axiom and Hilbert’s Axiom of Parallels which Hilbert called “Euclid’s Axiom.” This “Euclid’s Axiom” is the central tenet of Euclidian geometry and contradicts the saddle shaped non-Euclidean geometry. Hilbert’s Axiom of Parallels, “Euclid’s Axiom,” was a logical equivalent of the original Playfair’s axiom, which was the logical substitute for Euclid’s famous fifth postulate added by Playfair to Euclid’s geometry in 1795. The Facebook Note, The LITE Triangle Axiom, explains more about Pasch’s Triangle Axiom, Hilbert’s Axiom of Parallels, and a more elementary triangle axiom: https://www.facebook.com/notes/reid-barnes/the-lite-triangle-axiom/230992473620001/

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Remembering Stephen Hawking