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Implications of consistency of hyperbolic geometry 1. There are models within R 3 or even R 2 that satisfy the postulates of hyperbolic geometry (Hilbert IBC + HH (p. 259) + Dedekind). (We get to details around pp. 329–330.) This shows that hyperbolic geometry is consistent if our theory of R n is (the latter needing the real numbers and hence some level of set theory). That is, elementary linear algebra establishes the consistency of hyperbolic geometry just as surely as that of Euclidean geometry — which it does, because R 2 itself is a model of (Hilbert) Euclidean geometry (pp. 139–140 and the Chapter 3 projects). In fact, the hyperbolic models can be developed within axiomatic Euclidean geometry, so we don’t really need the consistency of the real numbers to reach the conclusion, just consistency of Euclidean geometry. An ironic consequence of the foregoing is that HE/EV can’t be proved within the Hilbert axioms + Dedekind (or your favorite continuity axiom), unless Euclidean geometry itself is inconsistent. In other words, if Saccheri et al. had succeeded in “vindicating ” Euclid by proving EV, they would have destroyed Euclidean geometr

Year: 2011
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