On Saturday, February 15, 2014 12:48:52 PM UTC-8, Ken Quirici wrote: > > I'm beginning to think that Euclid takes a huge number of 'obvious' facts for granted. I'm beginning to think Euclid's elements can only REALLY be made rigorous by making the apparently simple geometry a subset of R^2 real analysis - the whole kit-and-caboodle of calculus, continuity, &c.
The whole kit-and-caboodle of the real numbers is not necessary. I believe that most, if not all, of Euclid can be modeled in the "surd plane" (see also "surd field")
See Edwin E Moise's "Elementary Geometry from an Advanced Standpoint"
From what I recall, Euclid missed a bunch of stuff such as order properties.