Or dedekind complete ordered field - yes that is the mathematical definition of real numbers!!
This link is very good and explains it very beautifully in accessible fashion:
http://www.math.vanderbilt.edu/~schectex/courses/thereals/
It explains the contents of Robert Bartle's real analysis book in a single page. Author has some talent in explaining hard things to people!
Real numbers are not so real after all. They are mathematical constructs as much as say complex numbers.
So we have no reason to expect real riemannian manifolds will explain the universe [spacetime] - In my thinking it should be discrete.
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