Main Question or Discussion Point
I know this can't be done but I don't know why. It's been established that the Axiom of Choice is required to prove the well ordering of the reals. Why can't we say that for any pair of real numbers a,b in the set R there exists a Dedekind cut that falls between a and b such that always a>b or always a<b?. Does the AC somehow come into play here? I don't see it.