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## 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

http://www.math.sc.edu/~nyikos/sem.pdf

**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.http://www.math.sc.edu/~nyikos/sem.pdf

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