# completeness axiom for the real numbers

Do any treatises on real analysis take the following as the basic completeness axiom for the reals?

"Let $A$ and $B$ be set of real numbers such that (a) every real number is either in $A$ or in $B$; (b) no real number is in $A$ and in $B$; (c) neither $A$ nor $B$ is empty; (d) if $\alpha \in A$, and $\beta \in B$, then $\alpha < \beta$. Then there is one (and only one) real number $\gamma$ such that $\alpha \leq \gamma$ for all $\alpha \in A$, and $\gamma \leq \beta$ for all $\beta \in B$."

This appears as Theorem 1.32 in Walter Rudin's "Principles of Mathematical Analysis", and can be traced back to Dedekind's "Continuity and Irrational Numbers" (section V, subsection IV). Both Rudin and Dedekind derive this result from the construction of the reals via cuts of the rationals.

Authors who prefer to axiomatize the reals directly (instead of constructing them from the rationals) might be expected to take the above property as an axiom, but I haven't found anyone who does this. Instead, they all assume the least upper bound property as an axiom, or the nested interval property, or the convergence of Cauchy sequences.

I personally think the way to go is to take Rudin's Theorem 1.32 as an axiom (because it is simple and compelling) and then derive the least upper bound property (since it is more useful in practice than 1.32) and then get to work building up the apparatus of real analysis. But leaving aside the issue of whether this is the right way to go: have any authors taken this approach?

I should remark that the geometrical analogue of Theorem 1.32, characterizing the completeness of the line, appears to be well known to geometers (especially those interested in the foundations of geometry; see for instance Marvin Jay Greenberg's very nice article in the March 2010 issue of the Monthly).

• You don't need (b), because of (d). – lhf Apr 6 '10 at 1:07