# 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).

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

As Akhil says, yes. Another somewhat standard name for this axiom is the "Dedekind cut axiom". If you Google that with quotes, you will find some references.

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Yes- see Real and Complex Analysis, by C. Apelian and S. Surace. This is precisely what they call the Dedekind completeness property. For them, properties such as the least-upper-bound property are stated as theorems.

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Number Systems and the Foundations of Analysis by Mendelson follows the cut approach.

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A great book by my old teacher.Hope he's around awhile in retirement-I want to get him to teach his legendary logic course one last time. –  Andrew L May 7 '10 at 22:21

In his "A Course of Pure Mathematics" G H Hardy makes use of the Dedekind's theorem to prove almost all the usual theorems of real analysis and he prefers to avoid least upper bound property altogether. In my opinion this is more easily understandable and presentable. The least upper bound property looks high brow and has been championed by later authors unnecessarily.

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