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There appear to be two different nested interval properties for the reals with the punchline "... then the intersection of the intervals is non-empty", and I'd like to know their respective histories (and the appropriate terminology to use for distinguishing between them). Here I am not concerned with the issue of "singleton-icity" of the intersection; only non-emptiness.

In one version (which I believe is due to Cantor), the length of the intervals is assumed to shrink down to 0; in the other version, their lengths are not constrained.

One sense in which the two properties are inequivalent is that there are non-archimedean ordered fields that satisfy the first but not the second. (Side question: Which is the preferred spelling, "archimedean" or "archimedian"? The number of Google hits for each spelling is about 800,000, so we can't use the Google-hits heuristic to decide, though Google seems to think that the former is preferred.) Specifically, consider the ordered field of formal Laurent series in $x$ over the reals, "ordered by size at zero" (so that effectively we're adjoining a formal infinitesimal $x$ to the reals). This ordered field satisfies the first nested interval property but not the second. (Consider the closed intervals $[nx,1/n]$. Their intersection is empty, but this doesn't contradict the first version of the nested interval property because the lengths of these intervals don't shrink to 0; indeed all the lengths are greater than $x$.)

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Instead of asking google to count web pages, you could ask it to count uses in books… . One reason to do this is if you trust books more than web pages. Another is that google is claiming to actually count these uses, while it makes no claims about "hits." –  Ben Wieland Mar 13 '12 at 23:41
Dictionaries seem to prefer <a href="">archimedean</a>;. My feeling is that if you're going to pronounce it archiMEEdian then the i is good but it's nicer to say archimeDEEan and spell it with the e. –  Tom Goodwillie Mar 14 '12 at 0:36
Lang (Algebra) and Gillman-Jerison (Rings of Continuous Functions) also use "archimedean". –  Ralph Mar 14 '12 at 0:55
The standard term for your second property is "spherically complete". See and… –  David Speyer Mar 14 '12 at 0:55
Since there doesn't seem to be any standard nomenclature that distinguishes between them while implicitly acknowledging their similarity, I think I'll call them the Shrinking Interval Property and the Nested Interval Property in the article I'm writing (with a remark that the latter in a more general context is also called spherical completeness). –  James Propp Mar 16 '12 at 5:55

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