Surely this depends on the details of the number-to-name mapping. For example, in the Geeble language, the numbers 1, 2, 3, and so on, are "geeble", "geeblegeeble", and "geeblegeeblegeeble", and so on. The system is obviously universal, but the lengths of the names are all multiples of 6. (Well, multiples of 7 in Geeble, for orthographic reasons.) I have spent about ten minutes attempting to rigorize your eminently sensible conjecture, but with no luck. Back over to you. On Wed, Jan 14, 2009 at 2:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:

Allan wrote:

<< More "constructively", let a(0)=3, and then define a(n+1) as the smallest number whose name written out in English has a(n) letters. I suspect it's easy to prove your parenthesized comment "strictly monotonically increasing". ...

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Only now do I comprehend the idea of this sequence -- Thanks!

Question: Is it clear that the sequence is indefinitely extensible? I.e., couldn't there be a length for which (given a fixed universal naming system) there exists no English number name? (Or if there are some obvious small sizes missing, is it clear that each sufficiently large length has even one exemplar?

--Dan

_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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