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From: ttobler@unislc.uucp (Trent Tobler)
Newsgroups: comp.compression
Subject: Re: Integer not expressable in less than 13 words
Message-ID: <1991Apr25.234539.24276@unislc.uucp>
Date: 25 Apr 91 23:45:39 GMT
References: <15901@smoke.brl.mil>
Organization: unisys
Lines: 38

From article <15901@smoke.brl.mil>, by gwyn@smoke.brl.mil (Doug Gwyn):
> In article <1991Apr19.155442.4840@looking.on.ca> brad@looking.on.ca (Brad Templeton) writes:
>>After all, "least positive integer not expressable in less than 13
>>English words"   has 11 words, so in fact there is no such animal.
> 
> But then what about the following argument:
> 	There are a finite (although large) number of English words.
> 	The number of combinations of fewer than 13 words from
> 	the finite set of words is also finite (although very large).
> 	The combinations of fewer than 13 English words that express
> 	positive integers form a subset of the set of all possible
> 	combinations of fewer than 13 English words, and thus that
> 	subset is finite.
> 	Every finite set of positive integers has a least member.
> 	Therefore the (finite) set of positive integers expressed by
> 	that subset of English word combinations has a least member.
> It would seem that the phrase does describe some integer.

No, it is self referencial.  If 'The least positive integer not expressable
in less than thirteen english words' is allowed to represent the least
positive integer not expressable in less than thirteen english words, then
that number does not exist.  You try to make some correlation between this
and the labeling of english sentences.  The problem with this is that
it is not 'The least positive integer not expressable in less than eighteen
english words which specify a unique integer.'

> Yes, it's a paradox.

No, no paradox.  The number simply does not exist.

>         But it could be relevant when considering compression schemes
> that encode numeric values in terms of formulas.

Yes, it is relevent, and has been considered.  See Godel's theorem and
the turing halting problem.

--
		Trent Tobler - ttobler@csulx.weber.edu