Berry paradox without paradox.
 

Let n be the smallest positive integer not definable in under two words.

A friend of mine is unsure whether n can be found at all, and believes n is uncomputable in virtue of the equivocity of the English language (there are infinite ways of describing n, and one can therefore never be sure that any given description of a number is the shortest). The problem, he says, may seem more evident if you change "under two words" by "under three words", four, five, etc... (Obviously not going any higher than ten.)

Any advice ?

PS : This friend of mine studies philosophy...

[QUOTE=victor;131129]Let n be the smallest positive integer not definable in under two words.

A friend of mine is unsure whether n can be found at all, and believes n is uncomputable in virtue of the equivocity of the English language (there are infinite ways of describing n, and one can therefore never be sure that any given description of a number is the shortest). The problem, he says, may seem more evident if you change "under two words" by "under three words", four, five, etc... (Obviously not going any higher than ten.)

Any advice ?

PS : This friend of mine studies philosophy...[/QUOTE]If every number were to eventually get a name (like googol, Grahams number, etc.) then that name [i]could[/i] conceivably be only one word. So I think it comes down to how long do you want to wait to see how many numbers get one word names. At some point you have to make a cut-off and say "up to this point in time the smallest positive integer not definable in under two words is [whatever it is]".

Assuming a static vocabulary (i.e. the list of allowable "words" is constant) and unique interpretation (i.e. one specific sequence of word can define at most one number), then it is a simple matter of iterating thru all single-word and two-word sequences, and then finding out the smallest number not represented by them.

Isn't it? :unsure:

[quote=victor;131129]Let n be the smallest positive integer not definable in under two words.[/quote]

any integer can be defined by just writing it down (theoretically), i.e. with 1 string of digits.

and, there are certainly languages in which all numbers are spelled as one single word.

and and and....

Philosophy can be scientific, but often people in that field just produce nonsense.

[QUOTE=axn1;131135]Assuming a static vocabulary (i.e. the list of allowable "words" is constant) and unique interpretation (i.e. one specific sequence of word can define at most one number), then it is a simple matter of iterating thru all single-word and two-word sequences, and then finding out the smallest number not represented by them.[/QUOTE]

[QUOTE]not definable in under two words[/QUOTE]

"Under two" limits it to single words. With a fixed vocabulary, the task is rather trivial.

"Not more than two" is more interesting.
And adding one more really opens up the possibilities.

[quote=axn1;131135]Assuming a static vocabulary (i.e. the list of allowable "words" is constant) and unique interpretation (i.e. one specific sequence of word can define at most one number), then it is a simple matter of iterating thru all single-word and two-word sequences, and then finding out the smallest number not represented by them. Isn't it? :unsure:[/quote]

no, since there are 2-word combinations whose meaning is not clear and/or may depend on (i.e. vary with) the historical context (e.g. "known primes"...)
and, for some word combinations it cannot be said, by principle, what they represent, they are ill-defined, like "the set of all sets", or "this phrase is a lie", or so.
To be concrete, the "leastbiwordundefinable number" cannot exist.

PS: if vou don't like "leastbiwordundefinable", call these numbers "infraperfect". So you are looking for the "smallest infraperfect". It cannot exist.

This sounds similar to this thread:
[URL]http://www.mersenneforum.org/showthread.php?t=9546[/URL], in which one or more posts contain web links to a consistent methodology for speaking or writing any number. Everyone must agree on the rules before everyone can agree on an answer to this puzzle. That is what we did in the above thread...

There are ways to construct a single word for a number. I think that this can be done through about 21.