[FOM] 23 syllables

[henri galinon]

Dear FOMers,
  in his essay "The ways of paradox", Quine has a passage on Berry's  
paradox. At the very end of the passage (last remark of the following  
quote), he offers the reader a little amusing game (implicitly at  
least) by hinting at a solution. My question is : what is Quine's  
"solution" ?

  I quote the entire relevant passage :

" Ten has a one-syllable name. Seventy-seven has a five-syllable  
name. The seventh power of seven hundred seventy-seven has a name  
that, if we were to work it out, might run to 100 syllables or so;  
but this number can also be specified more briefly in other terms. I  
have just specified it in 15 syllables. We can be sure, however, that  
there are no end of numbers that resist all specification, by name or  
description, under 19 syllables. There is only a finite stock of  
syllables all together, and hence only a finite number of names or  
phrases of less than 19 syllables, whereas there are an infinite  
number of positive integers. Very well, then ; of those numbers not  
specifiable in less than 19 syllables, there must be a least. And  
here is our antinomy : the least number not specifiable in less than  
nineteen syllables is specifiable in 18 syllables. I have just so  
specified it.
The antinomy belongs to the same family as the antinomies that have  
gone before. For the key word of this antinomy, "specifiable", is  
interdefinable with "true of". It is one more of the truth locutions  
that would take on subscripts under the Russell-Tarski plan. The  
least number not specifiable-0 in less than nineteen syllables is  
indeed specifiable-1 in 18 syllables, but it is not specifiable-0 in  
less than 19 syllables ;  for all I know it is not specifiable-0 in  
less than 23."


Best,

H.G.
PhD student
Paris, France