[FOM] 23 syllables

Richard Heck rgheck at brown.edu
Thu Dec 7 16:33:11 EST 2006

henri galinon wrote:
> 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" ?
The quoted passage is below.

I'm not sure what the "amusing game" is supposed to be here. Quine's
point is simply that Berry's paradox can be solved using Tarski's
methods to whatever extent the Liar can be. As he says, "x specified by
A" here means, more or less: x is the one and only one object of which
the formula in one free variable A is true. So consider PA, and instead
of talking of syllables talk of the length of formulae. So, there being
only finitely many formulae containing one free variable and less than N
symbols total, there are only finitely many numbers that satisfy any
such formula uniquely; hence there are numbers that do not; hence there
is a least; and all of that can be proven in PA^T = PA plus a
Tarski-style truth theory. By the same token, there is a least number
not "specifiable" by a formula of PA^T. But if we choose N correctly,
there will be a formula of PA^T containing less than N symbols that
specifies the least number not so specifiable in PA itself. And that
number will be provably greater than the least such number not so
specifiable by a formula of PA. And so up up the hierarchy. Exactly what
N should be taken to be here could be calculated simply by writing down
the relevant formula of PA^T and counting.

Richard Heck
>   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
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Richard G Heck, Jr
Professor of Philosophy
Brown University
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