[FOM] First-order arithmetical truth

[Stephen Pollard]

Some of us claim to understand how omega is structured. Vladimir  
Sazonov is trying to show us that we are deluded. On October 14, he  
offered an argument that helped me understand his position a little  
better. I now regret not having responded.

I had mentioned the following version of the least number principle:   
"Among any positive integers there is always a least." This carries  
the following logical commitment: if anyone manages to refer to some  
positive integers, then I must concede that one of them is less than  
all the others.

Vladimir Sazonov wrote:

> This principle only looks ... self-evident as an extrapolation from  
> some finite "pictures" of our real world. As to the above general  
> form, I would consider it even meaningless (recall, e.g. the Berry  
> paradox on "The smallest positive integer not definable in under  
> eleven words").

VS and G. G. Berry ask us to consider the positive integers not  
definable in under eleven words. If, in making this request, they  
have referred to some positive integers, then I must concede that one  
of those integers is the smallest. It would be characterized by the  
phrase (*) "the smallest positive integer not definable in under  
eleven words." If (*) is a definition in the sense expressed by the  
occurrence of 'definable' in (*), we can derive a contradiction. I  
conclude: either VS and GGB have not referred to any positive  
integers or (*) is not a definition of the indicated type.

OK, that's hardly a revolutionary conclusion, and perhaps not even  
worth rehearsing on FOM, except to emphasize the point that it's  
really quite unlikely for a competent speaker of English to affirm,  
in good faith, a well-formed sentence of English that is meaningless.  
I would even say it's quite unlikely for a competent speaker of  
mathematical English to affirm, in good faith, a well-formed sentence  
of mathematical English that is mathematically meaningless (in a  
sense akin to 'physically meaningless' as employed by physicists).

Stephen Pollard
Professor of Philosophy
Division of Social Science
Truman State University
spollard at truman.edu