[FOM] Finitist concept of consistency?

[William Tait]

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On Aug 25, 2006, at 2:12 PM, Stephen Pollard wrote:

> Now here's the question. Have I recognized that the formal
> theory S is consistent? If I have not represented to myself the
> proposition that S is consistent, then I have not recognized that S
> is consistent. Can I represent this proposition to myself without
> adverting to all S-proofs, where the "all" ranges over infinitely
> many things?
..........
>  Tait argues that nothing
> in principle prevents finitists from verifying quantifier-free
> formulas that non-finitists would interpret as universal claims about
> the natural numbers. This should help us answer the question of what
> finitists are in a position to assert when they enunciate such a
> formula. I am not confident, however, that I understand what that
> answer would be under Prof. Tait's analysis.

I'm not comfortable with the notion of representing something to  
myself, but one can say this: If S is primitive recursive arithmetic,  
say, then the statement of its consistency is of the form fX=0, where  
X is a free variable and f is primitive recursive. On my analysis of  
finitism, fX is a finitist construction of a number from  the  
arbitrary number X, so in that sense, the finitist can represent the  
statement of consistency to himself. But he cannot 'represent to  
himself' its truth: This would require a construction that is not  
included in S.

Something in the first paragraph of Stephen Pollard's posting makes  
me think that the following remark is relevant to his concern: The  
statement that I (if I were a finitist) accept a proof of the  
proposition gX=0 because it is a proof in S is ambiguous. What is  
true is that, since it is a proof in S, I will accept it (even if I  
didn't know what the system S is, since the proof uses only  
constructions that are finitistically admissible). But being a proof  
in the system S is not part of my reason for accepting it:  If it  
were, I would be relying on the validity of the system S as a whole,  
and that as we noted already (in the case of S = PRA), is not  
finitistically justified.

Regards,

Bill Tait