[FOM] consistency and completeness in natural language

[Neil Tennant]

On Fri, 4 Apr 2003, Torkel Franzen wrote:

> Neil says:
> 
>   >> After all, whether or not we know that S is consistent, we
>   >> know that B(_n) is true if and only if n is a proof in S of (x)A(x).
> 
>   >This is incorrect. Suppose we don't know that S is consistent. I take 
>   >it, then, that you are countenancing the possibility that S is
>   >inconsistent. But if S is inconsistent, the "if" part of your claim fails.
> 
>   How do you arrive at this conclusion?

If S is inconsistent, then there will be some n such that n is a proof in
S of (x)A(x). (Extend the proof of the inconsistency of S with a single
step of ex falso quodlibet, to obtain (x)A(x) as the conclusion, and then
find the code number n of the resulting proof.) But B(_n) could be false,
for all that.

Neil