[FOM] consistency and completeness in natural language

[Torkel Franzen]

Neil says:

 >Because it is too swift to claim, as you do,

 > (1) B(_n) is true if and only if n is a proof in S of (x)A(x).

 >What the background result of representability-in-S of recursive functions 
 >ensures is only that (if S is consistent then)

 >	B(_n) is *provable-in-S*
 >	if and only if 
 >	n is a proof in S of (x)A(x).


  First, it is by no means too swift to claim (1). On the contrary,
exactly the same proof that shows, in (an extension by definitions of) PA, 

   (x)(B(x) <-> x is a proof in S of (x)A(x))

shows, given standard properties of "true", that for every n, B(_n) is
true if and only if n is a proof in S of (x)A(x).

  Second, suppose we don't want to use this proof, but instead, for
whatever reason, in our semantic argument want to use

(2)  B(x) represents "x is a proof in S of (x)A(x)" in T, and T
     is consistent.

  You seem to have the idea that the T here should be the same theory
as S. Why? 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).
If we wish to stick to your tortured and peculiar argument, we don't
even know that this is the case unless we know that S is consistent.

---
Torkel Franzen