[FOM] consistency and completeness in natural language

[Torkel Franzen]

 Neil says:

 >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.

  How? Please recall that

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

is provable in PA, and that this proof does not assume the consistency of S.
The Godel sentence (x)A(x) has the form "for every x, x is not a proof in S
of t", where the value of t can be proved to be (x)A(x) itself. B(x) is
"x is a proof in S of t".

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Torkel Franzen