[FOM] consistency and completeness in natural language

[Torkel Franzen]

Neil says:

>This is not the only straightforward justification. Here is another:

>	Each instance A(n) is of the form not-B(n), where B(n) represents
>	"n is a proof, in the formal system, of (x)A(x)". If, for any n,
>	B(n) were a theorem of the formal system, then there would be a
>	proof, in the formal system, of (x)A(x). But (assuming the system
>	is consistent) there is no proof, in the system, of (x)A(x). Hence
>	B(n) is not a theorem (for any n). But every p.r. statement is
>	provable or refutable in the system. Hence, for every n, B(n) is
>	refutable; whence, A(n) [i.e., not-B(n)] is provable.

To this argument, then, we add (on your reading of Dummett)

(1)     Since every theorem of the formal system S in question is true,
        and every A(n) is provable in S, it follows that every A(n) is true.

That every A(n) is true means that for no n is n a proof in S of (x)A(x).
But this is just what was used in the argument to show that every
A(n) is provable in S. If somebody were to present the above argument,
with the addition of (1), and was in fact not joking, I would be baffled.

  >The only interpretation of your claim that is consistent with G"odel's
  >incompleteness theorem would require the correctness of the step in
  >question to be proved in some *extension* of S.

  Merely an extension by definitions. There is no problem with
defining truth for Pi-0-sentences in S. Certainly somebody could
choose to put forward the semantic argument with the explicit
stipulation that an unrestricted truth predicate is used, but since
this is obviously unnecessary, there is no reason why the semantic
argument should be held to necessarily involve any concepts not
arithmetically definable.

  >Here again, by objecting to my diagnosis of the semantic arguers'
  >semantical argument, you support, in effect, the deflationary thesis of my
  >paper. The principle that I called U.R._p.r. (uniform reflection on
  >primitive recursive predicates) would be an adequate arithmetical
  >formalization of the allegedly "semantical" step above. 

  Sure. We don't need to invoke anything that goes logically beyond
arithmetic to formalize the informal argument whereby G is seen to be
true if S is consistent, or if some reflection principle holds. The
question of how to justify consistency extensions, or extensions by
reflection, is a different matter, and this is where "substantialism
vs deflationism" becomes a substantial issue.

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Torkel Franzen