FOM: CH and 2nd-order validity

[John Steel]

On Tue, 17 Oct 2000, Robert Black wrote:


> The point is that whatever your axioms are, for any sentence not decided by
> those axioms you can be (mis?)interpreted so that the sentence comes out as
> either true or false, and if the only constraint on correct interpretation
> is that the axioms should come out as true, the undecidable sentence will
> lack truth-value.
>

          This (the position that the only constraint ....) sounds like a
pretty thoroughgoing formalist position. For any set of people, the axioms
explicitly stated and agreed to by those people are at worst a recursive
set, so their theory T is axiomatizable.  Must the truth value of Con(T)
be undetermined by their usage of the language of T? 
          Is there any ambiguity in the language of Peano arithmetic?


John Steel