> [...] this is a requirement > in Gödel's paper that T should be about the concept of the natural > numbers.
I don't think that's so. Second order predicate calculus and the set theories ZF and NBG are both incomplete. (They are both incomplete in both senses: there is a formula phi (different for the different theories) such that neither phi nor ~phi is provable _and_ there is a formula (ditto) that is both true and unprovable. None of of those three theories is about the natural numbers. (The phrase 'the concept of' is just so much noise.)
> [...] > > The key point being is some of my opponents were (and still are) wrong > in adamantly insisting that Incompleteness is _only_ about the > un-provability (undecidability)in T (Gödel'sP) of Gödel's sentence > G and not about the truth value of G in the natural numbers.
Your emphasised 'only' is a red herring. Here are two facts. i) Gödel exhibited a formula, call it G, such that P neither proves G, nor does it prove ~G. ii) The G in question is true. Imagine that Gödel had written his paper without the introductory chat. Imagine, further, that he kept quiet thereafter. Then all one could attribute to Gödel would be i); but that would not stop some other person showing ii). Incompleteness in sense i) (as proved (more than once, as it happens) by Gödel) is a syntactic matter syntactically proved. The further _interpretation_ ii) is a different matter. > > Under the assumptions T is about the natural numbers and is consistent, > _G being true in the natural numbers and G isn't provable in T are_ > _materially and logically equivalent_ in his paper: one can _NEVER_ > find G being unprovable (undecidable) in T and yet being _false_ !
When you write "unprovable (undecidable)" does that indicate that you think the two concepts are the same? They are not. 0=/=S0 is unprovable (assuming consistent) but not undecidable. Your last paragraph is either false or much in need of clarification.
-- [Dancing is] a perpendicular expression of a horizontal desire. G.B. Shaw quoted in /New Statesman/, 23 March 1962