> On 08/04/2013 8:30 AM, Alan Smaill wrote: >> Nam Nguyen <firstname.lastname@example.org> writes: >> >>> Seriously, we should begin to abandon the idea that whatever is true >>> or false in the naturals can be structure theoretically proven, >>> verified. >>> >>> If we don't, we'd be in _no_ better position than where Hilbert >>> was with his All-mighty-formal-system, proving all arithmetic >>> true formulas. >>> >>> We'd be simply change the name "All-mighty-formal-system" >>> to "All-mighty-language-structure". But it's still an Incompleteness >>> (of the 2nd kind) that we'd encounter: the Incompleteness of language >>> structure interpretation of the abstract (non-logical) concept known >>> as the natural numbers. >> >> But it is known structure theoretically that if we have any 2 structures >> that satisfy Peano axioms, then they are isomorphic: a statement >> is true in one if and only if it's true in the other. > > Provided that the statement is true-able, or false-able, in the first > place.
What reasoning steps are allowed in showing properties of language structures?
> And cGC isn't that kind of statement.
You already said IIRC that you can express cGC using the language of first-order Peano arithmetic. That means that if cGC is true in *any* one language structure where the PA axioms are true, then it's true in *every* such structure; it cannot be true in one such structure and false in another.