On 4/9/2013 8:21 PM, Nam Nguyen wrote: > On 09/04/2013 3:06 AM, Alan Smaill wrote: >> Nam Nguyen <email@example.com> writes: >> >>> 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? > > A structure theoretical property is just an n-ary predicate set > of (n-tuples).
logic/theory -> predicate symbol with arity
model/structure -> relation over a domain with tuples
> Hence, the step of showing the property is quite simple:
Maybe. Maybe not.
It is a matter of *proving* that the stated interpretation of language symbols by the stipulated relations of the model description are, in fact, *evaluable truth conditions* that satisfy the statement of the axioms.
> to structure theoretically verify that, per a given formula F, a certain > set of n-tuples _is a subset of the underlying predicate set_ . > (Note a function set is also a predicate set).
Yes. Functions are representable as relations.
> > If we don't know how to verify whether or not a set is a subset of > another set, it's very much a forgone conclusion we simply can't > argue anything about language structure: about the natural numbers, > about Goldbach Conjecture, about cGC, etc... .
No. It is not. One *proves* using the meta-logic of the meta-language. A proof can be a mechanical verification under certain circumstances. But, a proof within the metalanguage can also be a proof in the usual sense about how the stipulations do, in fact, describe truth conditions for the formulas of the object language. If a verification requires an infinite number of steps, it will not be able to conform with the notion of a metalinguistic proof.