On 09/04/2013 3:06 AM, Alan Smaill wrote: > Nam Nguyen <firstname.lastname@example.org> writes: > >> On 08/04/2013 8:30 AM, Alan Smaill wrote: >>> Nam Nguyen <email@example.com> 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). Hence, the step of showing the property is quite simple: 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).
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... .
> >> 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.
If the set of ordered pairs you perceived as a language structure can be used to verify the truth of cGC at all.
Remember, a language structure is something you must create per the FOL definition of language structure. And until you _verify_ what you've created is indeed a language structure, some bets will be off (so to speak).
So, can you verify that what you perceive as the natural numbers be indeed a language structure for L(PA)?
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.