> 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... .
For the question at hand, the property of interest is a correlation between two language structures. For example, if a set X is a model for Peano arithmetic, with 0, successor, plus and times symbols, and a set Y is also a model, is it possible to identify the element of X corresponding to 0, and similarly for Y?
>> >>> 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.
Ihat not what I'm arguing. Do you think that there are any language structures that satisfy Peano's axioms?
> 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)?
The question here is rather whether there can be two *different* language structures, both satisfying Peano axioms, such that cGC is true in one and false in the other. You believe this to be the case, it seems.
The result mentioned says that this cannot happen -- it does not tell us which the answer is, though.