Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.


fom
Posts:
1,968
Registered:
12/4/12


Re: Matheology § 224
Posted:
Apr 10, 2013 3:59 AM


On 4/9/2013 8:21 PM, Nam Nguyen wrote: > On 09/04/2013 3:06 AM, Alan Smaill wrote: >> Nam Nguyen <namducnguyen@shaw.ca> writes: >> >>> On 08/04/2013 8:30 AM, Alan Smaill wrote: >>>> Nam Nguyen <namducnguyen@shaw.ca> 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 Allmightyformalsystem, proving all arithmetic >>>>> true formulas. >>>>> >>>>> We'd be simply change the name "Allmightyformalsystem" >>>>> to "Allmightylanguagestructure". But it's still an Incompleteness >>>>> (of the 2nd kind) that we'd encounter: the Incompleteness of language >>>>> structure interpretation of the abstract (nonlogical) 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 trueable, or falseable, in the first >>> place. >> >> What reasoning steps are allowed in showing properties of >> language structures? > > A structure theoretical property is just an nary predicate set > of (ntuples).
In general,
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 ntuples _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 metalogic of the metalanguage. 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.



