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Re: Matheology § 224
Posted:
Apr 10, 2013 8:58 PM


On 10/04/2013 1:59 AM, fom wrote: > 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.
But what is "metalogic of the metalanguage", in the context of FOL structure? Or is that at best just intuition and at worst just a buzzword?
> A proof can be a mechanical verification under certain circumstances.
Right.
> 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.
On the other hand one could very much _claim_ anything a proof too.
> If a verification > requires an infinite number of steps, it will not be able > to conform with the notion of a metalinguistic proof.
Which verification that would require "an infinite number of steps" you're referring here?
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI 



