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Topic: Matheology § 224
Replies: 6   Last Post: Apr 11, 2013 12:51 AM

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Registered: 12/4/12
Re: Matheology § 224
Posted: Apr 10, 2013 3:59 AM
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On 4/9/2013 8:21 PM, Nam Nguyen wrote:
> On 09/04/2013 3:06 AM, Alan Smaill wrote:
>> Nam Nguyen <> writes:

>>> On 08/04/2013 8:30 AM, Alan Smaill wrote:
>>>> Nam Nguyen <> 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).

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 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.

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