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

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fom

Posts: 1,968
Registered: 12/4/12
Re: Matheology § 224
Posted: Apr 10, 2013 11:47 PM
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On 4/10/2013 7:58 PM, Nam Nguyen wrote:
> 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 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.

>
> But what is "meta-logic of the meta-language", in the context of FOL
> structure? Or is that at best just intuition and at worst just a
> buzzword?


Tarski's 1933 paper "The Concept of Truth in
Formalized Languages" is the paper in which
Tarski formulated his definition of satisfaction.
You will find the basic sense of that statement
explained in his analyses and subsequent
formulations.

The model-theoretic notion of arithmetical
extension introduced with the 1956 publication
of "Arithmetical Extensions of Relational
Systems" by Tarski and Vaught changed certain
aspects of model-theory. But, that paper refers
to the 1933 paper with regard to the nature
of satisfaction.

As I said before, the production of textbooks
that fail to explain these matters sufficiently
is irresponsible. You should consider the
possibility that your understanding of what
constitutes appropriate dialogue is influenced
by your background sources.

If you wish to discount the history of the subject
matter then you need to start producing a proper
description of what you are doing for consideration
by others (it is not a received paradigm, others are
entitled to reject it if they do not accept your
explanations).

The fact that you can throw a few lines of
code together only means that you know how
to follow instructions. It does not mean
that you know how to think.









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