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

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namducnguyen

Posts: 2,674
Registered: 12/13/04
Re: Matheology § 224
Posted: Apr 11, 2013 12:51 AM
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On 10/04/2013 9:47 PM, fom wrote:
> 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.


So now the technical subject is "appropriate dialogue"?

Goodness!

>
> 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.

Is that supposed to be some kind of personal attack
in technical debate?

--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------



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