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

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Posts: 2,777
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 <> 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.

>> 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"?


> 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.


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