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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 10, 2013 8:58 PM
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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

> 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.

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.


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