Date: Apr 6, 2013 12:01 AM
Author: namducnguyen
Subject: Re: Matheology § 224

On 05/04/2013 9:38 PM, Nam Nguyen wrote:
> On 05/04/2013 9:30 PM, Nam Nguyen wrote:
>> On 05/04/2013 8:51 PM, Virgil wrote:
>>> In article <nlL7t.371032$O02.109836@newsfe18.iad>,
>>> Nam Nguyen <> wrote:

>>>> On 05/04/2013 6:11 PM, Virgil wrote:
>>>>> In article <_pJ7t.186674$rk7.149719@newsfe05.iad>,
>>>>> Nam Nguyen <> wrote:

>>>>>> In so far as a _perceived_ language structure would enable
>>>>>> us to interpret the concept of the natural numbers, such
>>>>>> a perception is a theology; in it, there are 2 offshoot
>>>>>> theologies which we'll _forever_ (i.e. even in principle of
>>>>>> logic) struggle to choose for acceptance:
>>>>>> - cGC being true
>>>>>> - ~cGC being true.

>>>>> According to Wikipedia
>>>>> CGC can be an abbreviation for:
>>>>> ¤ Chen Guangcheng a civil rights activist in the People's
>>>>> Republic
>>>>> of China who drew international attention to human rights issues in
>>>>> rural areas
>>>>> ¤ Canadian Grenadier Guards
>>>>> ¤ Cambridge Gliding Centre
>>>>> ¤ Canada Games Company
>>>>> ¤ The Capital Group Companies, an investment management
>>>>> organization
>>>>> ¤ the Canine Good Citizen certification
>>>>> ¤ Cerebellar granule cell
>>>>> ¤ Certified general contractor, a type of unlimited
>>>>> contractor in
>>>>> Florida, USA as opposed to registered (limited)
>>>>> ¤ Board-Certified Genetic Counselor
>>>>> ¤ United States Coast Guard Cutter
>>>>> ¤ Color Glass Condensate
>>>>> ¤ Comics Guaranty LLC, a grading service for the comic book
>>>>> collecting industry
>>>>> ¤ Conspicuous Gallantry Cross
>>>>> ¤ Constrained geometry complex
>>>>> ¤ Career Guidance Council, is a not-to-profit organization
>>>>> ¤ Consumer generated content, also known as Consumer generated
>>>>> media
>>>>> ¤ Co-operative Grocer Chain Japan, known as CGC Japan

>>>> Sure. Here cGc means the FOL formula written in L(PA) that would stand
>>>> as:
>>>> cGC <-> "There are infinitely many counter examples of the Goldbach
>>>> Conjecture".

>>> Then you presume that the Goldbach conjecture will never be settled?
>>> It has not been around as long as the FLT, which finally was settled in
>>> the affirmative.

>> Then you don't seem to understand the nature of cGC, depending on the
>> formulation of the Conjecture but being a _different_ formula.
>> For GC (the Goldbach conjecture), there naturally are 2 cases:
>> Case 1 - ~GC is true: we found _one specific even natural_ > 4 that
>> isn't a sum of two primes.
>> But that of course has no bearing on either cGC or ~cGC!
>> So you can't setttle cGC or ~cGC on the account that ~GC
>> is true. And ~GC can still be settled as true!
>> Case 2: GC is true in the naturals as the standard structure for L(PA),
>> and it's said NEG(PA |- GC) and NEG(PA |- ~GC).
>> But if GC is undecidable in PA, there's no proof left in FOL but
>> _structure theoretically verifying_ the truth value of GC in
>> this structure.
>> But how would you _verify_ GC be true in this structure?
>> So, what you have left is just a _pure unverified intuition_
>> which is nothing more or less than a mathematical (theology-
>> like) _belief_ : _no structure theoretical proof_ !
>> In summary, only in Case 1 could you settle GC, but _in both cases_
>> you still can _never_ settle cGC and ~cGC.

> Of course this is just a summary, a bird eye view.
> You can prove that it's impossible to structure theoretically
> verify the truth value of either cGC or ~cGC, using some logic
> anti-induction principles.
> The problem is that you, we, have been so "trained" on IP (Induction
> Principle) that we _don't even suspect_ IP has some loopholes.
> It does!

Seriously, we should begin to abandon the idea that whatever is true
or false in the naturals can be structure theoretically proven,

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.

We can not as a matter of logic prove the truth value of cGC and ~cGC
in the perceived natural numbers, as long as we have the concept of
the natural numbers.

cGC, ~cGC aren't in the same class of FLT, or even GC.

There is no remainder in the mathematics of infinity.