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 <namducnguyen@shaw.ca> wrote:

>>>

>>>> On 05/04/2013 6:11 PM, Virgil wrote:

>>>>> In article <_pJ7t.186674$rk7.149719@newsfe05.iad>,

>>>>> Nam Nguyen <namducnguyen@shaw.ca> 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,

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.

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.

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There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI

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