Date: Apr 5, 2013 11:30 PM
Subject: Re: Matheology § 224
On 05/04/2013 8:51 PM, Virgil wrote:
> In article <nlL7t.371032$O02.firstname.lastname@example.org>,
> Nam Nguyen <email@example.com> wrote:
>> On 05/04/2013 6:11 PM, Virgil wrote:
>>> In article <_pJ7t.firstname.lastname@example.org>,
>>> Nam Nguyen <email@example.com> 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
>> cGC <-> "There are infinitely many counter examples of the Goldbach
> 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
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.
There is no remainder in the mathematics of infinity.