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 >> 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.
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.