On 05/04/2013 9:30 PM, Nam Nguyen wrote: > 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.
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.
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.