On 12/08/2013 11:53 PM, quasi wrote: > Nam Nguyen wrote: >> >> More than once, I was asked what the difference between the >> Goldbach Conjecture and its weaker form that an odd number >> greater than 7 is the sum of three odd primes. >> >> The point is though the essences of the 2 conjectures are >> drastically different: _an odd number can not be defined >> without addition_ while an even number can (as per Def-03b >> above). > > Consider the following statements: > > GC_2: All sufficiently large even numbers can be expressed as > the sum of 2 primes. > > GC_4: All sufficiently large even numbers can be expressed as > the sum of 4 primes. > > GC_6: All sufficiently large even numbers can be expressed as > the sum of 6 primes. > > etc ... > > It seems you claim to have proved: > > "It impossible to know whether or not GC_2 is true."
That's not what I claimed here though: a counter example of Goldbach conjecture could still be verified (found) and it could be very large. As of to date, there's no logic to say that there's no such counter example.
Virgil challenged Nam:
> Then you are also automatically claiming that the Goldbach conjecture > can never be proved! > > Can you prove that?
Toward that my much more restricted claim to prove (as opposed to my own about cGC) would be:
(*) If the Goldbach conjecture is true in the natural numbers, then it's impossible to structure theoretically prove, verify it so.
And that's very much different from your:
"It impossible to know whether or not GC_2 is true."
Although I think you meant to say:
"It's impossible to know/verify [structure theoretically] GC_2 is true, if it is so true."
and this could be proven in the same manner as (*)
> > Would your proof method also suffice to prove the same for > GC_4? For GC_6? etc?
Before this, do you agree there has been a misunderstanding on what I had said to Virgil I could prove here, in relation to GC_2?
If you do agree, would your questions about GC_4 and GC_6 _still_ stand?
-- ----------------------------------------------------- There is no remainder in the mathematics of infinity.