
Re: A finite set of all naturals
Posted:
Aug 13, 2013 8:54 PM


On 13/08/2013 2:43 AM, quasi wrote: > Nam Nguyen wrote: >> quasi wrote: >>> Nam Nguyen wrote: >>>> 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 Def03b >>>>>> 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 ... >>>> >>>> ... 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. >>> >>> OK. >>> >>>>> 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? >>> >>> Sure, no problem  agreed. >>> >>>> If you do agree, would your questions about GC_4 and GC_6 >>>> _still_ stand? >>> >>> Yes. >> >> Then the answer is No: my proof wouldn't be sufficient for GC_4 >> or GC_6 (should they be true), but for a different reason compared >> to the weak Goldbach Conjecture. > > What reason?
Please see below.
> >> Note that in the case of G_4, (sum of 4 primes) => (sum of 2 evens); >> and in the case of G_6, (sum of 6 primes) => (sum of 2 odds). > > Is the above sentence supposed to be the reason why your proof > method can't generalize from G_2 to G_4 or G_6?
Yes.
As I've alluded to in my recent response to you, the conclusion of GC_2 isn't of the same essence as those of GC_4 and GC_6.
For instance, (sum of 4 primes) can be written as (sum of 2 evens) where there's _no_ 'prime' being mentioned. On the other hand, in (sum of 2 primes), we can't get rid of 'prime' in the expression, with the exception that (sum of 2 primes) is a sum of 1's but that's true but indistinguishable from any sums (odd, even, what have you).
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI

