Virgil
Posts:
7,021
Registered:
1/6/11


Re: A finite set of all naturals
Posted:
Aug 15, 2013 3:58 AM


In article <mL_Ot.85191$An7.2051@fx08.iad>, Nam Nguyen <namducnguyen@shaw.ca> wrote:
> On 14/08/2013 1:20 AM, quasi wrote: > > Nam Nguyen wrote: > >> quasi wrote: > >>> Nam Nguyen wrote: > >>>> 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. > >>> > >>> The conclusions are clearly not the same. > >>> > >>> After all, the numbers 2,4,6 are not the same. > >>> > >>> The question I asked was whether the proof technique you used > >>> to show that GC_2, if true, is not provable, would generalize > >>> to show the same for GC_4 or GC_6. The conclusions, while > >>> different, have some similarities, so it's not inconceivable > >>> that you could apply essentially the same reasoning for GC_4 > >>> or GC_6 as you did for GC_2. > > The problem is "have some similarities" is provably so vague that it > won't offer help in comparison, while "isn't of the same essence" > is actually defensible. > > For instance, let's consider: > > GC_7: All sufficiently large even numbers can be expressed as > the sum of 6 primes, which is zero. > > Obviously GC_7 is false but it definitely "has some similarities" since > it's still about the sum of (6) primes. > > On the other hand, a _sum of exactly 2 primes_ (Goldbach Conjecture, > GC_2) means we're talking about addition function reduction over > the subdomain _consisting of primes only_ , while in the case of > "sum of 4 primes" or "sum of 6 primes" the subdomain for addition > would contain no primes, or would contain more than just primes. > > So: different essence.
At least in your own mind. 

