antani
Posts:
116
Registered:
12/13/04


Re: A finite set of all naturals
Posted:
Aug 15, 2013 4:07 PM


In article <gS4Pt.42035$zp7.11594@fx03.iad>, Nam Nguyen <namducnguyen@shaw.ca> wrote:
> On 15/08/2013 1:58 AM, Virgil wrote: > > In article <mL_Ot.85191$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. > > Care to explain why such a simple observation couldn't be in your mind?
Must be because I am not as simple as you are.

