antani
Posts:
116
Registered:
12/13/04
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Re: A finite set of all naturals
Posted:
Aug 15, 2013 4:07 PM
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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 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 ...
> >>>>>>>>>>
> >>>>>>>>>> ... 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 sub-domain _consisting of primes only_ , while in the case of
> >> "sum of 4 primes" or "sum of 6 primes" the sub-domain 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.
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