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Topic: A finite set of all naturals
Replies: 32   Last Post: Aug 15, 2013 4:07 PM

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Virgil

Posts: 9,012
Registered: 1/6/11
Re: A finite set of all naturals
Posted: Aug 15, 2013 3:58 AM
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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.
--




Date Subject Author
8/12/13
Read Re: A finite set of all naturals
Ben Bacarisse
8/12/13
Read Re: A finite set of all naturals
Peter Percival
8/12/13
Read Re: A finite set of all naturals
fom
8/12/13
Read Re: A finite set of all naturals
Ben Bacarisse
8/12/13
Read Re: A finite set of all naturals
namducnguyen
8/12/13
Read Re: A finite set of all naturals
namducnguyen
8/12/13
Read Re: A finite set of all naturals
antani
8/12/13
Read Re: A finite set of all naturals
namducnguyen
8/13/13
Read Re: A finite set of all naturals
Marshall
8/13/13
Read Re: A finite set of all naturals
quasi
8/13/13
Read Re: A finite set of all naturals
namducnguyen
8/13/13
Read Re: A finite set of all naturals
quasi
8/13/13
Read Re: A finite set of all naturals
namducnguyen
8/13/13
Read Re: A finite set of all naturals
namducnguyen
8/13/13
Read Re: A finite set of all naturals
quasi
8/13/13
Read Re: A finite set of all naturals
namducnguyen
8/14/13
Read Re: A finite set of all naturals
quasi
8/14/13
Read Re: A finite set of all naturals
namducnguyen
8/14/13
Read Re: A finite set of all naturals
quasi
8/14/13
Read Re: A finite set of all naturals
namducnguyen
8/15/13
Read Re: A finite set of all naturals
namducnguyen
8/15/13
Read Re: A finite set of all naturals
Virgil
8/15/13
Read Re: A finite set of all naturals
namducnguyen
8/15/13
Read Re: A finite set of all naturals
antani
8/13/13
Read Re: A finite set of all naturals
antani
8/13/13
Read Re: A finite set of all naturals
Peter Percival
8/13/13
Read Re: A finite set of all naturals
Ben Bacarisse
8/13/13
Read Re: A finite set of all naturals
namducnguyen
8/14/13
Read Re: A finite set of all naturals
Peter Percival
8/14/13
Read Re: A finite set of all naturals
Shmuel (Seymour J.) Metz

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