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

 Messages: [ Previous | Next ]
 namducnguyen Posts: 2,777 Registered: 12/13/04
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 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?

>
>> 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

Date Subject Author
8/12/13 Ben Bacarisse
8/12/13 Peter Percival
8/12/13 fom
8/12/13 Ben Bacarisse
8/12/13 namducnguyen
8/12/13 namducnguyen
8/12/13 antani
8/12/13 namducnguyen
8/13/13 Marshall
8/13/13 quasi
8/13/13 namducnguyen
8/13/13 quasi
8/13/13 namducnguyen
8/13/13 namducnguyen
8/13/13 quasi
8/13/13 namducnguyen
8/14/13 quasi
8/14/13 namducnguyen
8/14/13 quasi
8/14/13 namducnguyen
8/15/13 namducnguyen
8/15/13 Virgil
8/15/13 namducnguyen
8/15/13 antani
8/13/13 antani
8/13/13 Peter Percival
8/13/13 Ben Bacarisse
8/13/13 namducnguyen
8/14/13 Peter Percival
8/14/13 Shmuel (Seymour J.) Metz