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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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 quasi Posts: 12,067 Registered: 7/15/05
Re: A finite set of all naturals
Posted: Aug 13, 2013 3:56 AM

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?

quasi

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