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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 15, 2013 9:39 AM

On 15/08/2013 1:58 AM, Virgil wrote:
> 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?

>>>>>>
>>>>>>

>>>>>>>> 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,
>> 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?

--
-----------------------------------------------------
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