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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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namducnguyen

Posts: 2,701
Registered: 12/13/04
Re: A finite set of all naturals
Posted: Aug 15, 2013 2:42 AM
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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.


--
-----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI


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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