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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: 2,699
Registered: 12/13/04
Re: A finite set of all naturals
Posted: Aug 13, 2013 3:06 AM
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On 13/08/2013 12:40 AM, Nam Nguyen wrote:
> On 13/08/2013 12:56 AM, 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.
>
> 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).


Iow, the Goldbach Conjecture is different from its weak version in
the hypothesis, while is different from GC_4 or GC_6 in the conclusion.

The essence of the Goldbach Conjecture is still distinct.


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