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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 14, 2013 2:33 AM

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.
>>>

>>>> For instance,
>>>> (sum of 4 primes) can be written as (sum of 2 evens)

>>>
>>> Not really.
>>>
>>> Firstly, the statements of GC_2, GC_4, GC_6 do not require
>>> all of the primes to be odd.

>>
>> You seem to have mislead me then with your clause "All sufficiently
>> large even numbers" in GC_2, GC_4, GC_6.
>>
>> Without loss of generality, given GC_2, GC_4, GC_6, let's assume all
>> the underlying primes here are odd. Do you agree?
>>
>> If you do agree, we'll continue further.

>
> Sure, no problem.
>

>>> Of course, for GC_2, the primes do have to be odd (except for
>>> the case 4 = 2+2), but for GC_4 or GC_6 there's nothing to
>>> prevent the prime 2 from being used (twice).

>
> As you can see below, I already allowed for the possibility
> that you were assuming that 'primes' meant 'odd primes'.

Ok. Thanks. I'll respond tomorrow; I have to go now.

Cheers.

>
>>> But even if GC_4 and GC_6 were modified so as to be restricted
>>> to odd primes only, the statement
>>>
>>> All sufficiently large even numbers can be expressed
>>> as the sum of 4 primes.
>>>
>>> is not logically equivalent (unless GC_2 has been proved)
>>> to the trivially true statement
>>>
>>> All sufficiently large even numbers can be expressed
>>> as the sum of 2 even numbers.
>>>

>>>> 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,

>>>
>>> How would you rewrite GC_4, maintaining logical equivalence,
>>> without using the word 'prime'?
>>>

>>>> 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).

>>>
>>> I have no idea what the above sentence means.

>
> quasi
>

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