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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: 2,777 Registered: 12/13/04
Re: A finite set of all naturals
Posted: Aug 14, 2013 2:10 AM

On 14/08/2013 12:53 AM, 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.

>
> 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).
>
> 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