Date: Aug 14, 2013 2:33 AM
Author: quasi
Subject: Re: A finite set of all naturals

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.
>>

>>> 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'.

>> 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