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