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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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quasi

Posts: 10,327
Registered: 7/15/05
Re: A finite set of all naturals
Posted: Aug 14, 2013 2:10 AM
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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.

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


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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