quasi
Posts:
12,062
Registered:
7/15/05


Re: A finite set of all naturals
Posted:
Aug 14, 2013 2:10 AM


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 Def03b >>>>>>> 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

