
Re: A finite set of all naturals
Posted:
Aug 15, 2013 2:42 AM


On 14/08/2013 1:20 AM, quasi wrote: > 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 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.
The problem is "have some similarities" is provably so vague that it won't offer help in comparison, while "isn't of the same essence" is actually defensible.
For instance, let's consider:
GC_7: All sufficiently large even numbers can be expressed as the sum of 6 primes, which is zero.
Obviously GC_7 is false but it definitely "has some similarities" since it's still about the sum of (6) primes.
On the other hand, a _sum of exactly 2 primes_ (Goldbach Conjecture, GC_2) means we're talking about addition function reduction over the subdomain _consisting of primes only_ , while in the case of "sum of 4 primes" or "sum of 6 primes" the subdomain for addition would contain no primes, or would contain more than just primes.
So: different essence.
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI

