
Re: A finite set of all naturals
Posted:
Aug 15, 2013 9:39 AM


On 15/08/2013 1:58 AM, Virgil wrote: > In article <mL_Ot.85191$An7.2051@fx08.iad>, > Nam Nguyen <namducnguyen@shaw.ca> wrote: > >> 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. > > At least in your own mind.
Care to explain why such a simple observation couldn't be in your mind?
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI

