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


On 14/08/2013 12:53 AM, 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. > >> 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.
> > 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 >
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI

