
Re: A finite set of all naturals
Posted:
Aug 14, 2013 2:33 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. >>> >>>> 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. > > Sure, no problem. > >>> 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). > > As you can see below, I already allowed for the possibility > that you were assuming that 'primes' meant 'odd primes'.
Ok. Thanks. I'll respond tomorrow; I have to go now.
Cheers.
> >>> 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

