quasi
Posts:
11,309
Registered:
7/15/05


Re: A finite set of all naturals
Posted:
Aug 14, 2013 2:33 AM


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'.
>> 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

