SPQR
Posts:
411
Registered:
8/12/11


Re: Who's up for a friendly round of debating CANTORS PROOF?
Posted:
Aug 19, 2011 11:37 PM


In article <9b20ea7ffdf44a2f81e4f7552989433c@m4g2000pri.googlegroups.com>, Graham Cooper <grahamcooper7@gmail.com> wrote:
> On Aug 20, 1:01 pm, SPQR <S...@roman.gov> wrote: > > In article > > <4ae2b135dda045bea2470562bb92c...@g39g2000pro.googlegroups.com>, > > Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > > > > On Aug 20, 10:51 am, SPQR <S...@roman.gov> wrote: > > > > In article > > > > <260ffb75d12442a680fd88ebdf8ad...@j14g2000prh.googlegroups.com>, > > > > Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > > > > On Aug 20, 8:55 am, SPQR <S...@roman.gov> wrote: > > > > > > In article > > > > > > <07c4f31cf4454de88738e3cc9ab8a...@w22g2000prj.googlegroups.com>, > > > > > > Graham Cooper <grahamcoop...@gmail.com> wrote: > > > > > > > > > > > You have not accounted for this anywhere in the proof. You > > > > > > > > > merely > > > > > > > > > wave your hands and say "BUT IT WORKS ON ANY DIAGONAL!" > > > > > > > > > > Actually, it was Cantor that said it, and what Cantor said was > > > > > > > > that > > > > > > > > his > > > > > > > > method works on any listing of binary sequences. He never said > > > > > > > > anything > > > > > > > > like what you claim. > > > > > > > > > sure they work on any listing of binary sequences taken as an > > > > > > > individual dataset. > > > > > > > > > but the claims of missing elements should hold up for any given > > > > > > > SET > > > > > > > as > > > > > > > well as any given LIST. > > > > > > > > The standard way to establish COUNTABILITY of a set is to list its > > > > > > elements, i.e., create a surjection from N onto the set. Cantor's > > > > > > argument sows that this is not always possible. > > > > > > > > When impossible the relevant set is called uncountable. as is the > > > > > > set > > > > > > of > > > > > > all functions from N to {m,w} > > > > > > > > > Cantor had 2 proofs didn't he? > > > > > > > > > Powerset and antidiagonal. > > > > > > > > Actually, he has three proofs, but his first one, of the > > > > > > uncountability > > > > > > of a real interval, is doubtless beyond you comprehension. > > > > > > > well it seems I cannot mention digits, permutations, antidiagonals, > > > > > reals and numerous other aspects as they are deemed unrelated to his > > > > > proofs. > > > > > > > You however, gave 2 missing reals to my LISTS > > > > > > > 0.01111111111100000000000... > > > > > and > > > > > 0.11111111111100000000000... > > > > > > > Can you explain why you chose those particular binary digits in the > > > > > first binary place? > > > > > > Because the differed from every one of the ones you listed. > > > > > Right. Let's try again in ternary to avoid the dual representation > > > problem. > > > > > Assume I have a ternary list of all reals, T0. > > > > > My Quantum Computer won't actually output T0, but if I input a > > > parameter 1, 2, 3, ... it will output some permutation of T0 like T1, > > > T2, T3... these LISTS are all from the same SET of reals! > > > > > We can assume Tn = Tn1.Rowswap(a,b) where a and b are two Naturals. > > > > > So rather than give you the full 9 partial lists, here are the 9 > > > DIAGONALS I computed from > > > > > D1 = 0. T1[1,1] . T1[2,2] . T1[3,3] . .... > > > D2 = 0. T2[1,1] . T2[2,2] . T2[3,3] . .... > > > > > in the usual fashion. > > > > > D1 = 0.00111111111222222222... > > > D2 = 0.01111111111222222222... > > > D3 = 0.02111111111222222222... > > > D4 = 0.10111111111222222222... > > > D5 = 0.11111111111222222222... > > > D6 = 0.12111111111222222222... > > > D7 = 0.20111111111222222222... > > > D8 = 0.21111111111222222222... > > > D9 = 0.22111111111222222222... > > > > > I can output infinitely many of the LISTS T1, T2, T3, ... (different > > > permutations of the same set of reals) and also output infinitely many > > > DIAGONALS D1, D2, D3, ... which correspond to LISTS T1, T2, T3, ... > > > > > So given these 9 diagonals from permutations of T0, do you still claim > > > to be able to calculate some missing reals? If so, what are they? > > > > > Herc > > > > In order for the Cantor diagonal argument to be correctly modifiable to > > a list of real number representations in positional notation, one must > > use a base of at least 4 or use successive pairs of digits instread of > > single digits. > > > > Thus you base three example is cooked unless you use pairs of digits. > > > > Taking digits in pairs, which one could do for any base from 2 on up, > > in the diagonal one replaces the nth pair according to the nth pair of > > the nth listed entry in the list, replacing every pair but 01 by 01 and > > replacing 01 by 10. > > > > For your list: > > D1 = 0.00 11 11 11 11 12 22 22 22 22 ... > > D2 = 0.01 11 11 11 11 12 22 22 22 22 ... > > D3 = 0.02 11 11 11 11 12 22 22 22 22 ... > > D4 = 0.10 11 11 11 11 12 22 22 22 22 ... > > D5 = 0.11 11 11 11 11 12 22 22 22 22 ... > > D6 = 0.12 11 11 11 11 12 22 22 22 22 ... > > D7 = 0.20 11 11 11 11 12 22 22 22 22 ... > > D8 = 0.21 11 11 11 11 12 22 22 22 22 ... > > D9 = 0.22 11 11 11 11 12 22 22 22 22 ... > > Then > > AD = 0.01 01 01 01 01 01 01 01 01 01 ... is the antidiagonal so far. > > > Lets say you have a diagonal > > D = 0.1222222222.... > > which equals 0.2000000....
But using a proper method, such as I described above but you apparently didi not read or could not comprehend, such a diagonal is impossible to produce, since every digit pair will necessarily be either a 01 pair or a 10 pair.
> How are you getting a dual representation in ternary?
Using digit pairing, one never does. > >  > > this is NOT A LIST of all reals. > > > > D1 = 0.00111111111222222222... > > > D2 = 0.01111111111222222222... > > > D3 = 0.02111111111222222222... > > > D4 = 0.10111111111222222222... > > > D5 = 0.11111111111222222222... > > > D6 = 0.12111111111222222222... > > > D7 = 0.20111111111222222222... > > > D8 = 0.21111111111222222222... > > > D9 = 0.22111111111222222222... > > > It would be appreciated if you actually read my posts first.
I did and do, but have ignored and will continue to ignore most of what is posted in them as irrelevant to the issue of uncountability and Cantor's proofs of it > > > Herc

