SPQR
Posts:
411
Registered:
8/12/11
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Re: Who's up for a friendly round of debating CANTORS PROOF?
Posted:
Aug 19, 2011 11:37 PM
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In article
<9b20ea7f-fdf4-4a2f-81e4-f7552989433c@m4g2000pri.googlegroups.com>,
Graham Cooper <grahamcooper7@gmail.com> wrote:
> On Aug 20, 1:01 pm, SPQR <S...@roman.gov> wrote:
> > In article
> > <4ae2b135-dda0-45be-a247-0562bb92c...@g39g2000pro.googlegroups.com>,
> > Graham Cooper <grahamcoop...@gmail.com> wrote:
> >
> >
> > > On Aug 20, 10:51 am, SPQR <S...@roman.gov> wrote:
> > > > In article
> > > > <260ffb75-d124-42a6-80fd-88ebdf8ad...@j14g2000prh.googlegroups.com>,
> > > > Graham Cooper <grahamcoop...@gmail.com> wrote:
> >
> > > > > On Aug 20, 8:55 am, SPQR <S...@roman.gov> wrote:
> > > > > > In article
> > > > > > <07c4f31c-f445-4de8-8738-e3cc9ab8a...@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 anti-diagonal.
> >
> > > > > > 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 = Tn-1.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 anti-diagonal 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
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