On Jun 22, 11:42 pm, Virgil <Vir...@home.esc> wrote: > In article > <3c04e794-89f9-4438-995f-d818a4a0c...@5g2000yqz.googlegroups.com>, > > > > > > Newberry <newberr...@gmail.com> wrote: > > On Jun 22, 1:14 pm, Virgil <Vir...@home.esc> wrote: > > > In article > > > <e6cef47a-8776-48b7-b21a-627a6366d...@g19g2000yqc.googlegroups.com>, > > > > Newberry <newberr...@gmail.com> wrote: > > > > On Jun 21, 9:38 pm, Virgil <Vir...@home.esc> wrote: > > > > > In article > > > > > <2896ff83-7d48-4bcb-80fa-ea38b8e1b...@40g2000pry.googlegroups.com>, > > > > > > Newberry <newberr...@gmail.com> wrote: > > > > > > On Jun 21, 6:11 am, Sylvia Else <syl...@not.here.invalid> wrote: > > > > > > > On 21/06/2010 1:39 PM, Newberry wrote: > > > > > > > > > Not sure why you think you had to tell us how the anti-diagonal > > > > > > > > is > > > > > > > > defined. You claimed you could CONSTRUCT it. Please go ahead and > > > > > > > > do > > > > > > > > so. > > > > > > > > I'm sure he will - right after you provide the list of reals. > > > > > > > > Sylvia. > > > > > > > Dear Sylvia, I did not claim that I could construct a list of reals, > > > > > > but Virgil claimed he could construct an anti-diagonal. > > > > > > To what list? > > > > > > An antidiagonal to a list of decimal representations of reals is > > > > > simple. > > > > > > Ignore any integer digits (to the left of the decimal point) in the > > > > > listed numbers and have 0 to the left of the decimal point in the > > > > > anti-diagonal. If the nth decimal digit of the nth listed number is 5, > > > > > then make the nth decimal digit of the antidiagonal 7, otherwise make > > > > > it > > > > > 3. > > > > > > This rule prevents it from being equal to any real in the listing. > > > > > > The above is only one of many effective rules for constructing an > > > > > antidiagonal different from each listed number. > > > > > How is this effective if the diagonal has infinite amount of > > > > information? > > > > The list of naturals has an "infinite amount of information" in many > > > senses, but a finite rule of construction contains it all. > > > I do not know how to quantify the amount of information the list of > > naturals carries. If it is not 0 then it is a few bits at most. > > > You talk nonsense and you know it. > > It makes perfect sense to me, and perhaps to some of those who read me. > > The list of naturals is > > > compressible the diagonal is generally incompressible. > > Whatever does "compressible" mean? > > The list of naturals can be generated by iteration of a finite formula. > > But Cantor's original anti-diagonal argument uses only binary sequences, > i.e, functions from N to a two member set. Cantor's two element sets was > {m,w}, but using {0,1} simplifies things even more. > > Given a list of infinite binary sequences with values in {0,1}, > b_0, b_1, b_2,..., with b_m_n being the nth term in the mth sequence. > the antidiagonal is a where a_n = 1 - b_n_n
Cantor's proof starts with the assumption that a bijection EXISTS, not that it is effective. From this assumtion alone you cannot construct the anti-diagonal as you and some other people strangely claim. Then it is reasonable to conclude that the anti-diagonal is a contradiction in terms just like R = {x | ~(x in x}.
> > > > > > > > > > If, as in Cantor's original argument, one has a list of binary > > > > > sequences, one takes the nth value of the antidiagonal to be the > > > > > opposite value from the nth value of the nth listed sequence.- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -
