In article <f9884d20-096d-4fd6-9f63-eae9eba11bb2@w31g2000yqb.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 23 Jun., 04:18, Sylvia Else <syl...@not.here.invalid> wrote: > > On 23/06/2010 11:33 AM, Sylvia Else wrote: > > > > > > > > > > > > > On 23/06/2010 11:03 AM, Virgil wrote: > > >> In article > > >> <2000e81b-7c5a-41be-b6af-98f96f2fb...@w31g2000yqb.googlegroups.com>, > > >> WM<mueck...@rz.fh-augsburg.de> wrote: > > > > >>> On 22 Jun., 21:34, Virgil<Vir...@home.esc> wrote: > > > > >>>>>> But (A0, A1, A2, ...) is obviously countable. Above you say it's > > >>>>>> "certainly > > >>>>>> not countable", but it is. > > > > >>>>> The set is certainly countable. But it cannot be written as a list > > > > >>>> But it HAS been written as a list (A0, A1, A2, ...), > > > > >>> Does this list contain the anti-diagonal of (..., An, ... A2, A1, A0, > > >>> L0)? > > > > >> Since (..., An, ... A2, A1, A0, L0) does not appear to be a list, why > > >> should there be any antidiagonal for it? > > > > > Ach! Let's scrap A0 - it's confusing. > > > > > If we let L_n be the nth element in the list L0, and An the > > > anti-diagonal of the An-1, An-2,...., A1, L_1, L_2, L_3,... > > > > > then > > > > > L_1 > > > A1 > > > L_2 > > > A2 > > > L_3 > > > A3 > > > L_4 > > > ... > > > > > is a list. I'm still thinking about that. > > > > > Sylvia. > > > > Hmm... > > > > A1 is the antidiagonal of L1 L2 L3... > > > > A2 is the antidiagonal of L1 A1 L2 L3... > > > > A3 is the antidiagonal of L1 A1 L2 A2 L3 L4... > > > > Each An is thus constructed from a list that is different from the list > > into which it is inserted. So the construction does not lead to a list > > that should contain its own anti-diagonal, and it doesn't. > > Ln = > > An > ... > A2 > A1 > A0 > L0 > > Does your bijection contain the anti-diagonal of > (..., An, ... A2, A1, A0, L0)? > If yes, then Cantor's argument fails as a list contains its > antidiagonal. > Reason: you list in your listing above only the lines of lists. That > is so because the antidiagonal of every list Ln belongs to another > list L(n+1)). > > b) Does it not? Then a list cannot list all anti-diagonals that > belong to the countabe set constructed in my argument.
Of course no list of reals (or binary sequences) can list all possible nonmembers of that list as there are uncountably many of them.
Note that for every permutation of a list one gets a different antidiagonal, and there are uncountably many permutations of an infinite list. >
