On 23 Jun., 13:54, Sylvia Else <syl...@not.here.invalid> wrote: > On 23/06/2010 8:34 PM, WM 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)? > > I don't understand why you've recast it back to that form.
That is a an abbreviation of the construction I proposed. Of course the "..." stand only for an infinite sequence of well defined digits at finite places.
> You can't > even form the anti-diagonal of that - what would the first digit of the > antidiagonal be?
What would the last digit of a normal Cantor-diagonal? Why should the first digit be more important than the last one? An infinite sequence of digits (that is not converging and not defined by a finite formula, like Cantor's diagonal sequence) is as undefined when the last digit is missing as it is when the first digit is missing.
