In article <88eea0Fh36U1@mid.individual.net>, Sylvia Else <sylvia@not.here.invalid> wrote:
> On 23/06/2010 10:40 PM, WM wrote: > > 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. > > Cantor doesn't rely on being able to identify a last digit. He's just > saying that no matter how far down the list you look, you'll find that > the element at that point doesn't match the anti-diagonal. But you can't > even begin to formulate his proof if you can't identify the first > element of the list (and hence first digit of the anti-diagonal) either. > > First and last are interchangeable, of course
Not for infinite sequences when one of them clearly exists and the other clearly doesn't.
The first ordinal number is 0. If first and last are so interchaneable, lets see WM interchange it with "the last ordinal number".
