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, but with your construction above, you can't specify either the first or the last.
