On 24/06/2010 6:08 AM, Virgil wrote: > 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.
I was just considering it a matter of perspective. If you have a single ended list, you can consider the end to be the first element or the last element. I can't see that the choice makes any difference mathematically, other than to invert whatever ordering rule applies to the list.
