In article <88fn10FqspU3@mid.individual.net>, Sylvia Else <sylvia@not.here.invalid> wrote:
> 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. > > Sylvia.
The elements of a list are standardly indexed by the naturals and so standardly inherit the ordering of the naturals. In this sense, at least, a non-empty list always has a first and an infinite list never has a last. But that standard order is merely a matter of custom, not necessity.
