In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 22 Jun., 14:34, Sylvia Else <syl...@not.here.invalid> wrote: > > On 22/06/2010 9:28 PM, WM wrote: > > > > > On 22 Jun., 11:49, Sylvia Else<syl...@not.here.invalid> wrote: > > > > > The list containing (L0, A0, A1, A2,...) would only then be a list Ln, > > > if its antidiagonal is the antidiagonal of some Ln. > > > > > But if (L0, A0, A1, A2,...) is not a list Ln, then something must have > > > been happened in between that was incompatible with the process of my > > > proof. > > > > Strictly speaking, by your method of construction, the lists are > > > > (L0), (A0, L0), (A1, A0, L0), (... , A2, A1, A0, L0) ... > > > > being respectively, > > > > L0, L1, L2, L3, ... > > Correct. > > > > For (A0, A1, A2, ...) to be a list that should contain its own > > anti-diagonal, > > I do not argue about (A0, A1, A2, ...). The list that I consider has > the form > > Ln = > > An > ... > A2 > A1 > A0 > L0 > > where L0 is an infinite list of all rational numbers. > This list Ln does not contain its own antidiagonal.
Lets see if I have this correctly, with only a slight change in notation.
L_0 is a listing of the rationals. L_(n+1) is a listing in which A_N, an antidiagonal of L_N, is prepended to list L_n to form a new list.
> > > it has to be the same as an Ln, yet I can see no reason > > to think it would be. > > It is not, need not cannot and should not.
If I have it right above, there there is clearly a listing of the A_n. > > > The fact that a contradiction would arise seems a > > powerful indicator that (A0, A1, A2, ...) would not be same as any Ln. > > A contradiction arises for every Cantor list. Cantor thought that he > had proved the uncountability of all definable binary sequences > (because he used only definable binary sequences in his list and > obtained a definable binary sequence as its antidiagonal).
Actually, nothing in Cantor's argument requires any of the binary sequences in his argument to be "definable". What Cantor says is "for ANY list of binary sequences".
The > contradiction arose when the definable sequences were recognized to be > countable.
Except that Cantor made no such restriction to definable sequences as WM does, so it is only WM's statements, not Cantors, which don't work as Cantor said they did.