On 23 Jun., 15:42, Sylvia Else <syl...@not.here.invalid> wrote: > On 23/06/2010 11:10 PM, WM wrote: > > > On 23 Jun., 14:51, Sylvia Else<syl...@not.here.invalid> wrote: > > >> 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. > > > That is wrong. Cantor uses the alleged "fact", that infinity can be > > completed, i.e., that that the infinite list can be finished. > > If he would only assume what you say, then the anti-diagonal could > > remain in the unknown part of the list. You know and appreciate that > > after *any* line number n there are infinitely many more lines? > > Yet it's clear that if you look at any later line m, you will still find > that it doesn't match the anti-diagonal.
But that does not help. It is as clear that at any later line you have yet seen less lines than you will have to see. Therefore, without finished infinity, you cannot see any line. Most of them, nearly all of them (infinitely many compared to finitely many) will never be seen. > > > > >> 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. > > > Isn't it enough, also in my case, to know that every antidiagonal has > > a first digit? In fact it has. Every antidiagonal is constructed from > > a list with a first line (that is the previous antidiagonal) and the > > remaining list. Every line has a finite number n. > > I agree that every anti-diagonal you added is well defined. But for your > proof to work you also have to look at the anti-diagonal of the list > after you've added all of the (infinitely many) constructed > anti-diagonals.
How should that be possible? How many there ever may have been constructed. The majority of infinitely many will remain to be constructed.
And even if infinitely many diagonals would have been added, then the resulting list would yield another diagonal. Why do you think that would be a complete list withot its own diagonal?
> To construct the first digit of that anti-diagonal you > have to look at the first element in the list. But it has no first > element
It is very naive to believe that there would result a list without a first line. In my construction every list has a first line and every diagonal will be placed in front of it. How should a list without first line occur?
- any element you might claim is the first is in fact preceded > by infinitely many other elements.
That is ridiculous! How should a list without first element come to existence?
> > Your proof falls apart if you cannot construct the anti-diagonal which > you claim should have been in the list.
If it is in the list, I can construct it and place it in front of the list. Why should that situation change? > > As I observed earlier, this problem can be obviated
What problem? There is no problem. There is an infinite sequence of lists and of antidiagonals. If the sequence is interrupted by a list without containing its antidiagonal, then Cantor's argument fails. If the sequence is not interrupted by a list containing its antidiagonal, then the countable set of antidiagonals is not listable.
> >> First and last are interchangeable, of course, but with your > >> construction above, you can't specify either the first or the last. > > > As I told you, my notation is only an abbreviation for the following > > definition: > > 1) Take a list L0 of all rational numbers. > > 2) Construct its antidiagonal A0. > > 3) Add it at position 0 to get (A0,L0) > > 4) Construct the antidiagonal A1. > > 5) and so on. > > With a resulting 'list' which is infinite at both ends.
No, you are completely in error. How should a missing first line come into existence? If it occured in the process of construction, I would stop and boast that the preceding list has no antidiagonal. Why should I continue? And, first of all, *how* should I continue? To create another blank line from a list without first line???
Can you tell me that? Can you suppose that you could or would like to produce such nonsense-list?