On Jun 23, 1:55 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > 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.
This is interesting. So where is the error in the argument?
> > >> 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? > > Regards, WM > > > > - Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -
