In article <email@example.com>, WM <firstname.lastname@example.org> 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.
Seeing them is not required. Nothing in, for example, FOL+ZFC requires sets or their members to be visible. > > > > > > > > >> 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.
Any of them need for the construction of a new antidiagonal can be constructed as needed.
> And even if infinitely many diagonals would have been added, then the > resulting list would yield another diagonal.
Thus no list can ever be "complete", and the set of elements of a list is never the whole,
> Why do you think that > would be a complete list withot its own diagonal?
There is no way to list or count every real as Cantor and others have proved in several ways and WM has yet to disprove successfully.
Given any list (counting) of reals (or binary seqeuences), there are at least as many unlisted (uncounted) as in the list. > > > > 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?
When one has as many new first lines as entries in the original list. > > > - 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 The point is that anti-diagonals are NOT in the list from which they are constructed. > > > > As I observed earlier, this problem can be obviated > > What problem? There is no problem.
There is with WM's world being imposed on anyone else's, as it won't fit. > > > > >> 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?
By straightforward construction. What WM really means to ask is how to get a pseudo-sequence without a first elemnt. One can easily do it as an infinite process completed.
Note that in the real world, there are myriads of infinite processes being completed all the time. For example all motions are infinite processes in a very real sense.