Re: abundance of irrationals!)

In article <fb701d3c.0504130316.4550d5d7@xxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx (W. Mueckenheim) wrote:

> Matt Gutting <tchrmatt@xxxxxxxxx> wrote in message
> news:<1113231873.04c9a13b99987c16d5428259b12945b9@teranews>...
> > W. Mueckenheim wrote:
> > > Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx> wrote in message
> > > news:<ITSnetNOTcom#virgil-DFC74B.18580709042005@xxxxxxxxxxxxxxxxxxxxxxxx>.
> > > ..
> > >
> > >
> > >>Where did you ever see a definition of
> > >> oo
> > >> SUM (1/2^k) = 1
> > >> k=1
> > >>which requires one to have k take on any value not a natural number?
> > >>The symbol "oo" here simply means "without any upper limit".
> > >>
> > >>In the same way for real limits "as x -> oo" one does not mean that x
> > >>ever REACHES "oo", merely that there is no upper bound for x.
> > >
> > >
> > > So you are convinced that 1/9 is a term the following sequence of partial
> > > sums?
> > >
> > > 0,000...
> > > 0,1000...
> > > 0,11000...
> > > 0,111000...
> > > ...
> > >
> > > Regards, WM
> >
> > I don't think Virgil said anything to imply such a statement. What he
> > appears
> > to have said is that 1/9 is the limit of a sum of increasing numbers of
> > terms.
> > There is no need for 1/9 itself to be included in the sequence.
> >
> > In your discussion of Cantor's diagonal proof, you display the number
> > 0.11111...
> > This number is the limit of the sequence 0.1,0.11,0.111, ... (that is, the
> > rows
> > of your list). It is precisely *because* this number 1/9 is not in the
> > sequence
> > of rows that we say it is not on the list..
>
> If it is not in the enumerated rows (I agree that is not there) then
> it is neither in the antidiagonal, because Cantor can find and
> exchange a digit in an enumerated row only.

But Cantor only needs to construct an antidiagonal to show that there is
a number not in the list. If an unlisted number is already known,
"antidiagonals" are irrelevant.

The point is to show there is a number not in the list!

That has been done.
.