Re: abundance of irrationals!)

In article <fb701d3c.0504221142.1006d1ed@xxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx (W. Mueckenheim) wrote:

> Matt Gutting <tchrmatt@xxxxxxxxx> wrote in message
>
> > > Right. But neither 0 not 1 are part of the list.
> > >
> > As implied above, this is not the point. By the way sequences and infinite
> > series are defined, the infinite sum SUM(i=1 to infinity) f(i) could simply
> > be regarded as another way of writing "1", just as "2+2" is another way
> > of writing "4".
>
> It is unimportant whether there is this or another definition of a
> limit. It does not matter for our sake.


Then why does WM refuse to either accept the standard definition or give
his own.

Do you, WM, accept the standard (for every epsilon there is an n0)
definition of limits of seqences (including series) as the ONLY
definition? IF not you are obligated to give your definition.

> Each line of the Cantor list
> is enumerated by a natural number. And the digits of the diagonal
> fitting into these lines are all enumerated and do not constitue any
> other limit.

Given a list of reals {a_n : n in N} (function from N to R) with each
a_n expressed decimally to any desired but finite number of decimal
places, to construct a (Cantor) number different from all of them.

We construct a sequence of n-digit pre-Cantor numbers, c_n, inductively
using the rule:
If the nth digit of f(n) is 7 then the nth digit of the c_n
is 2, otherwise it is 7.

We start with n = 1 in the obvious way, then given n = m, do the
construction for n = m+1 by appending one more digit.

By the inductive principle, this guarantees that there is now a c_n for
every n. It further guarantees that no c_n equals any of
{a_1,...,a_n}

Since the sequence of pre-Cantor numbers clearly converges to some real
number limit, we take that limit as the Cantor number.
The constructed Cantor number does not contain any zero digits or 9
digits, so that the number has only one decimal representation.

And also, by construction, it differs from every real in the list at at
least one decimal place and differs from all those with dual
representations in infinitely many decimal places for either
representation.

Therefore the limit of the pre-Cantor numbers is a real number not in
the list.
.

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