Re: Cantor Confusion


William Hughes schrieb:

mueckenh@xxxxxxxxxxxxxxxxx wrote:
William Hughes schrieb:


This is the argument you present.


1) The set of lines is also the potentially infinite set of natural
numbers.
2) Every element of the diagonal is in at least one line.
3) Every initial segment of the diagonal is (in) a line.

No, there is one initial segment of the diagonal that is
not in a line.

Then name the element of the diagonal please, which supports your
claim.


There is one initial segment of the diagonal which is not defined
by a single element.

The potentially infinite sequence

{1,2,3 ...}

is an initial segment of the diagonal, but it is not in a line, nor
does it have a largest element.

If it is not in any line then there must be at least one element of it
which is not in any line. If you say it exists, then you must say by
what element it differes from any finite initial segment.

(It is contained in the union of all lines, but the
union of all lines is not a line)

That is a void assertion unless you can prove it by showing that
element by which the union differes from all the lines.



4) The limit (n -->oo, for the number n of elements) of the diagonal is
oo.

Yes and it attains this limit (i.e the limit is a maximum as
well as a supremum)

5) The limit (m -->oo, for the number m of elements of a line) is less
than infinite.

No. The limit is oo. (The limit cannot be any finite number)

And it cannot be an infinite number. There is no infinite natural
number.

Why do you think it is a natural number?

The limit a of a sequence (a_n) is a number which is approximated by
the sequence with arbitrary precision.
For any eps > 0 we can find a number n_0, such that for any n > n_0
|a_n - a| < eps.
For any infinite number, call it omega, and any finite natural number n
we have
|n - omega| > 1. Therefore no infinite number omega can be the limit of
the sequence f(n) = n of natural numbers.

The limit of natural numbers does not have
to be a natural number. This limit is
not a natural number so it can be infinite.

This limit is not a natural number so it cannot be.

Regards, WM

.

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