Re: Cantor Confusion


William Hughes schrieb:

Each diagonal digit is formed by a line.

and by a column!

The number of diagonal digits is the number of lines.
The number of lines is a supremum. Clearly the number
of lines does not yield a diagonal digit.

Let A be the list of all finite strings.
Let D be a diagonal formed from this list.

The question is "How many digits does D have".

It is not enough for you to say that this question does not have a
defined
answer. You must also show that the usual answer (omega)
leads to a contradiction.

It does, because the number of columns of each line is finite, i.e.,
less than omega.

Let the number of digits in D be X. We don't know if X exists.
However,
it is clear that if X does exist it must be greater than any natural
number.
Hence:

- X does not exist
or
- X is at least as great as omega

Correct.


Why do you say incorrect? Given any integer N, we certainly have more
letters
than N.

But the smallest number which is larger than any integer is omega. And
we have excluded a line with omega letters.

But excluding a line with omega letters does not mean that D cannot
have omega letters.

D can be projected or mapped into a line. Then it is a line longer than
any line.

The number of letters in D is less than the number
of letters in a line only when this line has more letters than any
other line.
There is no line with more letters than any other line,
so no line we can use to limit D.

We only know it must be finite.

Correct. If it exists, it must be infinite (or omega). Only then the
number of lines can be omega. But we have excluded that the number of
letters is omega.


No, we have not excluded omega.

You wish to introduce a line with omega letters?
That yields what I always say: There is no infinite number of natural
numbers without an infinite number omega.

If it does not exist you cannot use it to limit the
diagonal.

If it does not exist, then you cannot build a diagonal of that length.

In which case you cannot use the fact that it does exist to show
a contradiction. Yes, you can assume the diagonal does not exist.
No, you do not get a contradiction if you assume it does exist.

You don't get a contradiction if you assume that a supremum is taken
which, by definition, is not taken?

OK. That's st theory.

Regards, WM

.

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