Re: Cantor and the binary tree



*** T. Winter wrote:

> > So you would also assign a computer to the last line, theoretically?
>
> What last line?
>
> > Otherwise, if it did not exist, for example, none of your computers
> > could reach it and the process would not terminate.
>
> Eh? Each of them would be able in finite time to determine the n-th
> digit. Even in a very short time.

But there would always remain a large part undone, even an infinite
part, that means: nearly all of the work to be done.
>
> > You can also describe the creation of the antidiagonal in this way.
> > Exchange a diagonal digit. From that line go through the list until you
> > find the first one below. From that line go through the list until you
> > find the first one below. Etc. I think, my steps are larger, so I will
> > be ready faster. That is the only difference.
>
> You can describe *any* number in such an iterative process. But not all
> iterative processes are equivalent with non-iterative processes. That is
> the difference.

*Any* tranposition to be performed is determined from the beginning
(given a certain initial well-ordering of the rationals of (0,1)). You
can say, for *any* transposition, when it will have to occur and what
will be the result. It is not an iterative process. It is equivalent to
Cantor's a_nn replaced by b_n.

It is impossible, however, to show the completed result, namely the
ordered set of rationals. It is equally impossible to show the
completed antidiagonal. The only difference is, that the latter is not
so obvious. Terefore some people believe in finished infinity, as yet.

> > Those are ideas which only for special values of x take on the
> > character of numbers. The equations connecting these ideas are
> > certainly as true as "circumference of circle is its diameter * pi" or
> > sqrt(2) * sqrt(2) = 2.
>
> Pray explain. You have lost me again. What are "the characters of
> numbers". And I mean numbers in your sense (they do not conform to
> numbers in the mathematical sense). Beating at terminology? Or what?

A number is an idea which can be put in oder (<) with any other number.
sqrt(2) and the same idea, with digit number 10^100 exchanged by 2,
cannot and never be put in this order.

Regards, WM

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