Re: Cantor and the binary tree

On 7 Jul 2005 02:38:08 -0700, mueckenh@xxxxxxxxxxxxxxxxx wrote:

>> > 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.

The only way to even make sense of the "completed results" is to use a
limit. While I have some ideas for canditates, I don't know of any
natural topology on sequences of orderings. And in the ideas I do
have, your sequences diverges (I.e. there is no "completed result") or
the "completed result" is not what you claim it is.

> It is equally impossible to show the
>completed antidiagonal.

It's not necessary to do so. All that is needed is proof of it's
existence.

>> > 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.

It has a unique position in the natural order on R. Whether or not we
prove a particular descripition of the number to be less than or
greater than it is another story. BTW, even the natural numbers have
this "problem".

Martin

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