Re: Cantor and the binary tree

In article <1120729088.073422.29510@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
>
>
> *** 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.

Have you read the part you snipped?

> > > 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 not. But you are too dense to notice.

> It is impossible, however, to show the completed result, namely the
> ordered set of rationals.

There is no completed result.

> 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.

There is no completed antidiagonal, that is you can not give all decimal
digits. On the other hand it is sufficient to show the existence of a
real that is different from all the reals on your list.

> > > 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.

You are indeed using your own personal definition of the concept of number.
In general in mathematics you start with a set of elements with the
operations + and * defined. When the operations satisfy some basic
properties (the set under + forms a group, * is distributive over +)
you can call the elements numbers. (Some mathematicians want a bit more
structure like associativity, or sometimes even commutativity, of *, but
not all want that.) When it is clear about what structure you are talking
you can use the term numbers as synonym for elements. Otherwise it is best
to qualify the numbers. Note that until now order relations play no role.
And indeed, the 5-adic numbers can not be consistently ordered, amongst
others, because the square root of -1 is a 5-adic number.

The only place where, in the definitions, ordering plays a role is when
you want to introduce transcendental numbers. For algebraic numbers
there is no need. And indeed, it is a cornerstone of Galois theory
that you can not algebraically distringuish the roots of an irreducible
polynomial.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

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