Re: Cantor and the binary tree



*** T. Winter wrote:


> > > But what has been stated again and again, such a process of infinitely
> > > many operations that depend on each other will never terminate.
> >
> > You are in error. Once the well ordering is given the operations do not
> > depend on each other, but they are all determined from the beginning,
> > exactly defined by the given prescription. Two different computers
> > executing that program would never deviate from each other. Therefore
> > you cannot escape to agree that this example is *logically* equivalent
> > to Cantor's diagonal.
>
> There is a *huge* logical difference. If you (theoretically) assigned
> infinitely many computers to the task of calculating the antidiagonal,
> the operation will terminate.

So you would also assign a computer to the last line, theoretically?
Otherwise, if it did not exist, for example, none of your computers
could reach it and the process would not terminate.

> When you do the same with your process
> of exchanges the operation will not terminate. BTW, you can describe
> your process slightly different. Take the first number of the well-ordered
> set of rationals. From then on go through the list until you find the
> first one that is smaller. From then on go through the list until you
> find one that is smaller again. Etc.

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.

> > > Yes, so pi exists in the mathematical sense, because it is well-defined.
> >
> > Concerning real numbers, definition and existence do not coincide.
> > That was assumed in the past only because mathematicians were not aware
> > of the principle limitations.
>
> Well, if pi does not exist, what is it? Is d/dx sin(x) only approximately
> cos(x)?

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.

Regards, WM

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