Re: Cantor Confusion


*** T. Winter schrieb:

In article <1168004631.146333.67680@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
> *** T. Winter schrieb:
....
> > > > (3) Infinite paths are in none of the finite trees, so they are
> > > > also not in the union.
>
> Infinite node numbers are not in any of the finite trees, if you cut
> them at a certain level (contrary to my initial definition with
> sequences of 111... and 000... ). Does the union of such "cut-off"
> trees not contain infinitely many nodes?

Yes, but not infinite node numbers.

What is the difference, please?

Is there a tree with more nodes than the union of all finite trees has?


> You said that in the union of all finite trees there is no infinite
> path.
> Would you say that in the union of all finite trees there is a path
> which contains infinitely many nodes?

No.

After which node does the first non-infinite = finite path of the union
end?

Because in none of the finite trees there is a path that contains
infinitely many nodes. So the two questions following are irrelevant.

None of the sequences {1,2,3,...,n} contains infinitely many numbers.
Nevertheless, you assert that the union of all of them contains
infinitely many numbers.

> > How can an infinite path get into the union of finite trees if none of the
> > finite trees contains one? By magic? All paths that are in the union of
> > finite trees after a finite number of steps go only either to the left or
> > to the right.
>
> How can an infinite set N get into the union of all finite initial
> segments?

It *is* the union. It does not contain any infinite element because none
of the constituents contains an infinite element. And if you think that
N elementof N, you are wrong.

The same is valid for the infinite paths in the complete tree. The
complete tree is the union of all finite trees. The infinite path is
the union of all respective finite paths.

> > Have you *looked* at V=L? But you are unwilling to look at it because
> > you are unwilling to believe it.
>
> I need no belief. I know (from your reaction on the tree, among other
> evidence) that there are no irrational numbers (except for a few names,
> if you wish to call them numbers), so I will not spend any effort to
> look at V=L. The result can only be an assertion that well-ordering is
> possible. If there really was a well-ordering accomplished, then we
> would no longer need the axiom but could use the well-ordering.

You are dense. The well-ordering is given under the conditions of the
axiom.

THEN GIVE IT PLEASE, HERE AND NOW ! Simply reproduce it.

If the axiom is false the well-ordering is not given.

How could it disappear after having existed? Who would force us to
forget it, if it had been given?

> (If you really have two parallels, then you do not need the axiom that
> parallels exist.)

How do you prove they are parallel?

By measuring their distance in a plane.

Note moreover that Euclid postulated
that there was only a single parallel. So in a finitistic world Euclid's
postulate is false. There is more than a single parallel.

False.

Consider your screen.

_________________

x

Draw more than one parallel to the line through the centre of the x.


> > The trees are not different with respect to edges and nodes, it is the set
> > of paths that you allow in the trees that are different.
>
> I have not to allow for anything. Every path which is possible does
> exist as soon as all its edges exist.

No, you are talking about the union of sets of paths. That union is
different from the complete set of paths, as is easily shown.

What counts is only: There cannot be more unions than elements.

> > In the union of
> > finite trees you allow only finite path, when you want completion you also
> > do allow infinite paths.
>
> That is wrong. IF infinite path do exist, THEN they exist in in the
> union of all trees, because there is NOTHING to be completed if already
> ALL is together.

In that case do *not* talk about unions of sets of paths.

I have a union of all finite trees. The result is an infinite tree
(which need not be an element of the union).
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
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