Re: Cantor Confusion

In article <1174055000.972069.261630@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

On 16 Mrz., 15:15, Carsten Schultz <cars...@xxxxxxxxx> wrote:
mueck...@xxxxxxxxxxxxxxxxx schrieb:





On 16 Mrz., 14:35, Carsten Schultz <cars...@xxxxxxxxx> wrote:
mueck...@xxxxxxxxxxxxxxxxx schrieb:

On 16 Mrz., 01:31, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1173954799.919385.61...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
For even binary trees ( where even here means all paths are of equal
length),
Only those are under discussion here.
the number of paths increases exponentially with number of
levels (lengths of a path). Adding 1 to the number of levels doubles
the
number of paths.
The tree is continuous because its nodes are connected by paths.
That is a distinctly non-standard meaning for "continuous" in
mathematics.
It shows, however, that the number of paths cannot jump from finite to
uncountable.
Using a word does not constitute proof.

And indeed sup_{n<aleph_0} 2^n = aleph_0 < 2^aleph_0,
so in this sense the function kappa |-> 2^kappa is not continuous. If
you can prove (not claim!) by using your tree that it is, then you will
finally have succeeded in showing that ZF is inconsistent.

Have fun,

I had already quite a lot.

I can imagine.

The function of all cross sections, f: n |--> 2^n, is "continuous" in
the sense that never a jump by more than a factor 2 can occur because
the nodes of the tree are connected by an untearable network. The
domain is the same as the range, namely N. That is fact, not by claim
but by construction of the tree. That's why I constructed it.

You constructed the tree to show that 2^{n+1} <= 2*2^n ? Well, that
really must have been fun. Ok, I agree on this. Now we know a property
of the function

f : N -> N
n |-> 2^n.

This does not tell us anything about 2^aleph_0.


aleph_0 is not a natural number.

Are you just discovering that?

Don't mistake the infinite number of finite paths with infinite paths.

An infinite path corresponds to an infinite set of finite paths in the
which the nodes of each path in the set are included as nodes in every
longer path in the set. There is one such infinite path for each such
maximal infinite set of finite paths and vice versa.

And the union of the infinite set f finite paths, as set of nodes will
be the set of nodes of the infinite path. And that is using the
legitimate meaning of union as in ZF or NBG.


In the union of all fiite trees every path has a finite length, given
by a natural number of nodes. Presently we are considering the union
of all such finite paths. (The union of all finite natural numbers is
an infinite union - nevertheless this union cotains only finite
numberrs.)

The the mathematical union of the set of all finite paths is merely the
set of nodes of the tree. But each infinite path requires a different
infinite subset of the set of all finite trees, as described above.

So WM is faced with something like the power set of his set of all
finite paths, not merely the set of finite paths itself. And WM cannot
handle it.
.