Re: Cantor Confusion

mueckenh@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.

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Carsten Schultz (2:38, 33:47)
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