Re: Cantor Confusion

In article <1173724460.248046.46960@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
On 12 Mrz., 16:40, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1173464401.633116.124...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx writes:
....
> > > > It is your last paragraph which is inconsistent
> > > > with the > other definitions. See how you *did* define:
> > > > > lim {n-->oo} |{1,2,3,...,n}| = aleph_0
> > > > which means (as |{1,2,3,...,n}| = n:
> > > > lim {n-->oo} n = aleph_0?
> >
> > Do you not have a comment on this?
>
> It is wrong. A limit is either approached to any positive eps or it
> isn't a limit.

We are not doing analysis here. But you now state that your definition:
lim {n->oo} |{1,2,3,...,n}| = aleph_0
is wrong? Strange as you use it.

It is not my definitin, but it is the definition of set theory. Of
course it is wrong.

Pray provide a reference from set theory where that definition of limit is
defined and used. If you can not provide such you are simply telling an
untruth here. The closest I can come up with is Hrbacek and Jech page
193 from which you can find:
lim{n->oo} |{1,2,3,...,n}| = lim{n->oo} n = aleph_0.
So either show a source where lim{n->oo} |{1,2,3,...,n}| is defined and
where lim{n->oo} n is not defined or admit defeat.

> lim {n-->oo} n = aleph_0 is neither stated by set theory -nor by
> anyone else.

See page 193 of Hrbacek and Jech where you will find a definition of
limit from which you can derive precisely that.

I read that book. Therefore I know their definitions and I knew
already that set theory works with limits when you disputed that.

I really have my doubts. Your first reference to limits in that book were
from a section where they explicitly had stated that the theorems and
definitions from that section were from topology.

Nevertheless their definition does not apply to the infinite set of
finite numbers.

Why does it not apply to the infinite set of finite numbers? Did you
really read the book? First a definition from page 147:
Functions whose domain is an ordinal alpha are called transfinite
sequences of length alpha.
Note that the function whose domain is the finite integers by this
definition is a transfinite sequence of length omega. Further, from
page 193:
Let <alpha_nu | nu < theta> be a transfinite sequence of ordinal
numbers of length theta.
( this applies to <n | n < omega> )
We say that the sequence is increasing
if alpha_nu < alpha_mu whenever nu < mu < theta.
( this also is applicable here )
If theta is a
limit ordinal number and if <alpha_nu | nu < theta> is an increasing
sequence of ordinals,
( both applicable )
we define
alpha = lim{nu -> theta} alpha_nu = sup{alpha_nu | nu < theta}
and call alpha the limit of the increasing sequence.
So that definition is applicable in this case, and by this definition:
lim{n -> omega} n = omega
or (with slight rewording):
lim{n -> oo} n = aleph_0.


(Further it is wrong because what they "call the
limit" is not a limit. But this is irrelevant here.)

Irrelevant, but interesting. Why is it not a limit? I think you
focus to much on the definition of limit from analysis, disallowing
the definition of limits from other branches.

The actually infinite set of finite numbers (without their supremum)
is by set theory defined as:

lim {n-->oo} |{1,2,3,...,n}| = aleph_0 (that expresses the actually
infinite set)

Pray show a source giving that definition.

lim {n-->oo} n < aleph_0 (that is because every number is finite).

See above how the definition on page 193 of the book by Hrbacek and Jech
does apply. And as that is the only place in set theory texts where I
actually *have* found a set theoretic definition of limit, I wonder where
you did find *your* versions of limit.

> > No, they are *not*. The cross sections are sets of nodes (by your own
> > definition), except for C(oo), which is a cardinal number (again by
> > your own definition). And I see *no* relation between aleph-0 and
> > the paths.
>
> The cross section C(n) = |L(n)| is the number of nodes of the level
> L(n).

I still see no relation between aleph-0 and the paths.

aleph_0 is an upper bound for any set of separated paths at a finite
level.

*finite paths*.

Shopuld there be more paths, then they had to cross at least
one level L(alpha) with an infinite number alpha.

Why?

But you have not proven that. You have only proven that there are as many
paths that *terminate* at nodes at that level as there are nodes at that
level.

No. The infinite paths do not terminate at any level. They cross the
level L(n) and then there are 2^n paths (-bundles).

Those are *terminating* paths. (A path-bundle can be seen as a terminating
path: it is a set of nodes containing a finite number of nodes.)

In any finite evel
there are countably many paths (-bundles).

Terminating path.

> If you disagree: What part of a path should not be covered by a node
> of a level L(n) with a finite n?

If you disagree, are there only two paths in the tree because the
cross-section C(1) contains only two nodes?

No. Up to the level L(1) there are only two separated path (-bundles).

Terminating paths.

Up to level L(n) ther are 2^n.

Terminating paths.

And only if there are levels with non-
natural number there can be more than countably many paths (-bundles).

Terminating paths. This says nothing about non-terminating paths.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.