Re: Cantor Confusion

On 19 Feb., 15:23, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1171890647.125846.84...@xxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx writes:

> On 19 Feb., 01:35, "*** T. Winter" <***.Win...@xxxxxx> wrote:
...
> > > The limit {1,2,3,...} = N does exist according to set theory. Why
> > > should the limit {1}, {1,2}, {1,2,3}, ... = N not exist?
> >
> > There you go again. Set theory does *not* use limits. I know of *no*
> > definition that gives the limit of a sequence of sets. So pray write
> > down here how *you* define the limit of a sequence of sets.
>
> We need not get involved into a discussion about limits of sets or the
> etymology of the limit ordinal number.

That is irrelevant, you are using above undefined concepts.

> Consider whether the following
> definitions of N in your opinion are correct or not.
>
> N = {n | n e N} = {0,1, 2, 3, ...}
> N = U {{0,1,2,...,n} | n e N} = U {{0}, {0,1}, {0,1,2}, ...}

Both are correct.

> If they are acceptable, then consider whether a set which contains all
> sets of the form {0,1,2,...,n} also contains N.

As a subset (because every set is a subset of itself), not as an element.

> And if every set which contains all sets of the form {0,1,2,...,n}
> contains N,

As a subset.

Fine. Every path of the tree is a special subset. (Not every subset is
a path.) But there are only countably many finite subsets and
countably many sets of subsets which belong to one and the same path.

> why does the union of finite trees T(n) not contain an
> infinite path?

I have never said that. I have stated that it *does* contain infinite
paths.

So the union of finite trees U(T(n)) contains (as subsets) the path
p(oo) and all its co-paths q(oo), ..., i.e., it contains P(oo)?

Regards, WM


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