Re: Cantor Confusion

In article <1169987266.609036.283220@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
On 28 Jan., 03:31, "*** T. Winter" <***.Win...@xxxxxx> wrote:
....
> We can boil down the discussion about trees to the following simple
> question, considering only one path, for instance the path p on the
> outmost left hand side of the tree. This path p (in terms of nodes) is
> the union of all paths of finite trees with length n, n in N.

If you consider the union of paths as the union of sets of nodes, you are
right.

If I consider the path length as the union of path length: =
{0,1,2,...,n-1}, I am right too.

Yes, for this path. I never did have problems when you were uniting paths
that you get infinite paths and all that that entails. But you are uniting
*sets* of paths. In that case you do not unite paths at all.

> then at least
> one of the paths in the union must be infinite.

Wrong. The union of a collection of finite sets can be infinite, and in
this case is.

Here we are not concerned with the cardinal number of the union of
infinitely many elements but with the length of the union of all
finite path lengths, where each path length is measured from the
common origin, namely 0 at the root of the tree.

This is extremely unclear. A path is a set of nodes (by your definition),
its length is the cardinality of that set. Let's say that path p_n is
the set of nodes {n_1, n_2, n_3, ..., n_n}, with path length (cardinality)
n. The union of those paths is {n_1, n_2, n_3, ...} with cardinality
aleph_0. But none of the paths in the union is infinite. If you do not
agree with this, come up with a definition of path length when a path
is a set of nodes.

No. You fail to see a crucial difference. Going back at natural numbers:
union(n in N) {1, 2, 3, ..., n} is N
but
union(n in N) {{1, 2, 3, ..., n}} does not contain N.
As your paths are equivalent to initial segments {1, 2, 3, ..., n} we find:
union(n in N) p_n is p_oo
but
union(n in N) {p_n} does not contain p_oo.

Fine. Apply this knowledge to the paths of the infinite tree T(oo).

Yes, each path in T(oo) is the union of a set of finite paths. And each
path in T(oo) is not in the union of the sets of finite paths. So I did
apply the knowledge you agreed to. What now?

> That is set theory. But it is wrong. The union of all finite paths of
> length n is infinite, but not actually. It has no upper bound.

A fundamentally misunderstanding. The union of a set of things is not
the set of the union of things.

In case of path lengths measured as I defined, the union of all
lengths is a length.

We were talking about paths here, methink, not about lengths? So I wonder
why you are stating this here.

And T(oo) does not contain any path? But how do you *define* "finite
without an upper bound"? Can you come up with mathematical definitions
of "actual infinitiy" and "potential infinity"?

Read my chapter 8. There (nearly) all is said what can be said about
that topic.

Still to do that. I begin to wonder.

Wrong, and that is not what you claimed. You claimed that P was a subset
of P_C. You claimed:
> 4) The set of paths in T(oo) is a subset of the countable set
> of finite sets of all paths in the finite trees.
P was the set of paths in T(oo). P(k) was the set of paths in the finite
tree T(k). P_C was the set of finite sets P(k).

If T(oo) is constructed as the union of all finite trees T(k), then
every path in P is a path which is in the union P(1) U P(2) U ... of
the elements of P_C.

Prove it. The union of P(i) contains finite paths only, *by the definition*.
On the other hand P contains infinite paths *by the definition*. So P
can not be contained in the union. Or prove that the union of sets of
finite paths can contain an infinite path.

Therefore P is a subset of this union P(1) U P(2) U ... of elements of
P_C.
Is this correct?

No.

But even this statement is indeed wrong:
> It is a subset of the union set P(1) U P(2) U ...
0.010101... is a path in the complete tree, but is not a path in any of
the P(k).


0.010101... is in the complete tree T = T(oo). (Correct me, if I
remeber that wrong, but here are as much opinions as are set
theorists.).

Right.

0.010101... is not in the union of all T(n), n in N, as
you say.

I do not say that. I explicitly state that it is in it. But it is *not*
in the union of all P(n). The union of all P(n) does contain elements
like "0.0", "0.01", "0.010", "0.0101", but *not* "0.010101...". The
union contains all finite initial segments of "0.010101...", but not
that one itself. In a similar way, N contains all the finite initial
segements of itself, but not N itself.

Therefore T(oo) = {T(n) | n in N} =/= T(oo). Isn't that a
contradiction?

Would be if I ever stated such a thing.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

Quantcast