Re: Review of Mueckenheims book.

On 2 Mrz., 14:03, "William Hughes" <wpihug...@xxxxxxxxxxx> wrote:
On Mar 2, 2:37 am, mueck...@xxxxxxxxxxxxxxxxx wrote:

Recall: a union of finite elements can contain an infinite
number of elements.

Of course. But a set of finitely many elements cannot have an infinite
powerset.

Call such a union an infinite
union of finite elements.

There are a countable number of finite paths.

A finite number of nodes does not suffer to construct an infinite
number of paths from them. Look here:
from three nodes
ooo
you can construct three paths
o
oo
ooo

Now let n be any finite number of nodes, then you can prove that there
are not more than n paths possible, can't you? We assumed an infinite
union of *finite* trees. The latter condition guarantees a finite
number of nodes for each set of paths to be costructed from.

There are a countable number of finite unions of finite paths.

You need not say finite unions, because a finite number of paths
cannot form an infinite unin, can it?

There are an uncountable number of infinite unions of finite paths.

Where? Not in the union of finite trees.

An infinite path is the union of an infinite number of finite paths,
hence an infinite union of finite paths.

It is assumed, by ***, that the infinite path p(oo) is the union of
all finite paths p(n) belonging to p(oo), i.e., having only nodes with
value 0. The fnite paths p(n), however, are belonging as subsets to
the finite trees T(n). Therefore only finitely many nodes are
available. There is no set of infinitely many nodes (like it is in
T(oo), perhaps).

There are an uncountable number of infinite paths.

That is clearly wrong. You may have heard it somewhere and now you
repeat it without thinking. Even the tree T(oo) contains only
countably many nodes with no doubt. Therefore its cross section is
countable in any case. Therefore there cannot exist uncountably many
paths, een if we start with T(oo). That is independent of the
discussion above. You should be able to get this result by own
thinking.

Regards, WM

.

Quantcast