Re: Review of Mueckenheims book.

On Mar 2, 8:51 am, mueck...@xxxxxxxxxxxxxxxxx wrote:
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.

Recall: each set of paths used to create an infinite path
consists of an infinite number of finite
paths. Therefore, while each path contains a finite number of nodes,
the set of paths used to create an ininite
path contains an infinite number of nodes.





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.

At this point we are just determinging (as you requested) the
number of unions of finite paths. This we have done.
There are an uncountable number of them.
We can now go on to determine how many of them are 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.


OK the infinite path p(oo) consists of the union of
all finite paths having only nodes with value zero.

Let P_0 be the set of finite paths having only nodes with value zero
(P_0 is an element of the power set of the finite paths)

i: Every element of P_0 contains a finite number of nodes

ii: P_0 contains only elements of P_0

iii: P_0 contains a finite number of nodes


iii does not follow from i and ii. (Indeed, P_0 contains an
infinite number of finite paths, so it contains an infinite
number of nodes.)

The fnite paths p(n), however, are belonging as subsets to
the finite trees T(n). Therefore only finitely many nodes are
available.


Yes i is true. (You spend a lot of effort on trivial
statements.)

There is no set of infinitely many nodes

and yes ii is true.

Knowing that i and ii are true does not show that iii is true.

(like it is in
T(oo), perhaps).



There are an uncountable number of infinite paths.

That is clearly wrong.


Not yet proven, but the set of infinite paths is a subset
of an uncountable set.


- William Hughes

.