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From: Virgil on 4 Sep 2006 14:50
In article <1157366667.646948.260240(a)e3g2000cwe.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> Virgil schrieb:
>
>
> > > It is a potentially infinite set, as described by Peano and by all the
> > > mathematicians before Cantor.
> >
> > Does removing 1 from {1, 2, 3, ..., w} change it from an actually
> > infinite to a potentially infinite set?
>
> No. omega has by far another quality (and quantity) than 1.
> >
> > > Look at my last post concerning the final proof that 0.111... with
> > > actually infinitely many digits does not exist.
> >
> > It does in the world of mathematics, which is the only one that counts.
> > Pun intended.
>
> If you cannot read and understand proofs you may assert this.

I can read and understand proofs of the level that "Mueckenh" is capable
of producing, if and when "Mueckenh"produces any (but I do not claim to
be able to read and understand such proofs as Wiles' of FLT).

But "Mueckenh" 's attempts at proofs have so far all been failures.
From: Virgil on 4 Sep 2006 15:00
In article <1157366851.547033.234870(a)m79g2000cwm.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> Virgil schrieb:
>
>
> > Cantor's "diagonal" in its original form applied only to the set of all
> > binary sequences, i.e ., infinite sequences whose terms were from a set
> > of two objects, and not to real numbers of decimal digits. This is
> > essentially the same as the set of all functions from N to {0,1}, where
> > N is understood to be an endless set of natural numbers.
> > His theorem says that the set of all such functions could not be listed
> > sequentially without leaving some out.
>
> And my tree shows that the complete set can be listed in form of a
> tree.

That every real can be represented by a path in an infinite binary tree
(and some of them even twice) does not mean that there is any function
from the set of naturals, or any other countable set, that surjects to
the set of naturals.
>
> > > What is the difference between my tree and Cantor's list.

The set of members of a list are, by definition countable.
The set of paths in an infinite binary tree, being bijectable with the
power set of the set of naturals, is incapable of having any surjection
from the naturals to it.

Or does "Mueckenh" claim that there is some set which he is able to
surject on its power set?

> > It is only the set of individualy infinite paths in that tree that are
> > uncountable, The set of all edges and the set of all nodes are both
> > countable.
>
> And the set of paths is provably not larger than the set of edges

Not in an "infinite" binary tree in which no path has a terminal node.
In such a tree, the number of paths through each and every node (and
edge) is uncountable.
From: Virgil on 4 Sep 2006 15:03
In article <1157366930.917679.213280(a)b28g2000cwb.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> Virgil schrieb:>
>
> > "Mueckenh" 's claim implies that any infinite set must surject onto its
> > power set. But that has been shown to be false.
>
>
> Learn the difference between a surjection and an intercession.

Intercession is what God sometimes does in answer to a prayer.

But God won't intercede to help "Mueckenh"in this matter,
not even in answer to "Mueckenh" 's prayers.
From: Virgil on 4 Sep 2006 15:06
In article <1157367096.604428.36330(a)i42g2000cwa.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> Dik T. Winter schrieb:
>
> > But if you wish, provide a definition of "number". As far as I know,
> > there is not one in mathematics.
>
> A number has only one of the following properties: It is larger than or
> smaller than or equal to any natural number.

A complex number is a number which need not have any of these properties.

And before one uses "larger", "smaller" or "equal", in any definition,
one must define them too.
From: Virgil on 4 Sep 2006 15:10
In article <1157367209.318653.75760(a)p79g2000cwp.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:


> Counting is the most primitive version of addition.

Counting is not addition at all.

If anything, addition is definable as a short cut for certain counting
problems, but not the reverse.

"Mueckenh" has, as he often does , again put the cart before the horse.
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