Re: An uncountable countable set

In article <1152182749.926627.310460@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

*** T. Winter schrieb:


> All sequences, even all single transpositions which I apply can be
> enumerated by natural numbers, just like the lines of Cantor's list.

Yup, so there are infinitely many.

Are yo sure? How could an enumeration by natural numbers supply
infinity?

If an enumeration rule exhausts the naturals, the set being enumerated
is Dedekind infinite.


There are only finite natural numbers.

But infinitely many (an endless supply) of those finite natural numbers.
In addition to cardinality we have to do here with ordinality. The
sequence
of transpositions you give has order type w * w.

No. There is a first element and there remains a first element.

The order type w*w indicates a well ordering, so any nonempty subset
has a first element. But this is not the order type of the set being
permuted, but of the set of permutations being applied to it.

> I have shown that the transpositions can be enumerated by natural
> numbers. There is no number oo. There is neither such a number in
> Cantor's list. All we do is enumerated. Othewise it would not be
> defined at all. Neither in Cantor's list. Therefore it is irrelevant
> whether or not something has to be "executed in order". We are in the
> countable domain and do not leave it.

But transpostitions do not always commute: As right operators on a list
(a b) (b c) = (a c b) but (b c) (a b) = (a b c)

So changing their order of execution can change their effect.

That makes no sense, again. The transpositions have to be execute in the
order given.



Of course. But that does not exclude that these transposition can be
executed and finished (if Cantor's list can be finished).

But it prohibits them from being executed out of their prescribed order.
Since the order of execution induces a well ordering of the set of
transpositions, there would have to be a first transpostion producing
any given effect. What is the first transposition on a well ordering of
the rationals producing an ordering that is dense?



If you do them in any other order the result can be different.
So that you *can* enumerate them does not mean that the result is
independent
from the order in which you perform them.

That is true but has no relevance.

Perhaps not to you, but considerable to your alleged theorem.
.