Re: An uncountable countable set


Virgil schrieb:

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.

That does not make their sizes infinite. It is completely irrelevant
here.

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.

Why should we change their order?

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?

What is the first real number which cannot be named by a finite set of
letlers? There are such numbers. Which is the first, e.g., by size, or
in an assumed well-ordering of the reals?

Regards, WM

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