Re: An uncountable countable set


Dik 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?

But you have moreover an infinite
sequence of such infinite sequences.

> OK? And as long as I can enumerate, the number reached is finite. OK?

I do not know what you intend to say here. But you can enumerate, yes, and
when you stop you have done a finite number of transpositions you have
done.

And when I do not stop I nevertheless have done a finite numer of
transpositions anyhow.

Yes. When you continue, you do not stop, so you do not reach a
number, OK?

I can never reach the number oo or omega or aleph_0 if you mean that.

Moreover, this way you will only do the transpositions in
the first sequence of transpositions, so you will never even start with
the second sequence of transpositions.

If I do not get finished with this first infinity, then also Cantor
does get get finished with his list. That is the consequnce I wnated to
point out.

> Hence I do not need more than a finite set of transpositions, though
> its cardinality cannot be given.

Nope.

There are only finite natural numbers.

> This case is in general denoted by
> "countably infinite" - but the number oo does *not* appear. Please
> consider these facts before you demand something to be proved for "oo".

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.

> 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.

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).

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.


> > > Leave Cauchy out of the play. There is no justification to reach all
> > > last digits of the diagonal number by his epsilons. His specification
> > > would only show that all columns of the list up to nn have been
> > > covered.
> >
> > Apparently you do not understand what Cauchy involves.
>
> I understand that Cauchy requires precision on a positive epsilon,
> arbitrarily small.

In fact: given sum{i = 1, oo} d_i/10^i, where 0 <= d_i < 10, by Cauchy
it follows that that denotes a real number. What the number is is
irrelevant, but it *is* a real number.

That is the theory of Meray and Cantor.

> It is completely differenet from Cantor's requirement. Cantor
> does not define any limit process (because he considers finite natural
> numbers for enumeration purposes only).

Perhaps. Please use my definition of the diagonal number above, which
does not definie a "limit process", but uses the definition of limit.

Cauchy requires the epsilon. The theory of irrationals was made by
Meray (C. MERAY: Remarques sur la nature des quantités définies par
la condition de servir de limites à des variables données, Revue des
Sociétés Savants 4 (1869) 280-289.) and Cantor (G. CANTOR: Über die
Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen,
Math. Ann. 5 (1872) 123-132.)

> > There are. The first sequence of transpositions defines the first
> > "infinite permutation". Indeed, the transpositions do not overlap.
> > The second sequence of transpositions defines the second "infinite
> > permutation", but that one overlaps the first "infinite permutation".
>
> No problem. Everything is determined.

If you change the order of those permutations the result will be different
after a finite number of such infinite permutations...

But why should I change the order???

> > By representing a list it is assumed that all members are known. If they
> > are not all known, it is not a list.
>
> And that all are enumerated by finite numbers. The same is valid for
> the set of all positive rationals.

Yes, you can enumerate them. But enumerating does *not* mean ordering.

Enumerating by 1,2,3,... does mean well-ordering with order type
omega.

It is true that when you order them consistent with the enumeration you
get a well-ordered set (of type w). This does *not* mean that any
ordering gives a well-ordered set (of type w). A straight counter-example
is the integers ordered in reverse order. That is not a well-ordered set.

From that example we see that it is impossible to maintain the
well-order and we see why: Because the actual infinite does not exist.
Would it exist, there was always a first element, a second, an so on.

Regards, WM

.

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