Re: An uncountable countable set


*** T. Winter schrieb:


> Because it is very relevant. You state that an infinite string could be
> indexed by finite strings. Complete indexing includes covering.

Nope. You claim the last, but it is false.

No, it is true.

The list of sequences of 1's

1 0.1
2 0.11
3 0.111
....
n 0.111...1
n+1 0.111...11
....

is constructed such that the segment from 1 to n does not contain
numbers which can completely index or cover the digit positions of a
number which does not belong to the segment, like the sequence n+1 and
all sequences which are larger, i.e., which have more 1's.

As 0.111... does not belong to the list, it does not belong to any
segment of the list. Hence its digit positions cannot be completely
indexed or covered by the sequences of the list.

Now you may claim, that the mathematics of finity does not hold in
infinity, but that there everything is different. But that is not
mathematics in my opinion. May be that it is mathematics in some one
other's opinion. Nevertheless it is not useful to continue the
discussion on this topic between us.

Each finite segment can be
covered, and each finite index can be covered. But the total is not
finite, but contains only finite digit positions.

That is, in my opinion, purest nonsense. But the believer will never
give up his belief. So let's finish this discussion.

>
> Because your assertion that K could be completely indexed and covered
> is equivalent to K being in the true list.

My assertion is that K can be completely indexed but not completely
covered. Again you misquote my assertion.

Indexing n implies covering all digit positions m =< n. Indexing all n
implies covering all m =< n.

> > > > That is just opinion.
..
>
> Likewise a number with infinitely many digit positions can not be
> indexed by a natural number.

Why not? Each and every digit position is finite, so can be indexed,
and, by *your* definition the number can be indexed. But there are
infinitely many digit positions, as there are infinitely many
natural numbers.

Yes. And all of them are in the true list. If n can be indexed, then
also n+1 can be indexed, because with n also n+1 is in the list. This
holds for every natural number.

Do you agree that every digit position which can be indexed by a
natural number is in the list?
Do you agree that every natural number which can index a digit position
is in the list?

If a number is not in the list, what does this fact say about its digit
positions?

> It is obvious that an infinite
> number of differences of 1 leads to an infinite number.

Yes. When you can add all those numbers. But the result is not a natural
number.

Exactly. Therefore there cannot exist an infinite number of natural
numbers.

> > > Important: There is no largest number. The axiom says only that the set
> > > is infinite.
> >
> > I said, there is no largest number. The axiom says the set does exist. In
> > and of itself the axiom does *not* state that the set is infinite. It is
> > a theorem that the set is not finite, and so must be infinite.
>
> Why should a set which is not finite be actually infinite (i.e. have a
> cardinal number)?

A set that is not finite is infinite, that is the definition of the word
infinite.

No. Every (potentially) infinite set is finite.

So yuor question is actually, why are there cardinal number?
Well, they come in handy when comparing infinite sets. And (by definition)
a cardinal number is the equivalence class of sets that can be put in
bijection with each other. A set being in bijection with another set is
easily shown to be an equivalence relation, and so we can split the sets
in equivalence classes. That is all elementary mathematics. And those
equivalence classes are called cardinal numbers. Just as the equivalence
classes of Cauchy sequences are called real numbers and the equivalence
classes of ordered pairs of integers are called rational numbers (with
some particular equivalence relation in mind). All extremely elementary.

And all extremely false too. In particular if you consider the infinite
binary tree and call its set of edges uncountable.

> > > > Again an assertion, without proof.
> > >
> > > The staircase is a proof.
> >
> > No.
>
> You may deny this, nevertheless it is true. If there is an actual
> number of steps, then the width is equal to the lenghts is aleph_0.

You keep asserting that.

If you say the length is infinite but the width is not infinite, then
simply exchange the axes of size of numbers and number of numbers. You
will easily see your error.

> What I do not understand is how you can believe that someone (except,
> perhaps, Virgil) could share your opinon that only in one dimension the
> infinity was actually realized, but not in the other.

Just because there is no last line. If there were a last line, it would
have width aleph-0, but there is no last line.

And the line next to the last line, and the line next to the next to
the last line? They all do not exist. I have the impression that almost
every line does not exist, except a few, but in no case more than
finitely many.

Regards, WM

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