Re: An uncountable countable set


*** T. Winter schrieb:


> Further: Indexing the digit number n is equivalent to covering the
> string up to digit number n. A finite string can never cover an
> infinite string.

But *that* is irrelevant.
Wrong.
Consider K = 0.111... . For each n we can
index digit number n of K and so cover K up to digit number n.

Of course you can index each n. But your "each n" stems from the true
list. And we know that 0.111... is not in the true list, because it is
distinguished from any element of the true list.

Therefore your assertion is correct but void of power to prove that any
digit of 0.111... can be indexed.

> Don't forget: The infinite set of natural numbers contans only natural
> numbers, i.e., finite strings of 1's in unary representation.

Yes, but there is no bound on the index positions, and hence the number
of index positions is infinite.

The number of digit positions is irrelevant. The digit positions are
all finite - in the true list. All numbers with merely finite digit
positions are in the true list. 0.111... is not there.

> Why can't you learn that the
> asserted infinity of the number of numbers is completely irrelevant.

That is just opinion.

That is not opinon. In oder to index or to cover, only the digit
postions of the numbers are relevant. It is completely irrelevant how
many othe numbers are there. Because we always consider only one
special number.

> What counts (in the true sense of the word) is how many digits a number
> has. And every natural number has only finitely many digits, regardless
> of how many numbers there are.

I never contradicted that. Why are you always saying that I claim
something which I do not claim?

You claimed just above that this fact is not relevant.

> Please stop your current intermingling of infinity in sizes of numbers
> and number of numbers.
> There is no ifinite size. from hat I conclude that there is no infinite
> set.

Please stop claiming you found an inconsistency with the axiom of infinity
when your basic assumption already is in contradiction with it.

My basic assumption is the existence of this axiom. I *derive* a
contradiction.

> > But if you claim that the set if finite numbers is finite, there is
> > a largest number.
>
> There is no largest number. The set of of natural numbers is infinite,

Eh? Above you wrote that there is *not* an infinite set. Contradicting
yourself?

Important: There is no largest number. The axiom says only that the set
is infinite.
.
>
> There is no infinite natural number. Because there is no infinite
> finite number - and every natural number is finite. Even the axiom of
> infinity does not state the contrary. (But if the axiom of infinity had
> to be satisfied, then a finite infinite number would be required.
> Compare the staircase.)

Again an assertion, without proof.

The staircase is a proof.

Can't you understand that in mathematics it is on occasion very interesting
to compare infinite sets?

Of course. For this reason I constructed the binary tree.

Regards, WM

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