Re: Simple enough for every reader?

Am Mon, 30 Jun 2025 20:21:09 +0200 schrieb WM:

On 29.06.2025 12:25, Mikko wrote:

On 2025-06-28 13:56:57 +0000, WM said:

On 28.06.2025 11:56, Mikko wrote:

On 2025-06-27 19:36:41 +0000, WM said:

On 27.06.2025 09:33, Mikko wrote:

On 2025-06-26 13:09:32 +0000, WM said:

>

If we subtract in the order that is used for enumerating then a
last one is necessary.

No, there is no last one in an infinite enumeration.

Then it is not finished or completed.

No, but it can be continued.

Yes it is actually. That is why it's called infinite. You have to take
it, like Cantor, as it's own "finished" thing. That is where you fail.

The notion set can only be applied to complete sets. i.e., sets which
cannot be continued.

Saying that every set is "complete" does not mean anything,

It means that no further element can be found later on.

Elements aren't "found", they either are or are not. You seem to mean
"finite".

All are removed when all are removed.

When done in natural order, then a last one is to be removed before
all are removed. ℕ \ {1, 2, 3, ...} = { }.

No. You said that every set is complete, so {1, 2, 3, ...}, which must
be a set in order to be valid for the context is complete and so is ℕ \
{1, 2, 3, ...}, which is just another way to say { }-
 

This cannot be accomplished

There is nothing to accomplish. What is is, that's all.

Then it cannot be. If it is that all natural numbers are subtracted in
their order, then it is that a last one is subtracted.

For the infinitieth time: no, that does not follow. There is no "last"
number, and yet you can "subtract" an infinity of elements - in an
infinity of "steps", naturally. It would be silly to require a finite
number of them after which an infinite set is exhausted.

Being completed is not a mathematical concept. An infinite sequence
just is infinite.

    But Cantor already knew that there are incomplete, i.e.,
potentially infinite sets like the set of all cardinal numbers. He
called them "absolutely infinite".

Nah. It just has a higher cardinality.

"gegenseitig eindeutige und vollständige Korrespondenz" [Cantor, p.
238]

 
Usually "eindeutige und vollständige" is expressed in English with
"one-to-one", or "gegenseitig eindeutige und vollständige
Korrespondenz"
is expressed as "bijection". The word "gegenseitig" is not really
necessaty but at the time the idea was new and therefore greater
clarity was needed.

 
Yes. But "vollständig" is important. Otherwise "countable" would have no
meaning.

How do you mean?

"Die sämtlichen Punkte l unsrer Menge L sind also in gegenseitig
eindeutige und vollständige Beziehung zu sämtlichen Punkten f der
Menge F gebracht," [Cantor, p. 241]

The same meaning and translation ("one-to-one" or "bijection") applies
here, too.

Yes. both bijection and one-to-one imply completeness.

Do you think N \ {0} and N \ {3} can not be bijected?

In all these example "eindeutig and vollständig" is an feature of the
correspondence, not of any set.

A bijection is a set too.

A set of pairs. An "incomplete bijection" would be a bijection between
finite subsets. That is not a disproof.

So no example of a set of being "complete".

Without completeness countability and uncountability both would be
meaningless

How so?

"Zu dem Gedanken, das Unendlichgroße [...] auch in der bestimmten Form
des Vollendet-unendlichen mathematisch durch Zahlen zu fixieren, bin
ich fast wider meinen Willen, weil im Gegensatz zu mir wertgewordenen
Traditionen, durch den Verlauf vieljähriger wissenschaftlicher
Bemühungen und Versuche logisch gezwungen worden," [Cantor, p. 175]

This basically says that there is no real difference between actual and
potential infinity.

No. Here he talks about the "Form des Vollendet-unendlichen". Vollendet
means completed.

Exactly. "... _also_ in the form of the 'completed'..." Nothing about
a difference from the "incomplete".

Thie says that sets are always complete, so what has been said about
uncompleted infinities either aplies to completed infinities as well or
does not apply to sets.

Uncompleted does not apply to sets. Therefore I use the notion
collection for the potentially infinite.

Too bad nobody else does. Does Cantor? With context, please.

This says that if there were a set of all cardinals that would create a
contradiction.

Yes. If all prime numbers could be known, the same contradiction would
arise.

Primes are not infinite themselves.

These are various ways to point out that sets are immutable, not in the
middle of a construction process.

Yes. But the collection of known prime numbers, for instance, is never
completed.

I'm pretty sure you can download that somwhere.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.