From Newsgroup: sci.math


"Wenn jede Zahl einer [...] Menge verdoppelt wird, dann ergibt sich eine
Zahl, die gr|||fer ist als jede Zahl der urspr|+nglichen Menge." (WM)

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

This message was cancelled from within Mozilla Thunderbird
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Moebius <invalid@example.invalid> wrote:

"Wenn jede Zahl einer [...] Menge verdoppelt wird, dann ergibt sich eine Zahl, die gr|||fer ist als jede Zahl der urspr|+nglichen Menge." (WM)

"If each number of a [...] set is doubled, then this results in a number
which is larger than each number of the original set." (WM).
OK, we can all see that this is false, but what was [...]?
--
Alan Mackenzie (Nuremberg, Germany).
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 20.08.2025 um 22:56 schrieb Alan Mackenzie:

Moebius <invalid@example.invalid> wrote:

"Wenn jede Zahl einer [...] Menge verdoppelt wird, dann ergibt sich eine
Zahl, die gr|||fer ist als jede Zahl der urspr|+nglichen Menge." (WM)

"If each number of a [...] set is doubled, then this results in a number which is larger than each number of the original set." (WM).

OK, we can all see that this is false, but what was [...]?

"festen, invaliablen" ["fixed, invarable"]

Are there sets [in math] which are NOT "fixed" and "invariable". :-P

Actually, WM claims that {2*n : n e IN} contains elements which are NOT
in IN. No, joke.

In other words, he claims that there is an n e IN such that 2*n !e IN.

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 20.08.2025 um 23:20 schrieb Moebius:

Am 20.08.2025 um 22:56 schrieb Alan Mackenzie:

Moebius <invalid@example.invalid> wrote:

"Wenn jede Zahl einer [...] Menge verdoppelt wird, dann ergibt sich eine >>> Zahl, die gr|||fer ist als jede Zahl der urspr|+nglichen Menge." (WM)

"If each number of a [...] set is doubled, then this results in a number
which is larger than each number of the original set." (WM).

OK, we can all see that this is false, but what was [...]?

"festen, invaliablen" ["fixed, invarable"]

Are there sets [in math] which are NOT "fixed" and "invariable". :-P

Actually, WM claims that {2*n : n e IN} contains elements which are NOT
in IN. No, joke.

In other words, he claims that there is an n e IN such that 2*n !e IN.

"Wenn jede nat|+rliche Zahlen verdoppelt wird, dann entsteht eine Zahl,
die mindestens doppelt so gro|f wie jede verdoppelte ist." (WM)

["If every natural number is doubled, then there originates a number
that is at least twice as large as each doubled number."]

Mit anderen Worten: "Verdopplung aller Elemente f|+hrt zu transfiniten
Zahlen." (WM)

[In other words: "Doubling of all elements leads to transfinite numbers."]

Go figure.

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 20.08.2025 22:56, Alan Mackenzie wrote:

Moebius <invalid@example.invalid> wrote:

"Wenn jede Zahl einer [...] Menge verdoppelt wird, dann ergibt sich eine
Zahl, die gr|||fer ist als jede Zahl der urspr|+nglichen Menge." (WM)

"If each number of a [...] set is doubled, then this results in a number which is larger than each number of the original set." (WM).

OK, we can all see that this is false

That is absolutely true if the set is actually infinite.

, but what was [...]?

If each number of an actually infinite set is doubled, then this results
in a number which is larger than each number of the original set. The
reason is that "all" means all and not only a potentially infinite
collection. For the potentially infinite collection of all definable
numbers this is wrong.

Regards, WM

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Alan Mackenzie formulated the question :

Moebius <invalid@example.invalid> wrote:

"Wenn jede Zahl einer [...] Menge verdoppelt wird, dann ergibt sich eine
Zahl, die gr%#er ist als jede Zahl der ursprnnglichen Menge." (WM)

"If each number of a [...] set is doubled, then this results in a number which is larger than each number of the original set." (WM).

OK, we can all see that this is false, but what was [...]?

If he really believes it, it might be to exclude the emptyset. IOW
non-empty might have been the intent. Even so, he should also say that
a set containing only zero should also be excluded. The statement is so
wrong that it hardly matters what the eliding was for.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 21.08.2025 13:53, FromTheRafters wrote:

Alan Mackenzie formulated the question :

Moebius <invalid@example.invalid> wrote:

"Wenn jede Zahl einer [...] Menge verdoppelt wird, dann ergibt sich
eine Zahl, die gr|||fer ist als jede Zahl der urspr|+nglichen Menge." (WM) >>

"If each number of a [...] set is doubled, then this results in a number
which is larger than each number of the original set." (WM).

OK, we can all see that this is false, but what was [...]?

If he really believes it,

It is not a belief but the meaning of completeness.

I consider sets of natural numbers. The only condition is that the sets
are complete, as Cantor and ZF claim. An example is Hilbert's hotel. If
it contains all rooms which can be tagged with natural numbers and the
rooms are inhabited by all guests which are named by natural numbers,
then it is impossible that all guests move from room n to room n+1. It
is impossible that a new guest of this kind appears.

That is the meaning of completeness.

Regards, WM

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am Thu, 21 Aug 2025 12:33:09 +0200 schrieb WM:

On 20.08.2025 22:56, Alan Mackenzie wrote:

Moebius <invalid@example.invalid> wrote:

"Wenn jede Zahl einer [...] Menge verdoppelt wird, dann ergibt sich
eine Zahl, die gr|||fer ist als jede Zahl der urspr|+nglichen Menge."
(WM)

"If each number of a [...] set is doubled, then this results in a
number which is larger than each number of the original set." (WM).
OK, we can all see that this is false

That is absolutely true if the set is actually infinite.

No infinite set of naturals has a largest element.
--
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.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am Thu, 21 Aug 2025 15:07:42 +0200 schrieb WM:

On 21.08.2025 13:53, FromTheRafters wrote:

Alan Mackenzie formulated the question :

"If each number of a [...] set is doubled, then this results in a
number which is larger than each number of the original set." (WM).
OK, we can all see that this is false, but what was [...]?

If he really believes it,

It is not a belief but the meaning of completeness.

I consider sets of natural numbers. The only condition is that the sets
are complete, as Cantor and ZF claim. An example is Hilbert's hotel. If
it contains all rooms which can be tagged with natural numbers and the
rooms are inhabited by all guests which are named by natural numbers,
then it is impossible that all guests move from room n to room n+1. It
is impossible that a new guest of this kind appears.
That is the meaning of completeness.

Nonsense long as you donrCOt clarify what rCRincompleterCY means.
Guest 1 can check out and every guest n can move into room n-1.
--
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.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 21.08.2025 15:57, joes wrote:

Am Thu, 21 Aug 2025 12:33:09 +0200 schrieb WM:

On 20.08.2025 22:56, Alan Mackenzie wrote:

Moebius <invalid@example.invalid> wrote:

"Wenn jede Zahl einer [...] Menge verdoppelt wird, dann ergibt sich
eine Zahl, die gr|||fer ist als jede Zahl der urspr|+nglichen Menge."
(WM)

"If each number of a [...] set is doubled, then this results in a
number which is larger than each number of the original set." (WM).
OK, we can all see that this is false

That is absolutely true if the set is actually infinite.

No infinite set of naturals has a largest element.

Completeness is accepted. When all elements are doubled, then half of
all fall out of the set.

Regards, WM


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 21.08.2025 15:57, joes wrote:

Guest 1 can check out and every guest n can move into room n-1.

But if all natural numbers are occupied in all naturally enumerated
rooms, then no further guest can enter and no further room can be occupied.

Regards, WM




--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am Thu, 21 Aug 2025 17:53:27 +0200 schrieb WM:

On 21.08.2025 15:57, joes wrote:

Guest 1 can check out and every guest n can move into room n-1.

But if all natural numbers are occupied in all naturally enumerated
rooms, then no further guest can enter and no further room can be
occupied.

No, a guest *leaves* and the hotel is still completely occupied.
It doesnrCOt matter with what. You obviously canrCOt check a new natural
into the hotel if all of them are already inside (assuming they have
identity), but you *can* check in any other object, say -1. ThatrCOs
beside the point.
--
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.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

After serious thinking WM wrote :

On 21.08.2025 15:57, joes wrote:

Am Thu, 21 Aug 2025 12:33:09 +0200 schrieb WM:

On 20.08.2025 22:56, Alan Mackenzie wrote:

Moebius <invalid@example.invalid> wrote:

"Wenn jede Zahl einer [...] Menge verdoppelt wird, dann ergibt sich
eine Zahl, die gr%#er ist als jede Zahl der ursprnnglichen Menge."
(WM)

"If each number of a [...] set is doubled, then this results in a
number which is larger than each number of the original set." (WM).
OK, we can all see that this is false

That is absolutely true if the set is actually infinite.

No infinite set of naturals has a largest element.

Completeness is accepted. When all elements are doubled, then half of all fall out of the set.

Regards, WM

I suggest steel toed shoes and a lead undergarment for the heaviest
elements.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/21/2025 3:33 AM, WM wrote:

On 20.08.2025 22:56, Alan Mackenzie wrote:

Moebius <invalid@example.invalid> wrote:

"Wenn jede Zahl einer [...] Menge verdoppelt wird, dann ergibt sich eine >>> Zahl, die gr|||fer ist als jede Zahl der urspr|+nglichen Menge." (WM)

"If each number of a [...] set is doubled, then this results in a number
which is larger than each number of the original set." (WM).

OK, we can all see that this is false

That is absolutely true if the set is actually infinite.

, but what was [...]?

If each number of an actually infinite set is doubled, then this results
in a number which is larger than each number of the original set. The
reason is that "all" means all and not only a potentially infinite collection. For the potentially infinite collection of all definable
numbers this is wrong.

1, 2, 3, ...

doubled:

2, 4, 6, ...

They are all natural numbers.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

joes wrote :

Am Thu, 21 Aug 2025 17:53:27 +0200 schrieb WM:

On 21.08.2025 15:57, joes wrote:

Guest 1 can check out and every guest n can move into room n-1.

But if all natural numbers are occupied in all naturally enumerated
rooms, then no further guest can enter and no further room can be
occupied.

No, a guest *leaves* and the hotel is still completely occupied.
It doesnrCOt matter with what. You obviously canrCOt check a new natural
into the hotel if all of them are already inside (assuming they have identity), but you *can* check in any other object, say -1. ThatrCOs
beside the point.

He seems to think that the set of naturals, being closed under n+1,
means the set is complete. It isn't though, he misunderstands almost everything.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 21.08.2025 18:43, joes wrote:

Am Thu, 21 Aug 2025 17:53:27 +0200 schrieb WM:

On 21.08.2025 15:57, joes wrote:

Guest 1 can check out and every guest n can move into room n-1.

But if all natural numbers are occupied in all naturally enumerated
rooms, then no further guest can enter and no further room can be
occupied.

No, a guest *leaves* and the hotel is still completely occupied.

This is a false result.

It doesnrCOt matter with what. You obviously canrCOt check a new natural
into the hotel if all of them are already inside (assuming they have identity),

Surprising that you understand this delicate point.

but you *can* check in any other object, say -1. ThatrCOs
beside the point.

You have understood: If all natnumbers already reside in the hotel, no natnumber is available to check in.
You have not understood: If the rooms tagged with all natnumbers are
occupied, no room tagged with a natnumber is available for check-in.

Or can you explain the different behaviour of guests and rooms?

Regards, WM



--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Chris M. Thomasson formulated on Thursday :

On 8/21/2025 3:33 AM, WM wrote:

On 20.08.2025 22:56, Alan Mackenzie wrote:

Moebius <invalid@example.invalid> wrote:

"Wenn jede Zahl einer [...] Menge verdoppelt wird, dann ergibt sich eine >>>> Zahl, die gr%#er ist als jede Zahl der ursprnnglichen Menge." (WM)

"If each number of a [...] set is doubled, then this results in a number >>> which is larger than each number of the original set." (WM).

OK, we can all see that this is false

That is absolutely true if the set is actually infinite.

, but what was [...]?

If each number of an actually infinite set is doubled, then this results in >> a number which is larger than each number of the original set. The reason >> is that "all" means all and not only a potentially infinite collection. For >> the potentially infinite collection of all definable numbers this is wrong.

1, 2, 3, ...

doubled:

2, 4, 6, ...

They are all natural numbers.

Both closed under their respective generating formulae.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 21.08.2025 22:22, FromTheRafters wrote:

Chris M. Thomasson formulated on Thursday :

On 8/21/2025 3:33 AM, WM wrote:

On 20.08.2025 22:56, Alan Mackenzie wrote:

If each number of an actually infinite set is doubled, then this
results in a number which is larger than each number of the original
set. The reason is that "all" means all and not only a potentially
infinite collection. For the potentially infinite collection of all
definable numbers this is wrong.

1, 2, 3, ...

doubled:

2, 4, 6, ...

They are all natural numbers.

Both closed under their respective generating formulae.

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too and only
half as many.

Regards, WM

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

WM <wolfgang.mueckenheim@tha.de> wrote:

On 21.08.2025 22:22, FromTheRafters wrote:

Chris M. Thomasson formulated on Thursday :

On 8/21/2025 3:33 AM, WM wrote:

On 20.08.2025 22:56, Alan Mackenzie wrote:

If each number of an actually infinite set is doubled, then this
results in a number which is larger than each number of the original
set. The reason is that "all" means all and not only a potentially
infinite collection. For the potentially infinite collection of all
definable numbers this is wrong.

1, 2, 3, ...

doubled:

2, 4, 6, ...

They are all natural numbers.

Both closed under their respective generating formulae.

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too and only
half as many.

Both are infinite sets with the same cardinality - there is a trivial
bijection between them.

Not sure what you mean by "complete", though. Presumably that would be "vollendet" in German. So your sets are both "vollendet" and
"unendlich".

There's something not quite consistent, there. My vote would be on
"complete" being nonsense here. They're infinite sets, how can they be complete? What is the criterion for distinguishing a "complete"
infinite set from an "incomplete" infinite set?

What does it even mean for a set to be "incomplete"?

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 21.08.2025 um 23:58 schrieb Alan Mackenzie:

Not sure what you mean by "complete", though. Presumably that would be "vollendet" in German. So your sets are both "vollendet" and
"unendlich".

There's something not quite consistent, there. [...]

It's a nonsensical notion (in this context) made up by *crank* W.
M|+ckenheim (Hochschule Augsburg).

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am Thu, 21 Aug 2025 22:17:57 +0200 schrieb WM:

On 21.08.2025 18:43, joes wrote:

Am Thu, 21 Aug 2025 17:53:27 +0200 schrieb WM:

On 21.08.2025 15:57, joes wrote:

Guest 1 can check out and every guest n can move into room n-1.

But if all natural numbers are occupied in all naturally enumerated
rooms, then no further guest can enter and no further room can be
occupied.

No, a guest *leaves* and the hotel is still completely occupied.

This is a false result.

Mind explaining why?

It doesnrCOt matter with what. You obviously canrCOt check a new natural
into the hotel if all of them are already inside (assuming they have
identity),

Surprising that you understand this delicate point.

Maybe you should adjust your impression of me.
What the guests are is irrelevant. They can be referred to by their
(natural) room number.

but you *can* check in any other object, say -1. ThatrCOs beside the
point.

You have understood: If all natnumbers already reside in the hotel, no natnumber is available to check in.
You have not understood: If the rooms tagged with all natnumbers are occupied, no room tagged with a natnumber is available for check-in.
Or can you explain the different behaviour of guests and rooms?

There is no difference. Every guest has a room. But guest 0 can check
out and everybody else move down such that there *still* is no free room.
--
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.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am Thu, 21 Aug 2025 22:26:49 +0200 schrieb WM:

On 21.08.2025 22:22, FromTheRafters wrote:

Chris M. Thomasson formulated on Thursday :

On 8/21/2025 3:33 AM, WM wrote:

On 20.08.2025 22:56, Alan Mackenzie wrote:

If each number of an actually infinite set is doubled, then this
results in a number which is larger than each number of the original
set. The reason is that "all" means all and not only a potentially
infinite collection. For the potentially infinite collection of all
definable numbers this is wrong.

1, 2, 3, ...

doubled:
2, 4, 6, ...

They are all natural numbers.

Both closed under their respective generating formulae.

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too and only
half as many.

ThatrCOs obviously impossible as then one number wouldnrCOt have a double.
--
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.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/21/2025 1:26 PM, WM wrote:

On 21.08.2025 22:22, FromTheRafters wrote:

Chris M. Thomasson formulated on Thursday :

On 8/21/2025 3:33 AM, WM wrote:

On 20.08.2025 22:56, Alan Mackenzie wrote:

If each number of an actually infinite set is doubled, then this
results in a number which is larger than each number of the original
set. The reason is that "all" means all and not only a potentially
infinite collection. For the potentially infinite collection of all
definable numbers this is wrong.

1, 2, 3, ...

doubled:

2, 4, 6, ...

They are all natural numbers.

Both closed under their respective generating formulae.

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too and only

half as many.

^^^^^^^^^^^^^^^^^

NO!!!


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/22/2025 12:07 PM, Chris M. Thomasson wrote:

On 8/21/2025 1:26 PM, WM wrote:

On 21.08.2025 22:22, FromTheRafters wrote:

Chris M. Thomasson formulated on Thursday :

On 8/21/2025 3:33 AM, WM wrote:

On 20.08.2025 22:56, Alan Mackenzie wrote:

If each number of an actually infinite set is doubled, then this
results in a number which is larger than each number of the
original set. The reason is that "all" means all and not only a
potentially infinite collection. For the potentially infinite
collection of all definable numbers this is wrong.

1, 2, 3, ...

doubled:

2, 4, 6, ...

They are all natural numbers.

Both closed under their respective generating formulae.

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too and only

half as many.

^^^^^^^^^^^^^^^^^

NO!!!

1, 2, 3, ...

doubled:

2, 4, 6, ...

Index them using naturals:

[1] = 2
[2] = 4
[3] = 6
...




--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 21.08.2025 23:58, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too and only
half as many.

Both are infinite sets with the same cardinality - there is a trivial bijection between them.

Bijections concern only definable numbers.

Not sure what you mean by "complete", though. Presumably that would be "vollendet" in German. So your sets are both "vollendet" and
"unendlich".

The set rao can be used as the set of guests in Hilbert's hotel. Likewise
the set rao can be used as the set of rooms in Hilbert's hotel. Then no
new guest can arrive and no further room can be occupied.

There's something not quite consistent, there. My vote would be on "complete" being nonsense here.

That is a good choice. But then infinity is only potential.

They're infinite sets, how can they be
complete?

Only by means of dark numbers.

What is the criterion for distinguishing a "complete"
infinite set from an "incomplete" infinite set?

In principle a complete set can only be proven complete by finding a
last element.

What does it even mean for a set to be "incomplete"?

In potential infinity, Hilbert's hotel can always accept a new guest in
form of a natural number. The set of guests is never complete.

Regards, WM


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 22.08.2025 03:15, Moebius wrote:

Am 21.08.2025 um 23:58 schrieb Alan Mackenzie:

Not sure what you mean by "complete", though.-a Presumably that would be
"vollendet" in German.-a So your sets are both "vollendet" and
"unendlich".

There's something not quite consistent, there.-a [...]

It's a nonsensical notion (in this context)

In every context of infinite sets. Therefore dark numbers are required.

made up by

Georg Cantor.

"There was no objection to a 'potential infinity' in the form of an
unending process, but an 'actual infinity' in the form of a completed
infinite set was harder to accept." [H.B. Enderton: "Elements of set
theory", Academic Press, New York (1977) p. 14f]


1.1 Cantor's original German terminology on infinite sets

The reader fluent in German may be interested in the subtleties of
Cantor's terminology on actual infinity the finer distinctions of which
are not easy to express in English. While Cantor early used
"vollst|nndig" and "vollendet" to express "complete" and "finished", the
term "fertig", expressing "finished" too but being also somewhat
reminiscent of "ready", for the first time appeared in a letter to
Hilbert of 26 Sep 1897, where all its appearances had later been added
to the letter.
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". The details of this enigmatic notion are
explained in section 1.2 (see also section 4.1. rCo Unfortunately it has turned out impossible to strictly separate Cantor's mathematical and
religious arguments.)


1.1.1 Vollst|nndig

"Wenn zwei wohldefinierte Mannigfaltigkeiten M und N sich eindeutig und vollst|nndig, Element f|+r Element, einander zuordnen lassen (was, wenn es
auf eine Art m||glich ist, immer auch noch auf viele andere Weisen
geschehen kann), so m||ge f|+r das Folgende die Ausdrucksweise gestattet
sein, da|f diese Mannigfaltigkeiten gleiche M|nchtigkeit haben, oder auch, da|f sie |nquivalent sind." [Cantor, p. 119]

"gegenseitig eindeutige und vollst|nndige Korrespondenz" [Cantor, p. 238]

"Die s|nmtlichen Punkte l unsrer Menge L sind also in gegenseitig
eindeutige und vollst|nndige Beziehung zu s|nmtlichen Punkten f der Menge
F gebracht," [Cantor, p. 241]

"Zwei wohlgeordnete Mengen M und N heissen von gleichem Typus oder auch
von gleicher Anzahl, wenn sie sich gegenseitig eindeutig und vollst|nndig unter beidseitiger Wahrung der Rangfolge ihrer Elemente auf einander
beziehen, abbilden lassen;" [G. Cantor, letter to R. Lipschitz (19 Nov
1883)]

"Zwei bestimmte Mengen M und M1 nennen wir |nquivalent (in Zeichen: M ~
M1), wenn es m||glich ist, dieselben gesetzm|n|fig, gegenseitig eindeutig
und vollst|nndig, Element f|+r Element, einander zuzuordnen." [Cantor, p. 412]

"doch gibt es immer viele, im allgemeinen sogar unz|nhlig viele Zuordnungsgesetze, durch welche zwei |nquivalente Mengen in gegenseitig eindeutige und vollst|nndige Beziehung zueinander gebracht werden
k||nnen." [Cantor, p. 413]

"eine solche gegenseitig eindeutige und vollst|nndige Korrespondenz hergestellt [...] irgendeine gegenseitig eindeutige und vollst|nndige Zuordnung der beiden Mengen [...] auch eine gegenseitig eindeutige und vollst|nndige Korrespondenz" [Cantor, p. 415]

"Zwei n-fach geordnete Mengen M und N werden '|nhnlich' genannt, wenn es m||glich ist, sie gegenseitig eindeutig und vollst|nndig, Element f|+r Element, einander so zuzuordnen," [Cantor, p. 424]


1.1.2 Vollendet

"Zu dem Gedanken, das Unendlichgro|fe [...] 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 vielj|nhriger wissenschaftlicher
Bem|+hungen und Versuche logisch gezwungen worden," [Cantor, p. 175]

"In den 'Grundlagen' formulire ich denselben Protest, indem ich an verschiedenen Stellen mich gegen die Verwechslung des
Uneigentlich-unendlichen (so nenne ich das ver|nnderliche Endliche) mit
dem Eigentlich-unendlichen (so nenne ich das bestimmte, das vollendete Unendliche, oder auch das Transfinite, |Lberendliche) ausspreche. Das Irrth|+mliche in jener Gauss'schen Stelle besteht darin, dass er sagt,
das Vollendetunendliche k||nne nicht Gegenstand mathematischer
Betrachtungen werden; dieser Irrthum h|nngt mit dem andern Irrthum
zusammen, dass er [...] das Vollendetunendliche mit dem Absoluten,
G||ttlichen identificirt, [...] Das Vollendetunendliche findet sich
allerdings in gewissem Sinne in den Zahlen NU+, NU+ + 1, ..., NU+NU+, ...; sie sind Zeichen f|+r gewisse Modi des Vollendetunendlichen und weil das Vollendetunendliche in verschiedenen, von einander mit der |nussersten Sch|nrfe durch den sogenannten 'endlichen, menschlichen Verstand' unterscheidbaren Modificationen auftreten kann, so sieht man hieraus
deutlich wie weit man vom Absoluten entfernt ist, obgleich man das Vollendetunendliche sehr wohl fassen und sogar mathematisch auffassen
kann." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]

"da nun jeder Typus auch im letzteren Falle etwas in sich Bestimmtes, vollendetes ist, so gilt ein gleiches von der zu ihm geh||rigen Zahl.
[...] 'Eigentlichunendlichem = Transfinitum = Vollendetunendlichem = Unendlichseiendem = kategorematice infinitum' [...] dieser Unterschied
|nndert aber nichts daran, da|f NU+ als ebenso bestimmt und vollendet anzusehen ist, wie Nau2," [G. Cantor, letter to K. La|fwitz (15 Feb 1884). Cantor, p. 395]

"Wir wollen nun zu einer genaueren Untersuchung der perfekten Mengen |+bergehen. Da jede solche Punktmenge gewisserma|fen in sich begrenzt, abgeschlossen und vollendet ist, so zeichnen sich die perfekten Mengen
vor allen anderen Gebilden durch besondere Eigenschaften aus." [Cantor,
p. 236]

Regards, WM

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 22.08.2025 09:48, joes wrote:

Am Thu, 21 Aug 2025 22:17:57 +0200 schrieb WM:

It doesnrCOt matter with what. You obviously canrCOt check a new natural >>> into the hotel if all of them are already inside (assuming they have
identity),

Surprising that you understand this delicate point.

Maybe you should adjust your impression of me.
What the guests are is irrelevant. They can be referred to by their
(natural) room number.

If we use the natural numbers as guests and as room numbers, then the
hotel breaks down. There is no new guest arriving and there is no
further room available.

There is no difference. Every guest has a room. But guest 0 can check
out and everybody else move down such that there *still* is no free room.

ChatGPT:
Sobald man aber rCo wie es die anschauliche Rede oft nahelegt rCo G|nste gleichsetzt mit den Zahlen selbst, kollabiert das Bild:

Dann ist klar: Kein rCRneuer GastrCL kann erscheinen, weil es keinen rCRau|ferhalb von NrCL gibt. Ebenso wird kein Zimmer frei, weil jedes schon von seiner eigenen Nummer belegt ist. In diesem Modell ist die Parabel schlicht unsinnig.

Efae Mit anderen Worten:
Das Hotel ist gar keine rCRVeranschaulichungrCL der Mengenlehre, sondern ein sprachliches Blendwerk, das nur funktioniert, weil man unterschwellig
den potentiellen Prozess des rCRUmziehensrCL denkt. Formal hat es keinerlei Erkl|nrungskraft, wenn man G|nste und Zimmer wirklich mit N identifiziert.

Regards, WM
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 22.08.2025 09:49, joes wrote:

Am Thu, 21 Aug 2025 22:26:49 +0200 schrieb WM:

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too and only
half as many.

ThatrCOs obviously impossible as then one number wouldnrCOt have a double.

If Cantor is right and rao is complete, then many natnumbers have no
doubles in rao. If all could be doubled and all results were natnumbers
too, then rao would not be complete but expansive.

Regards, WM

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/22/2025 12:58 PM, WM wrote:

On 22.08.2025 09:49, joes wrote:

Am Thu, 21 Aug 2025 22:26:49 +0200 schrieb WM:

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too and only
half as many.

ThatrCOs obviously impossible as then one number wouldnrCOt have a double.

If Cantor is right and rao is complete, then many natnumbers have no
doubles in rao. If all could be doubled and all results were natnumbers
too, then rao would not be complete but expansive.

Huh? Name a natural number that cannot have its value doubled? Sigh.

1, 2, 3, 4, 5, ...

Doubled:

2, 4, 6, 8, 10 ...

?


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/22/2025 12:15 PM, Chris M. Thomasson wrote:

On 8/22/2025 12:07 PM, Chris M. Thomasson wrote:

On 8/21/2025 1:26 PM, WM wrote:

On 21.08.2025 22:22, FromTheRafters wrote:

Chris M. Thomasson formulated on Thursday :

On 8/21/2025 3:33 AM, WM wrote:

On 20.08.2025 22:56, Alan Mackenzie wrote:

If each number of an actually infinite set is doubled, then this
results in a number which is larger than each number of the
original set. The reason is that "all" means all and not only a
potentially infinite collection. For the potentially infinite
collection of all definable numbers this is wrong.

1, 2, 3, ...

doubled:

2, 4, 6, ...

They are all natural numbers.

Both closed under their respective generating formulae.

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too and only

half as many.

^^^^^^^^^^^^^^^^^

NO!!!

1, 2, 3, ...

doubled:

2, 4, 6, ...

Index them using naturals:

[1] = 2
[2] = 4
[3] = 6
...

Notice a pattern? Say:

[1] = 2 = 1 * 2
[2] = 4 = 2 * 2
[3] = 6 = 3 * 2
[4] = 8 = 4 * 2
[5] = 10 = 5 * 2
[6] = 12 = 6 * 2
....

See? Name a natural numbers that is not compatible with this? Pretty Please?


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am Fri, 22 Aug 2025 21:52:00 +0200 schrieb WM:

On 22.08.2025 09:48, joes wrote:

Am Thu, 21 Aug 2025 22:17:57 +0200 schrieb WM:

It doesnrCOt matter with what. You obviously canrCOt check a new natural >>>> into the hotel if all of them are already inside (assuming they have
identity),

Surprising that you understand this delicate point.

It speaks volumes that you consider this delicate, but itrCOs beside the point.

Maybe you s

From Newsgroup: sci.math

On 23.08.2025 20:40, Chris M. Thomasson wrote:

On 8/23/2025 4:45 AM, WM wrote:

On 22.08.2025 22:11, Chris M. Thomasson wrote:

On 8/22/2025 12:15 PM, Chris M. Thomasson wrote:

On 8/22/2025 12:07 PM, Chris M. Thomasson wrote:

On 8/21/2025 1:26 PM, WM wrote:

On 21.08.2025 22:22, FromTheRafters wrote:

Chris M. Thomasson formulated on Thursday :

On 8/21/2025 3:33 AM, WM wrote:

On 20.08.2025 22:56, Alan Mackenzie wrote:

If each number of an actually infinite set is doubled, then >>>>>>>>> this results in a number which is larger than each number of >>>>>>>>> the original set. The reason is that "all" means all and not >>>>>>>>> only a potentially infinite collection. For the potentially >>>>>>>>> infinite collection of all definable numbers this is wrong.

1, 2, 3, ...

doubled:

2, 4, 6, ...

They are all natural numbers.

Both closed under their respective generating formulae.

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too and >>>>>> only

half as many.

^^^^^^^^^^^^^^^^^

NO!!!

1, 2, 3, ...

doubled:

2, 4, 6, ...

Index them using naturals:

[1] = 2
[2] = 4
[3] = 6
...

Notice a pattern? Say:

[1] = 2 = 1 * 2
[2] = 4 = 2 * 2
[3] = 6 = 3 * 2
[4] = 8 = 4 * 2
[5] = 10 = 5 * 2
[6] = 12 = 6 * 2
....

See? Name a natural numbers that is not compatible with this? Pretty
Please?

No such number can be identified.

Because it works with any natural number!

Not with -e/2. Note that every number smaller than -e is finite. -e is the first infinite number.

Such numbers are dark. If dark numbers did not exist but all
natnumbers as fixed set could be named, then there would be a last one.

Huh? wow.

Because sets are fixed and the natnumbers are ordered.

Regards, WM

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 23.08.2025 20:41, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 23.08.2025 17:49, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

If there is a new guest, then there must be a new room. Try logic.

You try logic, for a change - the logic appropriate to infinite sets.
It is somewhat different from that appropriate to finite sets.

No, logic remains true in all connections.

Increase on the left-hand side of a bijection implies increase on the
right-hand side.

That's barely meaningful.

It is simplest truth.

And if it is meaningful, it's not true, as
the Hilbert's Hotel mechanism demonstrates.

Hilbert's hotel is nonsense.

According to Cantor rao is complete.

I think Cantor was too bright to say anything like that.

You are wrong. Obviously you are too lazy to read or understand what I
wrote on that topic. Cantor on "abgeschlossen und vollendet":

"Zu dem Gedanken, das Unendlichgro|fe [...] 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 vielj|nhriger wissenschaftlicher
Bem|+hungen und Versuche logisch gezwungen worden," [Cantor, p. 175]

"In den 'Grundlagen' formulire ich denselben Protest, indem ich an verschiedenen Stellen mich gegen die Verwechslung des
Uneigentlich-unendlichen (so nenne ich das ver|nnderliche Endliche) mit
dem Eigentlich-unendlichen (so nenne ich das bestimmte, das vollendete Unendliche, oder auch das Transfinite, |Lberendliche) ausspreche. Das Irrth|+mliche in jener Gauss'schen Stelle besteht darin, dass er sagt,
das Vollendetunendliche k||nne nicht Gegenstand mathematischer
Betrachtungen werden; dieser Irrthum h|nngt mit dem andern Irrthum
zusammen, dass er [...] das Vollendetunendliche mit dem Absoluten,
G||ttlichen identificirt, [...] Das Vollendetunendliche findet sich
allerdings in gewissem Sinne in den Zahlen NU+, NU+ + 1, ..., NU+NU+, ...; sie sind Zeichen f|+r gewisse Modi des Vollendetunendlichen und weil das Vollendetunendliche in verschiedenen, von einander mit der |nussersten Sch|nrfe durch den sogenannten 'endlichen, menschlichen Verstand' unterscheidbaren Modificationen auftreten kann, so sieht man hieraus
deutlich wie weit man vom Absoluten entfernt ist, obgleich man das Vollendetunendliche sehr wohl fassen und sogar mathematisch auffassen
kann." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]

"da nun jeder Typus auch im letzteren Falle etwas in sich Bestimmtes, vollendetes ist, so gilt ein gleiches von der zu ihm geh||rigen Zahl.
[...] 'Eigentlichunendlichem = Transfinitum = Vollendetunendlichem = Unendlichseiendem = kategorematice infinitum' [...] dieser Unterschied
|nndert aber nichts daran, da|f NU+ als ebenso bestimmt und vollendet anzusehen ist, wie Nau2," [G. Cantor, letter to K. La|fwitz (15 Feb 1884). Cantor, p. 395]

"Wir wollen nun zu einer genaueren Untersuchung der perfekten Mengen |+bergehen. Da jede solche Punktmenge gewisserma|fen in sich begrenzt, abgeschlossen und vollendet ist, so zeichnen sich die perfekten Mengen
vor allen anderen Gebilden durch besondere Eigenschaften aus." [Cantor,
p. 236]

You think that all this is not very bright. You are right.

"Completeness"
is not defined on sets.

It is assumed in ZF: Sets are invariable. Further complete means
vollst|nndig:

"Wenn zwei wohldefinierte Mannigfaltigkeiten M und N sich eindeutig und vollst|nndig, Element f|+r Element, einander zuordnen lassen (was, wenn es
auf eine Art m||glich ist, immer auch noch auf viele andere Weisen
geschehen kann), so m||ge f|+r das Folgende die Ausdrucksweise gestattet
sein, da|f diese Mannigfaltigkeiten gleiche M|nchtigkeit haben, oder auch, da|f sie |nquivalent sind." [Cantor, p. 119]

"gegenseitig eindeutige und vollst|nndige Korrespondenz" [Cantor, p. 238]

"Die s|nmtlichen Punkte l unsrer Menge L sind also in gegenseitig
eindeutige und vollst|nndige Beziehung zu s|nmtlichen Punkten f der Menge
F gebracht," [Cantor, p. 241]

"Zwei wohlgeordnete Mengen M und N heissen von gleichem Typus oder auch
von gleicher Anzahl, wenn sie sich gegenseitig eindeutig und vollst|nndig unter beidseitiger Wahrung der Rangfolge ihrer Elemente auf einander
beziehen, abbilden lassen;" [G. Cantor, letter to R. Lipschitz (19 Nov
1883)]

"Zwei bestimmte Mengen M und M1 nennen wir |nquivalent (in Zeichen: M ~
M1), wenn es m||glich ist, dieselben gesetzm|n|fig, gegenseitig eindeutig
und vollst|nndig, Element f|+r Element, einander zuzuordnen." [Cantor, p. 412]

"doch gibt es immer viele, im allgemeinen sogar unz|nhlig viele Zuordnungsgesetze, durch welche zwei |nquivalente Mengen in gegenseitig eindeutige und vollst|nndige Beziehung zueinander gebracht werden
k||nnen." [Cantor, p. 413]

"eine solche gegenseitig eindeutige und vollst|nndige Korrespondenz hergestellt [...] irgendeine gegenseitig eindeutige und vollst|nndige Zuordnung der beiden Mengen [...] auch eine gegenseitig eindeutige und vollst|nndige Korrespondenz" [Cantor, p. 415]

"Zwei n-fach geordnete Mengen M und N werden '|nhnlich' genannt, wenn es m||glich ist, sie gegenseitig eindeutig und vollst|nndig, Element f|+r Element, einander so zuzuordnen," [Cantor, p. 424]

If the rooms are rao and the guests are rao, then nothing can happen.
Cantor's infinity is voll-endet. That proofs voll and endet.

Not possible, given that complete ("vollendet") isn't defined on sets.

Your education on that topic was not complete but has large gaps. But
you start to understand that set theory is purest rubbish.

Regards, WM


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 23.08.2025 20:33, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 21.08.2025 23:58, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too and only >>>> half as many.

Both are infinite sets with the same cardinality - there is a trivial
bijection between them.

Bijections concern only definable numbers.

No, bijections concern sets.

It appears so, but that is wrong. Meanwhile even ChatGPT has recognized
this:

Lemma (Invariante): F|+r alle n gilt #(O)(n+1)=#(O)(n). Also
#(O)(n)=#(O)(1) f|+r alle n.
Beweis: Jeder |Lbergang erh|nlt die Anzahl der OrCOs. QED.
Korollar: Falls #(O)(1) -| 0, so gilt #(O)(n) -| 0 f|+r alle n. Kein
endlicher Schritt eliminiert alle O.
Satz: F|+r jede Aufz|nhlung durch verlustfreien Austausch gilt
"n |A N_{>=1}: #(O)(n)=#(O)(1) -| 0.
Folglich existiert kein Schritt, in dem alle Eintr|nge X sind.
Beweis: Unmittelbar. QED.
Schluss (Dunkle Positionen): Wird eine vollendete Abbildung behauptet,
in der alle Eintr|nge X sind,
so bleibt rCo da kein Schritt dies erreicht und #(O) invariant ist rCo nur
die Aufl||sung: Die OrCOs
persistieren, sind jedoch nicht lokalisierbar. Dies sind die dunklen Br|+che.

Not sure what you mean by "complete", though. Presumably that would be
"vollendet" in German. So your sets are both "vollendet" and
"unendlich".

The set rao can be used as the set of guests in Hilbert's hotel. Likewise
the set rao can be used as the set of rooms in Hilbert's hotel. Then no
new guest can arrive and no further room can be occupied.

There's something not quite consistent, there. My vote would be on
"complete" being nonsense here.

That is a good choice. But then infinity is only potential.

Infinite sets are defined, not merely potentially. An infinite set is
a set which has a bijection with a proper subset of itself.

That is nonsense. Dedekind applied potentially infinite sets.

In principle a complete set can only be proven complete by finding a
last element.

That's complete nonsense. The elements of a set are not ordered in any
way, so there can't be a last one, except in a singleton set.

The elements of rao can be ordered, and then the disaster happens.

I suspect that there is no mathematical definition of "complete set" and "incomplete set". The terms are just meaningless.

Sets are assumed to be complete in set theory. Set theorists are proud
on their ability to accept infinity and completeness as not
contradictory although it means complete and incomplete simultaneously.

Regards, WM

Regards, WM


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/24/2025 12:50 PM, WM wrote:

On 23.08.2025 20:40, Chris M. Thomasson wrote:

On 8/23/2025 4:45 AM, WM wrote:

On 22.08.2025 22:11, Chris M. Thomasson wrote:

On 8/22/2025 12:15 PM, Chris M. Thomasson wrote:

On 8/22/2025 12:07 PM, Chris M. Thomasson wrote:

On 8/21/2025 1:26 PM, WM wrote:

On 21.08.2025 22:22, FromTheRafters wrote:

Chris M. Thomasson formulated on Thursday :

On 8/21/2025 3:33 AM, WM wrote:

On 20.08.2025 22:56, Alan Mackenzie wrote:

If each number of an actually infinite set is doubled, then >>>>>>>>>> this results in a number which is larger than each number of >>>>>>>>>> the original set. The reason is that "all" means all and not >>>>>>>>>> only a potentially infinite collection. For the potentially >>>>>>>>>> infinite collection of all definable numbers this is wrong. >>>>>>>>>

1, 2, 3, ...

doubled:

2, 4, 6, ...

They are all natural numbers.

Both closed under their respective generating formulae.

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too
and only

half as many.

^^^^^^^^^^^^^^^^^

NO!!!

1, 2, 3, ...

doubled:

2, 4, 6, ...

Index them using naturals:

[1] = 2
[2] = 4
[3] = 6
...

Notice a pattern? Say:

[1] = 2 = 1 * 2
[2] = 4 = 2 * 2
[3] = 6 = 3 * 2
[4] = 8 = 4 * 2
[5] = 10 = 5 * 2
[6] = 12 = 6 * 2
....

See? Name a natural numbers that is not compatible with this? Pretty
Please?

No such number can be identified.

Because it works with any natural number!

Not with -e/2. Note that every number smaller than -e is finite. -e is the first infinite number.

-e is not a largest natural number...

Such numbers are dark. If dark numbers did not exist but all
natnumbers as fixed set could be named, then there would be a last one.

Huh? wow.

Because sets are fixed and the natnumbers are ordered.

Are you a moron * 2???


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

WM <wolfgang.mueckenheim@tha.de> wrote:

On 23.08.2025 20:41, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 23.08.2025 17:49, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

If there is a new guest, then there must be a new room. Try logic.

You try logic, for a change - the logic appropriate to infinite sets.
It is somewhat different from that appropriate to finite sets.

No, logic remains true in all connections.

Increase on the left-hand side of a bijection implies increase on the >>>>> right-hand side.

That's barely meaningful.

It is simplest truth.

And if it is meaningful, it's not true, as the Hilbert's Hotel
mechanism demonstrates.

Hilbert's hotel is nonsense.
According to Cantor rao is complete.

I think Cantor was too bright to say anything like that.

You are wrong. Obviously you are too lazy to read or understand what I
wrote on that topic. Cantor on "abgeschlossen und vollendet":

[ 83 lines of difficult German snipped. ]
This is an English language newsgroup. Massive swathes of foreign
language are not helpful here. That you failed to translate them into
English, paraphrasing them, suggests that you don't really understand
them yourself.
A set simply contains its members. It is unclear what you mean when you describe a set as "complete" or "incomplete". I don't think those two
terms are even defined for sets. Feel free to provide definitions and
show I am wrong.

If the rooms are rao and the guests are rao, then nothing can happen.
Cantor's infinity is voll-endet. That proofs voll and endet.

Not possible, given that complete ("vollendet") isn't defined on sets.

Your education on that topic was not complete but has large gaps. But
you start to understand that set theory is purest rubbish.

It is not set theory which is rubbish, it is your misunderstanding of
important parts of it which is rubbish.
So, are you going to tell us what you think "complete" means when applied
to a set, or not?

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

WM <wolfgang.mueckenheim@tha.de> wrote:

On 23.08.2025 20:33, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 21.08.2025 23:58, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

If 1, 2, 3, ... is complete, then 2, 4, 6, ... is complete too and only >>>>> half as many.

Both are infinite sets with the same cardinality - there is a trivial
bijection between them.

Bijections concern only definable numbers.

No, bijections concern sets.

It appears so, but that is wrong. Meanwhile even ChatGPT has recognized
this:

[ Incoherent ChatGPT output in German snipped. ]
Bijections concern sets.
[ .... ]

There's something not quite consistent, there. My vote would be on
"complete" being nonsense here.

That is a good choice. But then infinity is only potential.

Infinite sets are defined, not merely potentially. An infinite set is
a set which has a bijection with a proper subset of itself.

That is nonsense.

It's the definition of an infinite set.

Dedekind applied potentially infinite sets.

Dedekind was working with fluid notions early on in the development of
set theory, before it had become firmly established.
As I've said before, the notion "potentially infinite" isn't used in
modern mathematics. It just complicates and confuses (as can be seen in
this newsgroup) without shedding any light on anything.

In principle a complete set can only be proven complete by finding a
last element.

That's complete nonsense. The elements of a set are not ordered in any
way, so there can't be a last one, except in a singleton set.

The elements of rao can be ordered, ....

Z has structure over and above its structure as a set. But a set, as
such, is not ordered.

.... and then the disaster happens.

Huh?

I suspect that there is no mathematical definition of "complete set" and
"incomplete set". The terms are just meaningless.

Sets are assumed to be complete in set theory.

You might as well say that sets are assumed to be multi-coloured. That's equally meaningless.

Set theorists are proud of their ability to accept infinity and
completeness as not contradictory although it means complete and
incomplete simultaneously.

Set theorists work with infinite sets. I would be surprised indeed if
the concept of "completeness" w.r.t. sets ever occurs to them.

Regards, WM
Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 24.08.2025 22:20, Chris M. Thomasson wrote:

On 8/24/2025 12:50 PM, WM wrote:

Not with -e/2. Note that every number smaller than -e is finite. -e is
the first infinite number.

-e is not a largest natural number...

No, but it is the smallest unnatural number. Therefore all smaller
positive integers are naturals.

Regards, WM

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 25.08.2025 13:40, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 23.08.2025 20:41, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 23.08.2025 17:49, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

If there is a new guest, then there must be a new room. Try logic.

You try logic, for a change - the logic appropriate to infinite sets. >>>>> It is somewhat different from that appropriate to finite sets.

No, logic remains true in all connections.

Have you understood that?

I think Cantor was too bright to say anything like that.

You are wrong. Obviously you are too lazy to read or understand what I
wrote on that topic. Cantor on "abgeschlossen und vollendet":

[ 83 lines of difficult German snipped. ]

Not difficult. Use Google or any AI to translate it. By the way, what do
you in Germany without understanding German?

This is an English language newsgroup. Massive swathes of foreign
language are not helpful here.

It would be helpful for all newbies in set theory to first understand
Cantor. Then they could see that their teachers are wrong.

A set simply contains its members. It is unclear what you mean when you describe a set as "complete" or "incomplete".

Learn it from Cantor. I have translated it for you:

"To the idea to consider the infinite large not only in the form of the unlimited growing and the closely connected form of the convergent
infinite series, introduced first in the seventeenth century, but also
to fix it by numbers in the definite form of the completed-infinite I
have been forced logically almost against my own will, because in
opposition to highly esteemed tradition, by the development of many
years of scientific efforts and attempts, and therefore I do not believe
that reasons could be raised which I would not be able to answer."
[Cantor, p. 175]

"In spite of significant difference between the notions of the potential
and actual infinite, where the former is a variable finite magnitude,
growing above all limits, the latter a constant quantity fixed in itself
but beyond all finite magnitudes, it happens deplorably often that the
one is confused with the other." [Cantor, p. 374]

"By the actual infinite we have to understand a quantity that on the one
hand is not variable but fixed and definite in all its parts, a real
constant, but at the same time, on the other hand, exceeds every finite
size of the same kind by size. As an example I mention the totality, the embodiment of all finite positive integers; this set is a self-contained
thing and forms, apart from the natural sequence of its numbers, a
fixed, definite quantity, an aphorismenon, which we obviously have to
call larger than every finite number." [G. Cantor, letter to A.
Eulenburg (28 Feb 1886)]

I don't think those two
terms are even defined for sets.

Sets in set theory are what Cantor above described: Complete. Otherwise
there could not be a complete bijection.

If the rooms are rao and the guests are rao, then nothing can happen.
Cantor's infinity is voll-endet. That proofs voll and endet.

Not possible, given that complete ("vollendet") isn't defined on sets.

Your education on that topic was not complete but has large gaps. But
you start to understand that set theory is purest rubbish.

It is not set theory which is rubbish, it is your misunderstanding of important parts of it which is rubbish.

Together with your "complete isn't defined on sets" this is really
hilarious.

So, are you going to tell us what you think "complete" means when applied
to a set, or not?

In connection with Hilbert's hotel we need that only for the set rao. When
any element is subtracted or any other element is added, then the
resulting set is no longer rao.

Regards, WM

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 25.08.2025 um 14:00 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

An infinite set is a set which has a bijection with a proper subset of itself.

This definition is due to Dedekind.

See: https://en.wikipedia.org/wiki/Dedekind-infinite_set

That is nonsense.

It's the definition of an infinite set.

Dedekind applied potentially infinite sets.

Absolute nonsense. (See the definition you mentioned above.) It seems to
me that WM is completely demented now or lying.

He used to quote:

"Cantor's work was well received by some of the prominent mathematicians
of his day, such as Richard Dedekind. But his willingness to regard
infinite sets as objects to be treated in much the same way as finite
sets was bitterly attacked by others, particularly Kronecker. There was
no objection to a 'potential infinity' in the form of an unending
process, but an 'actual infinity' in the form of a completed infinite
set was harder to accept." [H.B. Enderton: "Elements of Set Theory",
Academic Press, New York (1977) p. 14f]

As I've said before, the notion "potentially infinite" isn't used in
modern mathematics.

And it's NOWHERE used by Dedekind (while on the other hand he considered infinite "Systeme" (called /sets/ these days).

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 25.08.2025 17:06, Moebius wrote:

Am 25.08.2025 um 14:00 schrieb Alan Mackenzie:

As I've said before, the notion "potentially infinite" isn't used in
modern mathematics.

And it's NOWHERE used by Dedekind#

The set of his thoughts, his prime example, is potentially infinite.

Regards, WM

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 25.08.2025 um 17:06 schrieb Moebius:

Am 25.08.2025 um 14:00 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

An infinite set is a set which has a bijection with a proper subset
of itself.

This definition is due to Dedekind.

See: https://en.wikipedia.org/wiki/Dedekind-infinite_set

That is nonsense.

It's the definition of an infinite set.

Dedekind applied potentially infinite sets.

Absolute nonsense. (See the definition you mentioned above.) It seems to
me that WM is completely demented now or lying.

A (translated) quote from Dedekind's famous book "Was sind und was
sollen die Zahen?" (1888):

_-o 5 The Finite and the Infinite._

64. Explanation 12). A set S is called infinite if it is similar to a
proper part of itself (32); in the opposite case, S is called a finite
set.

[...]

66. Theorem. There are infinite sets.

Proof 13). My world of thought, i.e., the totality S of all things that
can be the object of my thought, is infinite. For if s signifies an
element of S, then the thought s' that s can be the object of my thought
is itself an element of S. If one regards it as an image phi(s) of the
element s, then the mapping phi of S determined thereby has the property
that the image S' is part of S; namely, S' is a proper part of S because
there are elements in S (e.g., my own self) which are different from
every such thought s' and therefore not contained in S'. Finally,
it is clear that if a, b are different elements of S, their images a',
b' are also different, so that the mapping phi is a distinct [bijective]
one (26). Hence, S is infinite. qed

________________________________________________________________________

12 If one does not want to use the concept of similar sets (32), one
must say: S is called infinite if there exists a proper part of S (6)
into which S can be distinctly [bijectively] mapped (26, 36). In this form,
I communicated the definition of the infinite, which forms the core of
my entire investigation, to Mr. G. Cantor in September 1882 and several
years earlier also to Messrs. Schwarz and Weber. All other attempts
known to me to distinguish the infinite from the finite seem to me to
have been so unsuccessful that I believe I can dispense with a critique
of them.

13 A similar consideration can be found in -o 13 of Bolzano's Paradoxes
of the Infinite (Leipzig 1851).

Right:

"Cantor's work was well received by some of the prominent mathematicians
of his day, such as Richard Dedekind. But his willingness to regard
infinite sets as objects to be treated in much the same way as finite
sets was bitterly attacked by others, particularly Kronecker. There was
no objection to a 'potential infinity' in the form of an unending
process, but an 'actual infinity' in the form of a completed infinite
set was harder to accept." [H.B. Enderton: "Elements of Set Theory", Academic Press, New York (1977) p. 14f]

As I've said before, the notion "potentially infinite" isn't used in
modern mathematics.

And it's NOWHERE used by Dedekind (while on the other hand he considered infinite "Systeme" (called /sets/ these days)).

Only a crank like WM may believe (and claim) that Dedekind was concerned
with "the potentially infinite"; though Dedekind's "proof" of the
existence of "infinite sets" is rather unlucky.

"I communicated the definition of the infinite, which forms the core of
my entire investigation, to Mr. G. Cantor in September 1882 and ..."
no one (except M|+ckenheim) noticed that this definition defines
'potential infinity'? <HOLY SHIT!>

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

WM <wolfgang.mueckenheim@tha.de> wrote:

On 25.08.2025 13:40, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 23.08.2025 20:41, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 23.08.2025 17:49, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

If there is a new guest, then there must be a new room. Try logic. >>>>>> You try logic, for a change - the logic appropriate to infinite sets. >>>>>> It is somewhat different from that appropriate to finite sets.

No, logic remains true in all connections.

Have you understood that?

Yes. It's too banal to have any useful meaning.

I think Cantor was too bright to say anything like that.

You are wrong. Obviously you are too lazy to read or understand what I
wrote on that topic. Cantor on "abgeschlossen und vollendet":

[ 83 lines of difficult German snipped. ]

Not difficult. Use Google or any AI to translate it. By the way, what do
you in Germany without understanding German?

This is an English language newsgroup. Massive swathes of foreign
language are not helpful here.

It would be helpful for all newbies in set theory to first understand Cantor. Then they could see that their teachers are wrong.

You just can't answer a point, any point, can you?

A set simply contains its members. It is unclear what you mean when you
describe a set as "complete" or "incomplete".

Learn it from Cantor. I have translated it for you:
"To the idea to consider the infinite large not only in the form of the unlimited growing and the closely connected form of the convergent
infinite series, introduced first in the seventeenth century, but also
to fix it by numbers in the definite form of the completed-infinite I
have been forced logically almost against my own will, because in
opposition to highly esteemed tradition, by the development of many
years of scientific efforts and attempts, and therefore I do not believe that reasons could be raised which I would not be able to answer."
[Cantor, p. 175]
"In spite of significant difference between the notions of the potential
and actual infinite, where the former is a variable finite magnitude, growing above all limits, the latter a constant quantity fixed in itself
but beyond all finite magnitudes, it happens deplorably often that the
one is confused with the other." [Cantor, p. 374]
"By the actual infinite we have to understand a quantity that on the one hand is not variable but fixed and definite in all its parts, a real constant, but at the same time, on the other hand, exceeds every finite
size of the same kind by size. As an example I mention the totality, the embodiment of all finite positive integers; this set is a self-contained thing and forms, apart from the natural sequence of its numbers, a
fixed, definite quantity, an aphorismenon, which we obviously have to
call larger than every finite number." [G. Cantor, letter to A.
Eulenburg (28 Feb 1886)]

Three long, waffling paragraphs. There is no definition of "complete
set" there, just a discussion of the outmoded and superfluous notions of "potentially infinite" and "actually infinite".
The fact is, you don't know what a "complete set" is. You can't define
it. It's a meaningless combination of words.

I don't think those two terms are even defined for sets.

Sets in set theory are what Cantor above described: Complete. Otherwise
there could not be a complete bijection.

Meaningless word salad. I challenge you yet again: define "complete set"
and "incomplete set", in the same spirit I defined "infinite set" for you
a few posts ago. You can't.

If the rooms are rao and the guests are rao, then nothing can happen. >>>>> Cantor's infinity is voll-endet. That proofs voll and endet.

Not possible, given that complete ("vollendet") isn't defined on sets.

Your education on that topic was not complete but has large gaps. But
you start to understand that set theory is purest rubbish.

It is not set theory which is rubbish, it is your misunderstanding of
important parts of it which is rubbish.

Together with your "complete isn't defined on sets" this is really hilarious.

What's not hilarious is your teaching stuff you don't understand.

So, are you going to tell us what you think "complete" means when applied
to a set, or not?

In connection with Hilbert's hotel we need that only for the set rao. When any element is subtracted or any other element is added, then the
resulting set is no longer rao.

Vague rambling examples aren't definitions.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Moebius <invalid@example.invalid> wrote:

Am 25.08.2025 um 17:06 schrieb Moebius:

Am 25.08.2025 um 14:00 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

An infinite set is a set which has a bijection with a proper subset >>>>> of itself.

This definition is due to Dedekind.
See: https://en.wikipedia.org/wiki/Dedekind-infinite_set

That is nonsense.

It's the definition of an infinite set.

Dedekind applied potentially infinite sets.

Absolute nonsense. (See the definition you mentioned above.) It seems to
me that WM is completely demented now or lying.

A (translated) quote from Dedekind's famous book "Was sind und was
sollen die Zahen?" (1888):

For people who count on their toes (depending on how you fix the typo.
;-).

_-o 5 The Finite and the Infinite._
64. Explanation 12). A set S is called infinite if it is similar to a
proper part of itself (32); in the opposite case, S is called a finite
set.
[...]
66. Theorem. There are infinite sets.
Proof 13). My world of thought, i.e., the totality S of all things that
can be the object of my thought, is infinite. For if s signifies an
element of S, then the thought s' that s can be the object of my thought
is itself an element of S. If one regards it as an image phi(s) of the element s, then the mapping phi of S determined thereby has the property
that the image S' is part of S; namely, S' is a proper part of S because there are elements in S (e.g., my own self) which are different from
every such thought s' and therefore not contained in S'. Finally,
it is clear that if a, b are different elements of S, their images a',
b' are also different, so that the mapping phi is a distinct [bijective]
one (26). Hence, S is infinite. qed ________________________________________________________________________
12 If one does not want to use the concept of similar sets (32), one
must say: S is called infinite if there exists a proper part of S (6)
into which S can be distinctly [bijectively] mapped (26, 36). In this form,
I communicated the definition of the infinite, which forms the core of
my entire investigation, to Mr. G. Cantor in September 1882 and several
years earlier also to Messrs. Schwarz and Weber. All other attempts
known to me to distinguish the infinite from the finite seem to me to
have been so unsuccessful that I believe I can dispense with a critique
of them.
13 A similar consideration can be found in -o 13 of Bolzano's Paradoxes
of the Infinite (Leipzig 1851).
Right:

"Cantor's work was well received by some of the prominent mathematicians
of his day, such as Richard Dedekind. But his willingness to regard
infinite sets as objects to be treated in much the same way as finite
sets was bitterly attacked by others, particularly Kronecker. There was
no objection to a 'potential infinity' in the form of an unending
process, but an 'actual infinity' in the form of a completed infinite
set was harder to accept." [H.B. Enderton: "Elements of Set Theory",
Academic Press, New York (1977) p. 14f]

As I've said before, the notion "potentially infinite" isn't used in
modern mathematics.

And it's NOWHERE used by Dedekind (while on the other hand he considered
infinite "Systeme" (called /sets/ these days)).

Only a crank like WM may believe (and claim) that Dedekind was concerned with "the potentially infinite"; though Dedekind's "proof" of the
existence of "infinite sets" is rather unlucky.

Set theory could not have, and did not come into existence instantly as a coherent whole. Giants like Cantor and Dedekind were formulating
something new, and in doing so sometimes went up dead ends.

"I communicated the definition of the infinite, which forms the core of
my entire investigation, to Mr. G. Cantor in September 1882 and ..."
no one (except M|+ckenheim) noticed that this definition defines
'potential infinity'? <HOLY SHIT!>

WM's trouble is he's stuck in the 1880s. Set theory has since
chrystallized out into a coherent whole. Not everything Cantor and
Dedekind proposed has become part of that whole. WM really ought to
study more modern set theory.
--
Alan Mackenzie (Nuremberg, Germany).
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 25.08.2025 19:14, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 25.08.2025 13:40, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 23.08.2025 20:41, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

On 23.08.2025 17:49, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

If there is a new guest, then there must be a new room. Try logic.

You try logic, for a change - the logic appropriate to infinite sets. >>>>>>> It is somewhat different from that appropriate to finite sets.

No, logic remains true in all connections.

Have you understood that?

Yes. It's too banal to have any useful meaning.

Why did you claim the contrary?

I think Cantor was too bright to say anything like that.

You are wrong. Obviously you are too lazy to read or understand what I >>>> wrote on that topic. Cantor on "abgeschlossen und vollendet":

[ 83 lines of difficult German snipped. ]

Not difficult. Use Google or any AI to translate it. By the way, what do
you in Germany without understanding German?

This is an English language newsgroup. Massive swathes of foreign
language are not helpful here.

It would be helpful for all newbies in set theory to first understand
Cantor. Then they could see that their teachers are wrong.

You just can't answer a point, any point, can you?

Since you don't believe me, listen to Cantor.

A set simply contains its members. It is unclear what you mean when you >>> describe a set as "complete" or "incomplete".

Learn it from Cantor. I have translated it for you:

"To the idea to consider the infinite large not only in the form of the
unlimited growing and the closely connected form of the convergent
infinite series, introduced first in the seventeenth century, but also
to fix it by numbers in the definite form of the completed-infinite I
have been forced logically almost against my own will, because in
opposition to highly esteemed tradition, by the development of many
years of scientific efforts and attempts, and therefore I do not believe
that reasons could be raised which I would not be able to answer."
[Cantor, p. 175]

"In spite of significant difference between the notions of the potential
and actual infinite, where the former is a variable finite magnitude,
growing above all limits, the latter a constant quantity fixed in itself
but beyond all finite magnitudes, it happens deplorably often that the
one is confused with the other." [Cantor, p. 374]

"By the actual infinite we have to understand a quantity that on the one
hand is not variable but fixed and definite in all its parts, a real
constant, but at the same time, on the other hand, exceeds every finite
size of the same kind by size. As an example I mention the totality, the
embodiment of all finite positive integers; this set is a self-contained
thing and forms, apart from the natural sequence of its numbers, a
fixed, definite quantity, an aphorismenon, which we obviously have to
call larger than every finite number." [G. Cantor, letter to A.
Eulenburg (28 Feb 1886)]

Three long, waffling paragraphs. There is no definition of "complete
set" there, just a discussion of the outmoded and superfluous notions of "potentially infinite" and "actually infinite".

So you don't understand what Cantor tells you. How could you understand
me??? Better stay silent.

Regards, WM
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 25.08.2025 20:03, Alan Mackenzie wrote:

Moebius <invalid@example.invalid> wrote:

Am 25.08.2025 um 17:06 schrieb Moebius:

Am 25.08.2025 um 14:00 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

An infinite set is a set which has a bijection with a proper subset >>>>>> of itself.

This definition is due to Dedekind.

See: https://en.wikipedia.org/wiki/Dedekind-infinite_set

That is nonsense.

It's the definition of an infinite set.

Dedekind applied potentially infinite sets.

Absolute nonsense. (See the definition you mentioned above.) It seems to >>> me that WM is completely demented now or lying.

A (translated) quote from Dedekind's famous book "Was sind und was
sollen die Zahen?" (1888):

For people who count on their toes (depending on how you fix the typo.
;-).

_-o 5 The Finite and the Infinite._

64. Explanation 12). A set S is called infinite if it is similar to a
proper part of itself (32); in the opposite case, S is called a finite
set.

[...]

66. Theorem. There are infinite sets.

Proof 13). My world of thought, i.e., the totality S of all things that
can be the object of my thought, is infinite. For if s signifies an
element of S, then the thought s' that s can be the object of my thought
is itself an element of S. If one regards it as an image phi(s) of the
element s, then the mapping phi of S determined thereby has the property
that the image S' is part of S; namely, S' is a proper part of S because
there are elements in S (e.g., my own self) which are different from
every such thought s' and therefore not contained in S'. Finally,
it is clear that if a, b are different elements of S, their images a',
b' are also different, so that the mapping phi is a distinct [bijective]
one (26). Hence, S is infinite. qed

________________________________________________________________________

12 If one does not want to use the concept of similar sets (32), one
must say: S is called infinite if there exists a proper part of S (6)
into which S can be distinctly [bijectively] mapped (26, 36). In this form, >> I communicated the definition of the infinite, which forms the core of
my entire investigation, to Mr. G. Cantor in September 1882 and several
years earlier also to Messrs. Schwarz and Weber. All other attempts
known to me to distinguish the infinite from the finite seem to me to
have been so unsuccessful that I believe I can dispense with a critique
of them.

13 A similar consideration can be found in -o 13 of Bolzano's Paradoxes
of the Infinite (Leipzig 1851).

Right:

"Cantor's work was well received by some of the prominent mathematicians >>> of his day, such as Richard Dedekind. But his willingness to regard
infinite sets as objects to be treated in much the same way as finite
sets was bitterly attacked by others, particularly Kronecker. There was
no objection to a 'potential infinity' in the form of an unending
process, but an 'actual infinity' in the form of a completed infinite
set was harder to accept." [H.B. Enderton: "Elements of Set Theory",
Academic Press, New York (1977) p. 14f]

As I've said before, the notion "potentially infinite" isn't used in
modern mathematics.

And it's NOWHERE used by Dedekind (while on the other hand he considered >>> infinite "Systeme" (called /sets/ these days)).

Only a crank like WM may believe (and claim) that Dedekind was concerned
with "the potentially infinite"; though Dedekind's "proof" of the
existence of "infinite sets" is rather unlucky.

Set theory could not have, and did not come into existence instantly as a coherent whole. Giants like Cantor and Dedekind were formulating
something new, and in doing so sometimes went up dead ends.

"I communicated the definition of the infinite, which forms the core of
my entire investigation, to Mr. G. Cantor in September 1882 and ..."
no one (except M|+ckenheim) noticed that this definition defines
'potential infinity'? <HOLY SHIT!>

Thoughts are never complete.

WM's trouble is he's stuck in the 1880s. Set theory has since
chrystallized out into a coherent whole. Not everything Cantor and
Dedekind proposed has become part of that whole. WM really ought to
study more modern set theory.

Try to understand this (by the way you can learn German from it): https://www.academia.edu/91188101/Proof_of_the_existence_of_dark_numbers_bilingual_version_

Regards, WM


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 25.08.2025 um 19:01 schrieb Moebius:

Am 25.08.2025 um 17:06 schrieb Moebius:

Am 25.08.2025 um 14:00 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

An infinite set is a set which has a bijection with a proper subset >>>>> of itself.

This definition is due to Dedekind.

See: https://en.wikipedia.org/wiki/Dedekind-infinite_set

That is nonsense.

It's well known that FOR *crank* W. M|+ckenheim infinite sets "are
nonsenes". :-)

It's the definition of an infinite set.

Indeed!

Dedekind applied potentially infinite sets.

<facepalm>

Absolute nonsense. (See the definition you mentioned above.) It seems
to me that WM is completely demented now or lying.

A (translated) quote from Dedekind's famous book "Was sind und was
sollen die Zahen?" (1888):

_-o 5 The Finite and the Infinite._

64. Explanation 12). A set S is called infinite if it is similar to a
proper part of itself (32); in the opposite case, S is called a finite
set.

[...]

66. Theorem. There are infinite sets.

Proof 13). My world of thought, i.e., the totality S of all things that
can be the object of my thought, is infinite. For if s signifies an
element of S, then the thought s' that s can be the object of my thought
is itself an element of S. If one regards it as an image phi(s) of the element s, then the mapping phi of S determined thereby has the property
that the image S' is part of S; namely, S' is a proper part of S because there are elements in S (e.g., my own self) which are different from
every such thought s' and therefore not contained in S'. Finally,
it is clear that if a, b are different elements of S, their images a',
b' are also different, so that the mapping phi is a distinct [bijective]
one (26). Hence, S is infinite. qed

________________________________________________________________________

12 If one does not want to use the concept of similar sets (32), one
must say: S is called infinite if there exists a proper part of S (6)
into which S can be distinctly [bijectively] mapped (26, 36). In this form,
I communicated the definition of the infinite, which forms the core of
my entire investigation, to Mr. G. Cantor in September 1882 and several
years earlier also to Messrs. Schwarz and Weber. All other attempts
known to me to distinguish the infinite from the finite seem to me to
have been so unsuccessful that I believe I can dispense with a critique
of them.

13 A similar consideration can be found in -o 13 of Bolzano's Paradoxes
of the Infinite (Leipzig 1851).

-o 13

If we have now come to agreement on which concept we shall associate with
the word rCyinfiniterCO and if we have also made clear the components from which we
compose this concept, then the next question is whether it also has objectivity,
i.e. whether there are also things to which it can be applied,
multitudes, which we
may call infinite in the sense defined? And I venture to affirm this categorically. In
the realm of those things which make no claim to reality but only to possibility, there
are indisputably multitudes which are infinite. The multitude of
propositions and
truths in themselves is, as may very easily be seen, infinite. For if we consider some
truth, perhaps the proposition that there are actually truths, or
otherwise any
arbitrary truth, which I shall designate by A, then we find that the proposition
that the words rCyA is truerCO express is different from A itself, for the latter obviously
has a completely different subject from the former. Namely, its subject
is the
whole proposition A itself. However, by the rule by which we derived
from the
proposition A, this different one, which I shall call B, we can again
derive from B
a third proposition C, and continue in this way without end. The
collection of
all these propositions in which each successive one stands in the
relationship just
given to the one immediately before it, in that it makes it its subject
and states
of it that it is a true proposition, this collectionrCoI sayrCocomprises a multitude
of parts (propositions) which are greater than every finite multitude.
For without
my reminder the reader may notice the similarity between the series of these propositions formed by the rule just given, and the series of numbers considered
in -o8. This is a similarity consisting in this, that to every term of
the latter there is
a term of the former corresponding to it, that therefore for every
number, however
large, there is also a number of distinct propositions equal to it, and
that we can
always form new propositions, or to say it better, that there are such propositions
in themselves regardless of whether we form them or not. Whence it
follows that
the collection of all these propositions has a plurality which is
greater than every
number, i.e. is infinite.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Well put, Bolzano. Bolzano clearly was a much better PHILOSOPHER than
Dedekind (well, actually, he was one). [He coined the word "Menge" by
the way.]

Ultimately, Zermelo made that /idea/ completely rigorous (in the context
of set theory) by just stating it as an axiom in set theoretic terms only:

ES({} e S & Ax(x e S -> {x} e S)).

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 25.08.2025 um 19:01 schrieb Moebius:

Am 25.08.2025 um 17:06 schrieb Moebius:

Am 25.08.2025 um 14:00 schrieb Alan Mackenzie:

WM <wolfgang.mueckenheim@tha.de> wrote:

An infinite set is a set which has a bijection with a proper subset >>>>> of itself.

This definition is due to Dedekind.

See: https://en.wikipedia.org/wiki/Dedekind-infinite_set

That is nonsense.

It's well known that FOR *crank* W. M|+ckenheim infinite sets "are
nonsense". :-)

It's the definition of an infinite set.

Indeed!

Dedekind applied potentially infinite sets.

<facepalm>

Absolute nonsense. (See the definition you mentioned above.) It seems
to me that WM is completely demented now or lying.

A (translated) quote from Dedekind's famous book "Was sind und was
sollen die Zahlen?" (1888):

_-o 5 The Finite and the Infinite._

64. Explanation 12). A set S is called infinite if it is similar to a
proper part of itself (32); in the opposite case, S is called a finite
set.

[...]

66. Theorem. There are infinite sets.

Proof 13). My world of thought, i.e., the totality S of all things that
can be the object of my thought, is infinite. For if s signifies an
element of S, then the thought s' that s can be the object of my thought
is itself an element of S. If one regards it as an image phi(s) of the element s, then the mapping phi of S determined thereby has the property
that the image S' is part of S; namely, S' is a proper part of S because there are elements in S (e.g., my own self) which are different from
every such thought s' and therefore not contained in S'. Finally,
it is clear that if a, b are different elements of S, their images a',
b' are also different, so that the mapping phi is a distinct [bijective]
one (26). Hence, S is infinite. qed

________________________________________________________________________

12 If one does not want to use the concept of similar sets (32), one
must say: S is called infinite if there exists a proper part of S (6)
into which S can be distinctly [bijectively] mapped (26, 36). In this form,
I communicated the definition of the infinite, which forms the core of
my entire investigation, to Mr. G. Cantor in September 1882 and several
years earlier also to Messrs. Schwarz and Weber. All other attempts
known to me to distinguish the infinite from the finite seem to me to
have been so unsuccessful that I believe I can dispense with a critique
of them.

13 A similar consideration can be found in -o 13 of Bolzano's Paradoxes
of the Infinite (Leipzig 1851).

-o 13

If we have now come to agreement on which concept we shall associate with
the word rCyinfiniterCO and if we have also made clear the components from which we
compose this concept, then the next question is whether it also has objectivity,
i.e. whether there are also things to which it can be applied,
multitudes, which we
may call infinite in the sense defined? And I venture to affirm this categorically. In
the realm of those things which make no claim to reality but only to possibility, there
are indisputably multitudes which are infinite. The multitude of
propositions and
truths in themselves is, as may very easily be seen, infinite. For if we consider some
truth, perhaps the proposition that there are actually truths, or
otherwise any
arbitrary truth, which I shall designate by A, then we find that the proposition
that the words rCyA is truerCO express is different from A itself, for the latter obviously
has a completely different subject from the former. Namely, its subject
is the
whole proposition A itself. However, by the rule by which we derived
from the
proposition A, this different one, which I shall call B, we can again
derive from B
a third proposition C, and continue in this way without end. The
collection of
all these propositions in which each successive one stands in the
relationship just
given to the one immediately before it, in that it makes it its subject
and states
of it that it is a true proposition, this collectionrCoI sayrCocomprises a multitude
of parts (propositions) which are greater than every finite multitude.
For without
my reminder the reader may notice the similarity between the series of these propositions formed by the rule just given, and the series of numbers considered
in -o8. This is a similarity consisting in this, that to every term of
the latter there is
a term of the former corresponding to it, that therefore for every
number, however
large, there is also a number of distinct propositions equal to it, and
that we can
always form new propositions, or to say it better, that there are such propositions
in themselves regardless of whether we form them or not. Whence it
follows that
the collection of all these propositions has a plurality which is
greater than every
number, i.e. is infinite.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Well put, Bolzano. Bolzano clearly was a much better PHILOSOPHER than
Dedekind (well, actually, he w a s one). [He coined the word "Menge"
by the way.]

Ultimately, Zermelo made that /idea/ completely rigorous (in the context
of set theory) by just stating it as an axiom in set theoretic terms only:

ES({} e S & Ax(x e S -> {x} e S)).

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Je suis tellement pour lrCOinfini actuel, qurCOau lieu drCOadmettre, que la nature lrCOabhorre, comme lrCOon dit vulgairement, je tiens qurCOelle lrCOaffecte
par-tout, pour mieux marquer les perfections de son Auteur.

rCo Leibniz, Opera omnia studio Ludov. Dutens., Tom. II, part x, p. 243

[I stand for actual infinity so much that instead of admitting that
Nature abhors
it, as it is commonly said, I hold that [Nature] assumes it everywhere,
in order to
signal better the perfections of its Author.]

Well...

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 25.08.2025 um 20:03 schrieb Alan Mackenzie:

Set theory could not have, and did not come into existence instantly as a coherent whole. Giants like Cantor and Dedekind were formulating
something new [...]

David Hilbert ("|Lber das Unendliche", 1926): "So wurde schlie|flich durch
die gigantische Zusammenarbeit von Frege, Dedekind, Cantor das
Unendliche auf den Thron gehoben und geno|f die Zeit des h||chsten
Triumphes. Das Unendliche war in k|+hnstem Fluge auf eine schwindelnde
H||he des Erfolges gelangt."

["Thus, through the gigantic collaboration of Frege, Dedekind, and
Cantor, the infinite was finally raised to the throne and enjoyed a time
of supreme triumph. The infinite had reached a dizzying height of
success in the boldest flight."]

Yeah, than the paradoxes emerged... :-P

Russell and Zermelo showed (around 1908) ways to to avoid that problems
[...]

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 25.08.2025 um 22:50 schrieb Moebius:

Am 25.08.2025 um 20:03 schrieb Alan Mackenzie:

Set theory could not have, and did not come into existence instantly as a
coherent whole.-a Giants like Cantor and Dedekind were formulating
something new [...]

David Hilbert ("|Lber das Unendliche", 1926): "So wurde schlie|flich durch die gigantische Zusammenarbeit von Frege, Dedekind, Cantor das
Unendliche auf den Thron gehoben und geno|f die Zeit des h||chsten Triumphes. Das Unendliche war in k|+hnstem Fluge auf eine schwindelnde
H||he des Erfolges gelangt."

["Thus, through the gigantic collaboration of Frege, Dedekind, and
Cantor, the infinite was finally raised to the throne and enjoyed a time
of supreme triumph. The infinite had reached a dizzying height of
success in the boldest flight."]

Yeah, then the paradoxes emerged... :-P

Russell and Zermelo showed (around 1908) ways to to avoid that problems. [...]

Actually, I like Quine's /New Foundations/ (NF) VERY MUCH.

It just consists of 2 (!) axioms, the first of which exists in ALL axiom systems for "set theory" (I know of):

- Extensionality

and

- a restricted axiom schema of comprehension.

In my opinion ... NO MODERN AXIOM SYSTEM is closer to Frege's original conception. (After all, UNRESTRICTED comprehension leads to an
inconsistency, as is well known. Hence "comprehension" MUST BE
restricted in some way or other.)

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 25.08.2025 um 22:50 schrieb Moebius:

Am 25.08.2025 um 20:03 schrieb Alan Mackenzie:

Set theory could not have, and did not come into existence instantly as a
coherent whole.-a Giants like Cantor and Dedekind were formulating
something new [...]

David Hilbert ("|Lber das Unendliche", 1926): "So wurde schlie|flich durch die gigantische Zusammenarbeit von Frege, Dedekind, Cantor das
Unendliche auf den Thron gehoben und geno|f die Zeit des h||chsten Triumphes. Das Unendliche war in k|+hnstem Fluge auf eine schwindelnde
H||he des Erfolges gelangt."

["Thus, through the gigantic collaboration of Frege, Dedekind, and
Cantor, the infinite was finally raised to the throne and enjoyed a time
of supreme triumph. The infinite had reached a dizzying height of
success in the boldest flight."]

Yeah, then the paradoxes emerged... :-P

Russell and Zermelo showed (around 1908) ways to to avoid that problems. [...]

Actually, I like Quine's /New Foundations/ (NF) VERY MUCH.

It just consists of 2 (!) axioms [or rather an axiom and an axiom
schema], the first of which exists in ALL axiom systems for "set theory"
(I know of):

- Extensionality

and

- a restricted axiom schema of comprehension.

In my opinion ... NO MODERN AXIOM SYSTEM is closer to Frege's original conception. (After all, UNRESTRICTED comprehension leads to an
inconsistency, as is well known. Hence "comprehension" MUST BE
restricted in some way or other.)

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 25.08.2025 um 17:06 schrieb Moebius:

| Dedekind applied potentially infinite sets. [M|+ckenidiot]

Hint: "Schon die beiden um die Grundlagen der Mathematik hochverdienten Mathematiker Frege und Dedekind haben rCo unabh|nngig voneinander rCo das aktual Unendliche angewandt und zwar zu dem Zwecke, die Arithmetik
unabh|nngig von aller Anschauung und Erfahrung auf reine Logik zu
begr|+nden und durch diese allein zu deduzieren. Dedekinds Bestreben ging sogar soweit, die endliche Anzahl nicht der Anschauung zu entnehmen,
sondern unter wesentlicher Benutzung des Begriffes der unendlichen
Mengen rein logisch abzuleiten."

[The two mathematicians Frege and Dedekind, who made significant
contributions to the foundations of mathematics, had
alreadyrCoindependently of each otherrCoapplied the actual infinite for the purpose of grounding arithmetic in pure logic, independent of all
intuition and experience, and deducing it solely through it. Dedekind's endeavor even went so far as to derive finite number not from intuition,
but rather to derive it purely logically, essentially using the concept
of infinite sets.]

(D. Hilbert, |Lber das Unendliche. 1926)

Hint: "... applied the actual infinite ...".

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

M|+ckenheim: "Dedekind applied potentially infinite sets"

Hilbert: "Dedekind ... applied the actual infinite".

Now, whom should I trust? A mathematical crank which has not even
published ONE (peer reviewd) mathematical paper, or one of the greatest mathematicians of al times? Well ... hard. to decide.

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Am 25.08.2025 um 23:21 schrieb Moebius:

Am 25.08.2025 um 17:06 schrieb Moebius:

| Dedekind applied potentially infinite sets. [M|+ckenidiot]

Hint: "Schon die beiden um die Grundlagen der Mathematik hochverdienten Mathematiker Frege und Dedekind haben rCo unabh|nngig voneinander rCo das aktual Unendliche angewandt und zwar zu dem Zwecke, die Arithmetik unabh|nngig von aller Anschauung und Erfahrung auf reine Logik zu
begr|+nden und durch diese allein zu deduzieren. Dedekinds Bestreben ging sogar soweit, die endliche Anzahl nicht der Anschauung zu entnehmen,
sondern unter wesentlicher Benutzung des Begriffes der unendlichen
Mengen rein logisch abzuleiten."

[The two mathematicians Frege and Dedekind, who made significant contributions to the foundations of mathematics, had alreadyrCo independently of each otherrCoapplied the actual infinite for the purpose
of grounding arithmetic in pure logic, independent of all intuition and experience, and deducing it solely through it. Dedekind's endeavor even
went so far as to derive finite number not from intuition, but rather to derive it purely logically, essentially using the concept of infinite
sets.]

(D. Hilbert, |Lber das Unendliche. 1926)

Hint: "... applied the actual infinite ...".

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

M|+ckenheim: "Dedekind applied potentially infinite sets"

Hilbert: "Dedekind ... applied the actual infinite".

Now, whom should I trust? A mathematical crank which has not even
published ONE (peer reviewed) mathematical paper, or one of the greatest mathematicians of al times? Well ... hard to decide.

Especially after reading a contemporary textbook converning set theory:

"Cantor's work was well received by some of the prominent mathematicians
of his day, such as Richard Dedekind. But his willingness to regard
infinite sets as objects to be treated in much the same way as finite
sets was bitterly attacked by others, particularly Kronecker. There was
no objection to a 'potential infinity' in the form of an unending
process, but an 'actual infinity' in the form of a completed infinite
set was harder to accept." [H.B. Enderton: "Elements of Set Theory",
Academic Press, New York (1977) p. 14f]

.
.
.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/25/2025 5:29 AM, WM wrote:

On 24.08.2025 22:20, Chris M. Thomasson wrote:

On 8/24/2025 12:50 PM, WM wrote:

Not with -e/2. Note that every number smaller than -e is finite. -e is
the first infinite number.

-e is not a largest natural number...

No, but it is the smallest unnatural number. Therefore all smaller
positive integers are naturals.

There is an infinite number of them. You think that there is less then
wrt doubled:

1, 2, 3, ...

2, 3, 6, ...

Why? Are you mad?


--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/25/2025 8:14 AM, WM wrote:

On 25.08.2025 17:06, Moebius wrote:

Am 25.08.2025 um 14:00 schrieb Alan Mackenzie:

As I've said before, the notion "potentially infinite" isn't used in
modern mathematics.

And it's NOWHERE used by Dedekind#

The set of his thoughts, his prime example, is potentially infinite.

oh my. Do you consider the natural numbers to be potentially infinite?
Sigh. They are infinite...
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 25.08.2025 23:56, Chris M. Thomasson wrote:

On 8/25/2025 8:14 AM, WM wrote:

On 25.08.2025 17:06, Moebius wrote:

Am 25.08.2025 um 14:00 schrieb Alan Mackenzie:

As I've said before, the notion "potentially infinite" isn't used in
modern mathematics.

And it's NOWHERE used by Dedekind#

The set of his thoughts, his prime example, is potentially infinite.

oh my. Do you consider the natural numbers to be potentially infinite?

Those which Dedekind or you can think of as individuals disinguished
from all others are potentially infinite.

Regards, WM

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/26/2025 6:55 AM, WM wrote:

On 25.08.2025 23:56, Chris M. Thomasson wrote:

On 8/25/2025 8:14 AM, WM wrote:

On 25.08.2025 17:06, Moebius wrote:

Am 25.08.2025 um 14:00 schrieb Alan Mackenzie:

As I've said before, the notion "potentially infinite" isn't used in >>>>> modern mathematics.

And it's NOWHERE used by Dedekind#

The set of his thoughts, his prime example, is potentially infinite.

oh my. Do you consider the natural numbers to be potentially infinite?

Those which Dedekind or you can think of as individuals disinguished
from all others are potentially infinite.

When does the "actual infinity" of natural numbers become only
"potentially infinite"?
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

Chris M. Thomasson explained on 8/26/2025 :

On 8/26/2025 6:55 AM, WM wrote:

On 25.08.2025 23:56, Chris M. Thomasson wrote:

On 8/25/2025 8:14 AM, WM wrote:

On 25.08.2025 17:06, Moebius wrote:

Am 25.08.2025 um 14:00 schrieb Alan Mackenzie:

As I've said before, the notion "potentially infinite" isn't used in >>>>>> modern mathematics.

And it's NOWHERE used by Dedekind#

The set of his thoughts, his prime example, is potentially infinite.

oh my. Do you consider the natural numbers to be potentially infinite?

Those which Dedekind or you can think of as individuals disinguished from >> all others are potentially infinite.

When does the "actual infinity" of natural numbers become only "potentially infinite"?

A long time ago, but we got over it.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 26.08.2025 23:34, Chris M. Thomasson wrote:

On 8/26/2025 6:55 AM, WM wrote:

oh my. Do you consider the natural numbers to be potentially infinite?

Those which Dedekind or you can think of as individuals disinguished
from all others are potentially infinite.

When does the "actual infinity" of natural numbers become only
"potentially infinite"?

Never. The numbers you can think of are a potentially infinite
collection: You can always think of a greater number but the thought
numbers are never complete. Actual infinity according to Cantor is complete.

Regards, WM
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/27/2025 6:08 AM, WM wrote:

On 26.08.2025 23:34, Chris M. Thomasson wrote:

On 8/26/2025 6:55 AM, WM wrote:

oh my. Do you consider the natural numbers to be potentially infinite?

Those which Dedekind or you can think of as individuals disinguished
from all others are potentially infinite.

When does the "actual infinity" of natural numbers become only
"potentially infinite"?

Never. The numbers you can think of are a potentially infinite
collection: You can always think of a greater number but the thought
numbers are never complete. Actual infinity according to Cantor is
complete.

So, you must be along the lines of:

Entity A, thinks of 1, 2, 3

The actual infinity is 1, 2, 3, 4, ...

Entity A is potentially infinite? Is that it? Thanks.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/26/2025 4:58 PM, FromTheRafters wrote:

Chris M. Thomasson explained on 8/26/2025 :

On 8/26/2025 6:55 AM, WM wrote:

On 25.08.2025 23:56, Chris M. Thomasson wrote:

On 8/25/2025 8:14 AM, WM wrote:

On 25.08.2025 17:06, Moebius wrote:

Am 25.08.2025 um 14:00 schrieb Alan Mackenzie:

As I've said before, the notion "potentially infinite" isn't used in >>>>>>> modern mathematics.

And it's NOWHERE used by Dedekind#

The set of his thoughts, his prime example, is potentially infinite.

oh my. Do you consider the natural numbers to be potentially infinite?

Those which Dedekind or you can think of as individuals disinguished
from all others are potentially infinite.

When does the "actual infinity" of natural numbers become only
"potentially infinite"?

A long time ago, but we got over it.

;^) ROFL! :^)
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 28.08.2025 01:33, Chris M. Thomasson wrote:

On 8/27/2025 6:08 AM, WM wrote:

On 26.08.2025 23:34, Chris M. Thomasson wrote:

On 8/26/2025 6:55 AM, WM wrote:

oh my. Do you consider the natural numbers to be potentially infinite? >>>>

Those which Dedekind or you can think of as individuals distinguished >>>> from all others are potentially infinite.

When does the "actual infinity" of natural numbers become only
"potentially infinite"?

Never. The numbers you can think of are a potentially infinite
collection: You can always think of a greater number but the thought
numbers are never complete. Actual infinity according to Cantor is
complete.

So, you must be along the lines of:

Entity A, thinks of 1, 2, 3

The actual infinity is 1, 2, 3, 4, ...

Entity A is potentially infinite? Is that it? Thanks.

1, 2, 3, is finite. If entity A can always think of the next number,
then its set is potentially infinite.

Regards, WM

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/28/2025 12:31 AM, WM wrote:

On 28.08.2025 01:33, Chris M. Thomasson wrote:

On 8/27/2025 6:08 AM, WM wrote:

On 26.08.2025 23:34, Chris M. Thomasson wrote:

On 8/26/2025 6:55 AM, WM wrote:

oh my. Do you consider the natural numbers to be potentially
infinite?

Those which Dedekind or you can think of as individuals
distinguished from all others are potentially infinite.

When does the "actual infinity" of natural numbers become only
"potentially infinite"?

Never. The numbers you can think of are a potentially infinite
collection: You can always think of a greater number but the thought
numbers are never complete. Actual infinity according to Cantor is
complete.

So, you must be along the lines of:

Entity A, thinks of 1, 2, 3

The actual infinity is 1, 2, 3, 4, ...

Entity A is potentially infinite? Is that it? Thanks.

1, 2, 3, is finite. If entity A can always think of the next number,
then its set is potentially infinite.

The natural numbers are actually infinite... Can't you see that? ;^)

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 28.08.2025 21:02, Chris M. Thomasson wrote:

On 8/28/2025 12:31 AM, WM wrote:

On 28.08.2025 01:33, Chris M. Thomasson wrote:

On 8/27/2025 6:08 AM, WM wrote:

On 26.08.2025 23:34, Chris M. Thomasson wrote:

On 8/26/2025 6:55 AM, WM wrote:

oh my. Do you consider the natural numbers to be potentially
infinite?

Those which Dedekind or you can think of as individuals
distinguished from all others are potentially infinite.

When does the "actual infinity" of natural numbers become only
"potentially infinite"?

Never. The numbers you can think of are a potentially infinite
collection: You can always think of a greater number but the thought
numbers are never complete. Actual infinity according to Cantor is
complete.

So, you must be along the lines of:

Entity A, thinks of 1, 2, 3

The actual infinity is 1, 2, 3, 4, ...

Entity A is potentially infinite? Is that it? Thanks.

1, 2, 3, is finite. If entity A can always think of the next number,
then its set is potentially infinite.

The natural numbers are actually infinite... Can't you see that? ;^)

The known prime numbers are not actually infinite. Can you understand that?

Regards, WM
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

It happens that WM formulated :

On 28.08.2025 21:02, Chris M. Thomasson wrote:

On 8/28/2025 12:31 AM, WM wrote:

On 28.08.2025 01:33, Chris M. Thomasson wrote:

On 8/27/2025 6:08 AM, WM wrote:

On 26.08.2025 23:34, Chris M. Thomasson wrote:

On 8/26/2025 6:55 AM, WM wrote:

oh my. Do you consider the natural numbers to be potentially
infinite?

Those which Dedekind or you can think of as individuals distinguished >>>>>>> from all others are potentially infinite.

When does the "actual infinity" of natural numbers become only
"potentially infinite"?

Never. The numbers you can think of are a potentially infinite
collection: You can always think of a greater number but the thought >>>>> numbers are never complete. Actual infinity according to Cantor is
complete.

So, you must be along the lines of:

Entity A, thinks of 1, 2, 3

The actual infinity is 1, 2, 3, 4, ...

Entity A is potentially infinite? Is that it? Thanks.

1, 2, 3, is finite. If entity A can always think of the next number, then >>> its set is potentially infinite.

The natural numbers are actually infinite... Can't you see that? ;^)

The known prime numbers are not actually infinite. Can you understand that?

The known prime numbers don't form a ZFC set. ZFC sets contain well
defined objects. If you pick an object, it should be clear whether or
not that object should be included in the ZFC set. Undefined or
undefinable objects are excluded from being in a ZFC set. All prime
numbers less than some number do form a ZFC set even if Francis the
Talking Mule cannot count that high.

Can you understand that?

--
I didn't think so.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/29/2025 1:47 AM, WM wrote:

On 28.08.2025 21:02, Chris M. Thomasson wrote:

On 8/28/2025 12:31 AM, WM wrote:

On 28.08.2025 01:33, Chris M. Thomasson wrote:

On 8/27/2025 6:08 AM, WM wrote:

On 26.08.2025 23:34, Chris M. Thomasson wrote:

On 8/26/2025 6:55 AM, WM wrote:

oh my. Do you consider the natural numbers to be potentially
infinite?

Those which Dedekind or you can think of as individuals
distinguished from all others are potentially infinite.

When does the "actual infinity" of natural numbers become only
"potentially infinite"?

Never. The numbers you can think of are a potentially infinite
collection: You can always think of a greater number but the
thought numbers are never complete. Actual infinity according to
Cantor is complete.

So, you must be along the lines of:

Entity A, thinks of 1, 2, 3

The actual infinity is 1, 2, 3, 4, ...

Entity A is potentially infinite? Is that it? Thanks.

1, 2, 3, is finite. If entity A can always think of the next number,
then its set is potentially infinite.

The natural numbers are actually infinite... Can't you see that? ;^)

The known prime numbers are not actually infinite. Can you understand that?

Some say that in the infinity of naturals there are infinite primes... ?

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 29.08.2025 18:12, FromTheRafters wrote:

It happens that WM formulated :

The known prime numbers are not actually infinite. Can you understand
that?

The known prime numbers don't form a ZFC set. ZFC sets contain well
defined objects.

So do potentially infinite collections too.

If you pick an object, it should be clear whether or
not that object should be included in the ZFC set. Undefined or
undefinable objects are excluded from being in a ZFC set.

Not at all! The natural numbers are a ZF-set. Most of them are undefined
as individuals and remain so forever.

All prime
numbers less than some number do form a ZFC set

Yes, every definable prime number is the greatest element of a finite ser.

Can you understand that?

That is the basis of my discovery of dark numbers.

Regards, WM



--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 29.08.2025 20:58, Chris M. Thomasson wrote:

Some say that in the infinity of naturals there are infinite primes... ?

They are wrong. There are only finite primes - infinitely many though.

Regards, WM
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: sci.math

On 8/30/2025 6:59 AM, WM wrote:

On 29.08.2025 20:58, Chris M. Thomasson wrote:

Some say that in the infinity of naturals there are infinite primes... ?

They are wrong. There are only finite primes - infinitely many though.

I should have wrote that there are infinitely many primes in the sea of infinitely many natural numbers. Also, the number of digits in the
primes keeps getting larger... Any better? :^)
--- Synchronet 3.21a-Linux NewsLink 1.2