Re: Simple enough for every reader?

On 29.06.2025 12:25, Mikko wrote:

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

On 28.06.2025 11:56, Mikko wrote:

On 2025-06-27 19:36:41 +0000, WM said:
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On 27.06.2025 09:33, Mikko wrote:

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

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

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

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Then it is not finished or completed.

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No, but it can be continued.

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That is potential infinity. But Cantor claimed complete enumeration.

 There is no mathematical definiton of "complete enumeration"

The definition of bijection requires completeness.

so it is
possible that Cantor's enumeartion is "complete" is one sense and
"incomplete" in another.

No. Cantor claims all, every and complete:
"The infinite sequence thus defined has the peculiar property to contain the positive rational numbers completely, and each of them only once at a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]
"thus we get the epitome (ω) of all real algebraic numbers [...] and with respect to this order we can talk about the th algebraic number where not a single one of this epitome () has been forgotten." [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 116]
"such that every element of the set stands at a definite position of this sequence"

 

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

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

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

All are removed when all are removed.

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

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

This cannot be accomplished

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

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

 

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

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    1.1 Cantor's original German terminology on infinite sets
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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 "vollständig" 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. – Unfortunately it has turned out impossible to strictly separate Cantor's mathematical and religious arguments.)

 There is nothing religious in Cantor's arguments. The only traces of
his religious motivations are in the choice of his symbols, in paricular
aleph and omega.

You are wrong. Here are only few pages of my Book Transfinity:
4.1 Cantor on theology
"it is clear that the theological considerations by which Cantor motivated his notion of the actual infinite, were metaphysical in nature." [A. Heyting: "Technique versus metaphysics in the calculus", in Imre Lakatos (ed.): "Problems in the philosophy of mathematics", North Holland, Amsterdam (1967) p. 43]
"Cantor is probably the last great exponent of the Newtonian attitude with respect to religion." [H. Meschkowski, W. Nilson: "Georg Cantor Briefe", Springer, Berlin (1991) p. 15]
"it was a certain satisfaction for me, how strange this may appear to you, to find in Exodus ch. XV, verse 18 at least something reminiscent of transfinite numbers, namely the text: 'The Lord rules in infinity (eternity) and beyond.' I think this 'and beyond' hints to the fact that  is not the end but that something is existing beyond." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]
"Compare the concurring perception of the whole sequence of numbers as an actually infinite quantum by St Augustin (De civitate Dei. lib. XII, ch. 19) [...] While now St Augustin claims the total, intuitive perception of the set (), 'quodam ineffabili modo', a parte Dei, he simultaneously acknowledges this set formally as an actual infinite entity, as a transfinitum, and we are forced to follow him in this matter." [G. Cantor, letter to A. Eulenburg (28 Feb 1886)]
"It can be absolutely ascertained that St Thomas only with great doubts and half-heartedly adhered to the received opinion concerning the actually infinite numbers, going back to Aristotle. [...] Thomas' doctrine 'It can only be believed but it is not possible to have a proof that the world has begun' is known to appear not only in that opusculo but also [...] in many other places. This doctrine however would be impossible if the Aquinatus had thought that the theorem 'there are no actually infinite numbers' was proven. Because from this sentence (if it was true), it would demonstrably follow with greatest evidence that an infinite number of hours could not have passed before the present moment. The dogma of the begin of the world (a finite time ago) could not have been defended as a pure dogma." [G. Cantor, letter to C.F. Heman (2 Jun 1888)]
"Your understanding of the relation of the two propositions:
I. 'The world including the time has begun before a finite time interval or, what is the same, the duration of the world elapsed until now (e.g., measured by hours) is finite.'
which is true and a Christian dogma and:
II. 'There are no actually infinite numbers.'
which is false and pagan and therefore cannot be a Christian dogma –
I say you have not the correct idea about the relation of these two propositions. [...]
The truth of proposition I does not at all imply, as you seem to assume in your letter, the truth of proposition II. Because proposition I concerns the concrete world of creation; proposition II concerns the ideal domain of numbers; the latter could include the actual infinite without its necessarily being included in the former. [...]
The pagan wrong proposition II, even without possessing the property of being a dogma acknowledged by the church or ever having been in that possession, has, because of its dogma-like popularity, done unmeasurable damage to Christian religion and philosophy, and one cannot, in my opinion, thank holy Thomas of Aquino too effusively that he has clearly marked this proposition as definitely doubtful." [G. Cantor, letter to C.F. Heman (21 Jun 1888)]

 

    1.1.1 Vollständig
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"Wenn zwei wohldefinierte Mannigfaltigkeiten M und N sich eindeutig und vollständig, Element für Element, einander zuordnen lassen (was, wenn es auf eine Art möglich ist, immer auch noch auf viele andere Weisen geschehen kann), so möge für das Folgende die Ausdrucksweise gestattet sein, daß diese Mannigfaltigkeiten gleiche Mächtigkeit haben, oder auch, daß sie äquivalent sind." [Cantor, p. 119]

 Above "vollständig" qualifies the verb "zuordnen" so the meaning may
differe from what it would mean when qualifying the word "set" or
any word that refers to all or some sets. It could be traslated as
"fully" or "completely", meaning that no member of either set is
unpaired.

Yes.

 

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

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

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

 

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

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

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

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

A bijection is a set too.

A correspondence can be expressed
with a set but was not in the above exmples (because "corresspondence"
was well understood at the time but "set" was not).

Set is not understood today. It was understood by Cantor: Unter einer "Menge" verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unsrer Anschauung oder unseres Denkens (welche die "Elemente" von M genannt werden) zu einem Ganzen.

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

Without completeness countability and uncountability both would be meaningless

 

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

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

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

"da nun jeder Typus auch im letzteren Falle etwas in sich Bestimmtes, vollendetes ist, so gilt ein gleiches von der zu ihm gehörigen Zahl. [...] 'Eigentlichunendlichem = Transfinitum = Vollendetunendlichem = Unendlichseiendem = kategorematice infinitum' [...] dieser Unterschied ändert aber nichts daran, daß  als ebenso bestimmt und vollendet anzusehen ist, wie 2," [G. Cantor, letter to K. Laßwitz (15 Feb 1884). Cantor, p. 395]
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"Wir wollen nun zu einer genaueren Untersuchung der perfekten Mengen übergehen. Da jede solche Punktmenge gewissermaßen in sich begrenzt, abgeschlossen und vollendet ist, so zeichnen sich die perfekten Mengen vor allen anderen Gebilden durch besondere Eigenschaften aus." [Cantor, p. 236]

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

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

 

    1.1.3 Fertig
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"Die Totalität aller Alefs ist nämlich eine solche, welche nicht als eine bestimmte, wohldefinirte fertige Menge aufgefaßt werden kann. [...] 'Wenn eine bestimmte wohldefinirte fertige Menge eine Cardinalzahl haben würde, die mit keinem der Alefs zusammenfiele, so müßte sie Theilmengen enthalten, deren Cardinalzahl irgend ein Alef ist, oder mit anderen Worten, die Menge müßte die Totalität aller Alefs in sich tragen.' Daraus ist leicht zu folgern, daß unter der eben genannten Voraussetzung (einer best. Menge, deren Cardinalzahl kein Alef wäre) auch die Totalität aller Alefs als eine best. wohldefinirte fertige Menge aufgefaßt werden könnte." [G. Cantor, letter to D. Hilbert (26 Sep 1897)]

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

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

Here "fertig" can be translated as "completed".

Yes.

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

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

ZFC has one word for the meaning of completeness: set.

 ZFC (or plain ZF) does not specify any meaning for "set".

So it is. But being complete is the precondition of all sets.
Regards, WM