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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Mon, 30 Jun 2025 20:21:09 +0200
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On 29.06.2025 12:25, Mikko wrote:
> On 2025-06-28 13:56:57 +0000, WM said:
> 
>> On 28.06.2025 11:56, Mikko wrote:
>>> On 2025-06-27 19:36:41 +0000, WM said:
>>>
>>>> On 27.06.2025 09:33, Mikko wrote:
>>>>> On 2025-06-26 13:09:32 +0000, WM said:
>>>>
>>>>>>
>>>>>> If we subtract in the order that is used for enumerating then a 
>>>>>> last one is necessary.
>>>>>
>>>>> No, there is no last one in an infinite enumeration.
>>>>
>>>> Then it is not finished or completed.
>>>
>>> No, but it can be continued.
>>
>> 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.
>>
>> 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.
>>
>>     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 
>> "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
>>
>> "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]
> 
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